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On the Integrability of Scalar Partial Differential Equations

in Two Independent Variables: Some Geometric Aspects

-4 Thesis

Submitted to the College of Graduate Studies and Research

in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

in the

Depart ment of Mathematics and S tatistics

McGill University

Montréal: Québec, Canada

b~

Enrique G. Reyes

October 1998

01998 E. G. Reyes

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For Margarita and Pedro, my parents and friends.

For Julia and Gonzalo.

For Claudia and her cello, and Miaskovsky.

For Judith, Myriam; and Pedro.

Acknowledgment s

\\arrneçt thanks to Prof. Niky Kamran for expertly guiding rny efforts along the road

of matbernatical research. He has been a most patient, dedicated, and encouraging

supervisor.

kfany thanks also to the following people:

l l y instructors at hIcGill and UQAM for helping to make my stay in LIontréaI

rnemorable: Ian Anderson (visiting from Utah State University). Ram 'ilurty. Paul

Koosis. 1-ictor Havin. Jacques Hurtubise. Lisa Jeffrey. and Franqois Lalonde.

hly friends a t the Department of Mathematics & Statistics (McGill C'niversity)

for companionship, encouragement, and also conmiseration when it was appropriate:

Rusandra Moraru (wbom 1 also thank for her invaluable help in translating the ab-

stract) : hlasoud Asgharian-Dastenaei: ~ Ia lco lm Harper, Gelila Tilahum. Rob Milson.

hlartin J u r S (visiting from Utah State University), Reem kassawi: Jonathon Funk?

Richard Squire, Claudio Herrnida, Mario Fadda, and George Katsenos.

hly other Alontréal friends: Sadeq Rahimi, Tamara Padis, Rawley, Svetlana

Philippol-a, José Hamra, David & Maeve Ainsworth. Yuke Wang: Carla Avila. and

.luan Carlos Folla.

hIy friends and instructors a t the Department of 'lathematics & Statistics (Uni-

versity of Saskatchewan) for early and much needed encouragement, and for gen-

erously sharing their insights with me: James .4. Brooke, George Patrick, Jacek

Szmigielski, Mamood Koshkam, Keith Taylor, Christine Soteros, John Martin. E.D.

Tyvmchatyn: and Andrew Carson.

My family in Santiago for being essential to it all.

Thanks, finally, to my fiiends a t large, people real or othenvise, whose lifepaths

have crossed mine and left a mark, and/or whose deeds have influencecl me:

J.S. Bach, Emilio Salgari, Julius Verne, Issac Asimov, Ray Bradbun;, H.P. Love-

craft, D. Shostakovitch, Pi. Miaskovsky, Marin Marais, Jordi Savall, Silvio Rodriguez,

Soren Kierkegaard, Herman Hesse, Don Quijote, Los Jaibas, -4rvo Part, Claudio ,Ar-

rau, Claudio Teitelboim, Salvador Dali, Pablo Picasso, Mikhail Bulgakov, Charles

Bukomski, Pablo Neruda, Raul Zurita, Rolando Chuaqui, Kicanor Parra- Edgar -41-

lan Poe, Father Brown, G.K. Chesterton: Irene Mikemberg, Gloria Schwarze, Renato

Lewin (who also indirectly gave me the idea of compilating this k t ) , Victoria Mar-

shall, Rolando Rebolledo, Claudio Fernandez, Kurt Godel, Oscar Castro, Themo

Lobos, Herge, Uderzo, Goscinny, Morris, Pepo, Quino, Marcela Paz, A. de Saint-

Esupéry, -4nton Chekhov, F. Dostoyevsky, Antonio Machado, Federico Garcia Lorca,

Xndrés Segovia, Yolanda Retamal, Ftigoberto PeÎia, Luis Lillo, Jorge Soto, Orlando

Cebalios, Oscar Lopez, Enzo Hernandez, Victor Albornoz, Fernando Rosselot, Lucilla

Capedeville, Sergio Henriquez, Patricia Miranda, Edison del Canto, Cristina Tor-

res, Emil, Ivo, Hernan Quintana, blonica Espina, Bahman Masood, Paul SpafTord,

Guillermo Candia, Marco Corgini, Ange1 Barrios, Edrnundo Leiva, Elianita Covarru-

bias, bfario Benedetti, Woody -411en, Fay Minifie, Marcela Bravo, Gisselia Harting.

the St. Thomas More College community (Saskatoon), Anna PLkhmatova, Pamela

Gillies, R.C. Bowen, Kurt Tischler, Pablo Milanes, Mônica Batista, E,rnesto Carde-

nal, Madredeus, Veronica Escobar, Charles Chaplin, Erika Wald, Rafael Benguria,

Fernando Lund, Manuel del Pino, Maria Lorca, Eliana Berrios, Manuel ,4humada,

Hilda C6rd0t.a~ Julio Pino, Rene Peralta (who lectured me when 1 was about to give

up) Luis Dissett, Boris Hiche, Catherine Forester, Tom & Betty Powell, Fay Minifie,

Charles Edebiri, Kashtin, Percy Paul, Vincent Van Gogh, Dalibor Atlaguich, Manuel

Elgueta, Michele iMackasey Paul, Chevez, Tristan, Carmen A c u ~ , -4nibal Fuentealba,

Anibal, Ruth, El Eternauta, Marcelo Lomeiïa, Charles Baudelaire, e-e. cummings,

the Princess Leia.

Abstract

The problem of integrability of scalar partial differential equations in two in-

dependent variables is investigated from a geometric viewpoint. The structure of

"equations describing pseudo-spherical sufaces" introduced by S S - Chern and Keti

Tenenblat is taken as the starting point, and the fact that every equation which

describes pseudo-spherical surfaces is the integrability condition of a sf(2, RI-linear

problem is esploited througbout.

A classification of evolution equations of the form ut = F ( x , t: u, ,.., ~ , m ) which de-

scribe one-parameter families of pseudo-spherical surfaces ( "kinematic integrabili ty" )

is pexforrned, under a natural a priori assumption on the form of the associated family

of linear problems. The relationship between the class of equations which results, and

the class of equations which are formally integrable in the sense of Mikhailov, Sha-

bat and Sokolov, is studied. It is s h o w that every second order formally integrable

evolution equation is kinematically integrable, and it is also shown that this result

cannot be estended as proven to third order formally integrable evolution equations.

-A special case is proven, however, and moreover, the Harry-Dym: cylindrical KdV,

and a family of equations solved by inverse scattering by Calogero and Degasperis,

are shown to be kinematically integrable.

The theory of coverings due to Krasil'shchik and Vinogradov is introduced, and

(local, nonlocal) conservation laws and (generalized, nonlocal) symmetries of kine-

iii

matically integrable equations are investigated within this framework: theorems on

the existence of generalized and/or nonlocal symmetries are proven, and several se-

quences of local and/or nonlocal conservation laws are constructed. The relationship

between the Cavalcante, Chern, and Tenenblat approach (consenation la- from

pseudo-spherical structure) and the more familiar "Riccati equation" approach (con-

servation laws from the associated linear problem) is analysed. An appendk on

noniocal Hamiltonians for evolution equations is aiso included.

The reIationship between Chern and Tenenblat "intrinsic" point of view, and the

"estrinsic" approach due to Sym is studied, and more conservation laws for equa-

tions describing pseudo-spherical surfaces are found. It is also shown that within

the "estrinsic" framework, a new class of equations, interpretable as two-parameter

deformations of the equations in the Chern-Tenenblat class, can be introduced. It

is pointed out that these deformations are themselves the integrability condition of

sl(2, R)-linear problems, and conservation laws for them are briefly considered.

Résumé

Le problème d'intégrabilité d'équations à dérivées partielles scalaires à deux variables

indépendentes est étudié du point de tue géométrique. La structure dl "équations

décrivant des surfaces pseudo-sphériques" , introduite par S.S.Chern et Keti Tenen-

blat, sert de point de départ. Le fait que toute équation décrivant une surface pseudo-

sphérique est la condition d'intégrabilité d'un problème sZ(2, R)-linéaire sera exploité

tout au long de cette étude.

Une ~Iassification est donnée pour des équations d'évolution. de la forme ut =

F ( x , t? u, . . . , u , m ) , qui décrivent des familles, à un paramètre: de surfaces pseudo-

sphériques. Celle-ci suppose que la famille de problèmes linéaires associée satis-

fakquant à sa forme, une condition naturelle au problème. La relation entre la

classe d'équations ainsi obtenue et la classe d'équations formellement intégrables, au

sens de Mikhailov? Shabat et Sokolov, est étudiée. Il est démontré que toute équation

d'évolution formellement intégrable de second ordre est cinématiquement intégrable.

11 est aussi prouvé que ce résultat, ainsi démontré, ne peut pas être étendu a a s

équations d'évolution formellement intégrables de troisième ordre. Une extention?

dans un cas pariculier, est cependant donnée. De plus, Iéquations de Harry-Dym.

KdV cylindriques, et une famille d'équations résolues par "scattering' inverse par

CaIogero et Desgarperis: sont montrées comme étant cinématiquement intégrables-

La théorie de recouvrements de Krasil'shchik et Vinogradov est introduite et sert

de contexte pour l'étude de lois de conservation (locales, non-locales) et de symétries

(généralisées, non-locales) d'équationes cinématiquement intégrables: des théorèmes

sur I'esistence de srnétries généralisées et/ou non-locales sont prouvés, et plusieures

lois de conservation locales et/ou non-locales sont construites. La relation entre

l'approche Cavalcante, Chern et Tenenblat (lois de conservations issues de struc-

tures pseudo-sphériques) et l'approche plus connue de "l'équation Ricatti" (lois de

conservation issues du problème linéaire associé) est analysée. Cette partie comprend

aussi un appendice sur les hamiltoniens non-locaus d'équations d'évolution.

Le lien entre l'approche "intrinsèque" de Chern et Tenenblat et l'approche "es-

trinsèque" due à Sym est estudié, et plus de lois de conservation pour des équations

différentielles décrivant une surface pseudo-sphérique sont trouves. Dans ce con-

teste L'estrinsèque", il est démontré que l'on peut introduire une nouvelle classe

d'équations, ces dernières pouvant être interprétées comme étant des déformations, à

deus paramètres, des équations de la classe Chem-Tenenblat. Ces déformations sont,

elles-même, la condition d'intégrabilité de problèmes sl(2, R)-linéaires, dont les lois

de consemation sont brièvement considérées.

Statement of originality

The results and examples presented in this work are new unless esplicitly said other-

wise in the main text. In particular, the followïng has been obtained by the author:

1. Classification of arbitrary scalar evolution equations in tlvo independent vari-

ables which are strictly kinematically integrable (Theorems I to 5 and Corollary

1. Also in Reyes [1998]).

2. Forma1 integrability implies kinematic integrabiiity for general second order

autonomous evolution equations (Theorem 6. Also in Reyes [1998]).

3. Forma1 integrability implies kinematic integrability for a special case of third

order autonomous evolution equations (Theorem 8. Also in Reyes [l998]).

4. Counter-exampte for a generalization of 3 (Theorem 7. -41~0 in Reyes [1998]).

5. Generation of sequences of conservation iaws for strictly kinematically integrable

equations (Theorem 9. Also in Reyes [1998]).

6. Relation between classical generation of conservation laws for kinematically in-

t egrable equations and the Chern-Tenenblat method (Section 4.2).

7. Geometric characterization and proof of existence of (generalized, nonlocal)

symmetries for kinematically integrable equations (Propositions 7: 8, 9: and

Theorem 10).

vii

S. Construction of (in principle nonlocal) conservation laws of kinematically inte-

grable equations via deformations of the Chern-Tenenblat family of conserva-

t ion laws (Proposition 10).

9. Construction of conservation laws of kinematically integrable equations wit hin

S m ' s estrinsic approach (Propositions 13 and 14).

10. Introduction of the class of equations describing Calapso-Guichard surfaces

(Section 6.3) .

Contents

Acknowledgments

Abstract

Résumé

S tatement of originality

Contents

iii

v

vii

ix

Introduction 1

1 Prelude 12

1.1 A geometric frarnework for scalar equations . . . . . . . . . . . . . . 12

1.2 Kinematically integrable equations . . . . . . . . . . . . . . . . . . . 20

2 Evolution equations describing pseudo-spherical surfaces 25

2.1 Characterization results . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . 40

3 Formal and kinematic integrability 47

3.1 Formai symmetries and forma1 integrability . . . . . . . . . . . . . - . 47

- 3.2 On the integrability of second order equations . . . . . . . . . . . . . 50

3.3 On the integrability of third order equations . . . . . . . . . . . . . . 57

4 Conservation laws and kinematic integrability 62

. . . . . . . . . . . . . . . . . 4.1 Conservation laws via Riccati equations 63

. . . . . . . . . . . . . . . . . . . 4.1.1 Example: Burgers' equation 67

. . . . . . . . . . 4.2 Conservation laws: The Chem - Tenenblat method 70 .. . . . . . . . . . . . . . . 4.2.1 Example: The Lund-Regge equation I CI

.. . . . . . . . . . . . . . . . . . . . . . 4.3 Coverings and conservation laws i I

5 Generalized syrnmetries and kinematic integrability 86

5.1 Generalized syrnmetries and infinitesimal deformations . . . . . . . . 87

. . . . . . . . . . . . . . . . . . 5.2 Symmetries and coverings . . . . .. 91

5.2.1 Example: Konlocal symmetries of Burgers' equation . . . . . . 97

. . . . . . . . . . . . . . . 5 -3 Symme tries and nodocal conservation laws 101

. . . . . . . . . . . . . . . . . . 5.3.1 Example: The KdV hierarchy 103

Appendix: Nonlocal Hamiltonians for evolution equations 106

6 Immersed surfaces and kinematic integrability 112

. . . . . . . . . . . . . . . . . . . 6.1 Immersing pseudo-spherical surfaces 113

. . . . . . . . . . . . . . . . 6.2 Conservation laws induced by immersions 117

6.3 Equations which describe Calapso - Guichard surfaces . . . . . . . . 123

. . . . . . . . . . . 6.4 Consemat ion laws and Calapso-Guichard surfaces 130

7 Summary. further comments and open problerns 132

References 138

"Weil, however it may be, 1 wiil subrnit'

(Schubert, quoted by P. van Nieuwenhuizen)

Introduction

This thesis represents the author's first steps in the world of integrable systems and

the forma1 geometry of differential equations. The recurrent themes of this research

have been to understand the connections among some of the different approaches to

the phenornenon of integrability of scalar partial differential equations in two indepen-

dent variables, and to study whether these connections shed a new light or indicate

generalizations of some of these viewpoints.

-4n approach which appears to be very general and yet relatively straightformard to

use is Chern and Tenenblat's [1986] notion of "equations describing pseudo-spherical

surfaces". A pleasant feature of this theory is that it can thought of as a "bridge"

betw-een the classical theory of two (and also three) dimensional Riemannian met-

rics, and the theory of scalar equations which are integrable in the sense of being

the integrability condition of a one-parameter family of Iinear problems. The nest

paragraphs will clarify this metaphor.

Definition 1 A diflerential equation Z = O for a real ualued function u(x, t ) is said to

describe pseudo-spherical surfaces if there exist smooth fvnctions faa, a = 1,2 .3: 3 =

1,2, depending on x, t : u and its derivatives, such that the one-forms w" = faldx + ff2dt satisfy the structure equations of a surface of constant Gaussian curuature equal

t6 - 1 with netn'c (wl)* + ( w ~ ) ~ and connection one- fom w3, namely

du1 = u3 A W ~ , dw2=w'hw3, and d u 3 = w L h w Y 2

if and only if u(x, t) i s a solution of E = 0.

This is precisely the structure which was considered for the first time by Chern and

Tenenblat (19861, motivated by Sasaki's (19'791 observation that the equations nhich

are the necessary and sufficient condition for the integrability of a linear problem

of AKNS type (.4blowitz et al. [1974]) do descnbe pseudo-spherical surfaces. As

suggested above, however, the connection betmeen pseudo-spherical surfaces and in-

tegrability of differential equations goes well beyond the AKNS framework.

Definition 2 (After Faddeev and Takhtajan [1987]) A differential equation for a real

valued function u(x , t ) is kinematically integrable i f zt i s the integrability condition of

a one-parameter farnily of Zinear problems

such that for each II E C, U(q) and V(7) are sZ(2, R) -ualued functions of x, t u, and

i ts derivatives up to a fznite order.

Thus, an equation is kinematically integrable if it is equivalent to the zero curvature

condition

where trace U(7) = trace V(q) = O for each 77. In addition, a differential equation

mil1 be said to be stn'ctly kinematically integrable if it is kinematically integrable and

the diagonal entries of the matrix U(77) introduced above are .rl and -q. For esample,

equations integrable by the AKNS method are kinematically integrable in the strict

sense, but those studied by Wadati, Konno and Ichikawa [1979] are not. If an equation

is integrable in the strict sense, it is customary to cal1 71 the "spectral" parameter,

in agreement with the fact that

incegrability condition of a linear

a strictly kinernatically integrable equation is the

problem of the form

in which Uo = (/(II) - qdiag(1, -1): and that therefore one may hope to solve it b~

inverse scattering techniques.

Matters being so: the bridge rnetaphor is explained by the following fact. -4 differ-

ential equation is kinematically integrable if and only if it describes a one-parameter

family of pseudo-spherical surfaces, which, in agreement with the classical tradition

(Eisenhart [1909], for esample), are allowed to be cornplex, and strict kinematic in-

tegrability amounts to the condition fil = 7. Indeed, Equations (1) hold if and onlv

if the linear problem

is integrable, and conversely, anÿ family of linear problems (2) satisfying (3) deter-

mines a one-parameter family of orie-forrns w" satisfj6ng (1) by setting, using an

obvious notation, w1 = + U12)dx + (Kl + K2)dt, u2 = 2Ulldx + and

L." (hl - K 2 ) d x + (G1 - G Z ) d t .

One has therefore a direct connection between classical differential geometq- and

the modern theory of integrable equations in one space variable. It is very satisfactory

that this connection aIso proves fruitful, as anticipated by the adjective used to qualify

it at the beginning of this introduction. The most obvious gain is the fact that

equat ions which are the integrability condition of sl(2, R)-valued linear pro blems

with spectral parameter are well known and important (one only needs to think of

the AKNS family, KdV, sine-Gordon, Liouville, and Burgers equations, for exarnple)

so that if one can understand their properties as equations describing pseudo-spherical

surfaces, one can hope to obtain a better or complementary understanding of their

properties as kinematically integrable equations.

A non trivial example of this interaction to be presented here is motivated by

the fact that there exists an approach to integrability which does not require lin-

ear problems for its formulation, namely, the formal syrnrnetry approach introduced

by Mikhailov, Shabat and Sokolov [1991] among others. -4n equation is said to be

formally integrable if it possesses a forma1 symmetry of infinite rank. the precise defi-

nition of which is given in Chapter 3. Formai integrability captures essential aspects

of mhat the sentence "The equation E = O is integrable" should mean. Indeed, if an

equation is formally integrable, it possesses an infinite number of (generalized and/or

nonlocal) sprnetr ies (See Krasil'shchik and Vinogradov [1984], h4ikhailov, Shabat

and Sokolov [1991], and Olver [1991]) and also an infinite number of local conserva-

tion Iaivs, which, nevertheless, may well be trivial. It is of course natural to ask if

formally integrable equations are kinernaticalty integrable. For example. it is known

that the generalized Burgers' equation, ut = u,, + u,u + h(x), is formally integrable

(Olver [ lgi i ]) . 1s it also the integrability condition of a (parameter dependent) linear

problem?

It will be proven in this Thesis that in fact, every second order evolution equation

ut = F (x, U, u,, u,,) which possesses a formal syrnmetry of infini te rank describes

a one-parameter family of pseudo-spherical surfaces. In particular, the family of

linear problems associated to the generalized Burgers' equation will be esplicitly con-

structed. The proof is a posteriori and basically consists of looking at kinematic

integrability from the Chem and Tenenblat's [1986] point of view. It runs as folloivs.

First , one classifies al1 the equations of the form ut = F (x, t , + u,, u,,, ..., U , L ) which

describe pseudo-spherical surfaces under the assumption that f21 = 77, a non-zero pa-

rameter. Second, one finds explicitly the pseudo-spherical structures associated with

the second order equations appearing in the exhaustive (up to contact transforma-

tions) lists of formally integrable evolution equations provided by hlikhailov, Shabat

and Sokolov [1991]. It turns out that al1 of them are strictly kinematically integrable.

and it follows, therefore, that every formally integrable second order equation is kine-

matically integrable, although not necessarily in the strict sense.

One irnrnediately wonders if the analogous theorem is true for third order equa-

t i ~ n s . It ni11 be seen in this work that the equation ut = (u, + 15)~u,,: 6 E R:

one of the two equations of the form ut = f ( t , U, u,)~,, which are formally inte-

grable (Abellanas and Galindo [1985]), is not kinematically integrable in the strict

sense. This is an interesting result in itself, and illustrates again the characteris-

tic of "bridge" atributed to Chern and Tenenblat's structure. Of course, it also

means that the proof of the implication "formal integrabiiit'; * kinematic inte-

grability" cannot be estended from second to third order equations in complete

generality. I t does estend to a subclass of formally integrable third order q u a -

tions, howet-er. hdeed, one can prove that every third order equation of the form

ut = I L - ~ I L , ~ + a2(x, U, U ~ ) U : ~ + ai (2, u, u,)u, + ao(x, u, u,) which possesses a forma1

symmetry of infinite rank, describes a one-parameter family of pseudo-spherical sur-

faces. It is obvious that one can still ask whether the equation ut = (u, + is

kinematically integrable in the general sense. It appears that to ansmer this question

in the present context requires the use of classification results more general that the

ones to be presented here. It ni11 not be considered in this rvork.

The classification results used in the proof of the theorems mentioned above are

presented in Chapter 2. The cornparison between forma1 and kinematical integrability

is presented in Chapter 3. These two chapters are based on a recent article by the

aut hor (Reyes [1998]) and conform the first part of this Thesis.

The themes mentioned a t the beginning of this introduction can be also recognized

in the subsequent two chapters of this report. Some classical subjects of integrable

systems will be studied in the Chern and Tenenblat's [1986] context. In the light of

the reIation between their structure and kinematic integrability, what one 1%-ould like

to do first is to compare Chern and Tenenblat's work with the early investigations (See

for esample Wadati, Sanuki, and Konno [1973]) on equations which are integrable in

the kinematical sense. This is done here at the fundamental level of the problem of

existence of conservation laws.

Conservation laws for equations describing pseudo-spherical surfaces have been

studied already by Chern and Tenenblat (19861, and a constructive method for corn-

puting them has been developed by Cavalcante and Tenenblat [1988]. Now, on the

other hand, there is a classical method to compute conservation laws of AKNS equa-

tions, sa?. The "Riccati equation" method presented in the paper by Wadati, Sanuki,

and Konno mentioned abow. Of course one wonders if application of the Cavalcante

and Tenenblat's (19881 algorithm will produce the same conservation laws. This ques-

tion has been asked already by Wyller [1989] in the context of the derivative nonlinear

Schrodinger equation, but it appears it has not been considered until now. One can

indeed analyse this issue in its full generality, and answer it in the affirmative, in a

sense to be made precise in Chapter 4.

Thus, one possesses at least two algorithmic met hods to generate conservation

laws of kinematically integrable equations which, as examples show, may be of a

local or nonlocal nature. They can be studied on a same footing in the contest

of the theory of coverings developed by Krasil'shchik and Vinogradov [1984, 1989j.

-4 short introduction to this theory, and some comments on how it illuminates the

constructions mentioned above, appear in Section 4.3.

Symmetries of equations describing pseudo-spherical surfaces are considered nest.

-4s before, one is interested in two aspects of the theory. How to understand (gen-

eralized, nonlocal) symmetries within this class of equations, and how to use this

understanding to illuminate the theory of kinematically integrable equations. In-

teresting phenornena appear in this research. The first result to be reported is the

formalization and verification of the following intuitive fact: symmetries (which may

be generalized or even nonlocai) of equations describing pseudo-spherical surfaces are

in a one-to-one correspondence with the symmetries of the pseudo-spherical struc-

tures determined by them. A characterization of (generalized, nonlocal) symmetries

can then be obtained. Its importance becomes quickly apparent, for it allows one

to prove theorems about the existence of an infinite number of generaIized (again,

maybe nonlocal) symmetries and (apparently new) nonlocal conservation l a w for

kinematically integrable equations.

The results on conservation laws and symmetries of equations describing pseudo-

spherical surfaces summarized above form the second part of this thesis. Consemation

laws are treated in Chapter 4. Symmetries are considered in Chapter 5. An append i~

(based on some interesting work by Hojman [1996]) on the Hamiltonian formalism

for evolution equations (which do not necessariIy describe pseudo-spherical surfaces)

is included after Chapter 5. It is presented here only because the construction of such

a Hamiltonian structure depends on the esistence of (nonlocal) conservation Law

and generalized (or nonlocal) symmetries, facts mhich for kinernatically integrable

equations follow from the analysis carried out in Chapters 4 and 5 , and also because

Hojman's construction can be formalized within the framework of covering t heory.

This appendis must be considered of a provisional character, insofar as no atternpt

to provide the reader with an algorithm to obtain symmetries is made.

The last part of this mork is about the relation between Chern and Tenenblat's

[1986] t heory and other geometric approaches to kinematic integrabilit- One is t hink-

ing mainly of Sym's [1985] theory of "soliton surfaces", and Lund and Regge's [1976]

discovery of the integrable equation named after them. Of course, the Lund-Regge

equation is, in actual fact, a system of equations for tnro dependent variables, but

clearly Definitions 1 and 2 may be estended mutatis mutandis to this more general

case. This extension d l be used hereafter without further notice.

The works by Syrn [1985] and Lund and Regge [1976] differ from the point of view

taken in this Thesis in that they consider surfaces immersed in higher dimensionaI

spaces as their starting point. The Chern-Tenenblat approach is, on the other hand,

intrinsic. Another way of expressing this difference is to Say that while the basic

ingredient in the Chern and Tenenblat theory is the equation itself, the fundamental

object in Sym's theory is the immersion. Thus, Syrn [1985] (and Lund and Regge

[19ï6]) starts From the Gauss-Weingarten linear system determining the immersion:

and impose a priori constraints on the components of the metric (which are assumed

tombe functions of sorne dependent variables uQ(x, t ) ) so that the Gauss-Coddazi equa-

tions become nonlinear differential equations for uQ(x, t ) . The system of equations

which results will be kinematically integrable as long as one can introduce a "spectral

parameter" in the Gauss-Weingarten iinear problem. One is then led to consider

the following problem, called by Sym [1985] the "generalized Lund-Regge problem".

What (systems of) equations for variables u" appear as Gauss-Coddazi equations

of a Gauss-Weingarten linear problem with spectral parameter? This question has

interesting consequences for the differential geometry of surfaces, of course: If one

knows that the Gauss-Coddazi equations are the integrability condition of a linear

problem with spectral parameter, one can hope to integrate the Gauss-Weingarten

linear system, and thereby find examples of immersed surfaces for which their funda-

mental forms are constrained a priori. For instance, one can find examples of compact

surfaces of constant mean curvature by integrating the sinh-Gordon equation (See

Pinkall and Sterling [1989] and references therein). The surfaces obtained in this

fashion have been called "integrable surfaces" by Bobenko [1991].

One would like to have the freedom to use an immersion approach if so wished, but

one wouid also like to ask that the class of equations to which is applicable be a t least

as large as the class of equations describing pseudo-spherical surfaces. In other words:

every equation describing a one-parameter family of pseudo-spherical surfaces should

be a solution of the generalized Lund-Regge problem. One would expect that new

properties of these equations become apparent within such a viewpoint. Fortunately?

to imnierse pseudo-spherical surfaces in a flat three dirnensional space is easy. If

one starts with an equation E = O for u(x, t ) describing pseudo-spherical surfaces by

means of a coframe {a1, w2} and connection one-form w3, one can, whenever u(z, t )

is a solution of E = O, locally and isometrically immerse these surfaces into a flat

space equipped with a metric of signature (+, +, -). Two gains are immediately obtained. First, one can exhibit a one-parameter

family of conservation laws of equations describing pseudo-spherical surfaces, one of

which is Chern and Tenenblat's. Of course, this means that if a given equation is

kinematically integrable, so that the functions faB introduced in Definition 1 depend

a priori on an extra parameter 71, one can find, in principle, a two-parameter family

of conservation laws for it. Second, one can introduce a new class of equations for

dependent variables u(x; t ) , h(x, t ) , and c(x, t ) , which can be interpreted as two-

parameter (h(x, t ) and c(x, t ) ) deformations of equations describing pseudo-spherical

surfaces, and which are t hernselves the integrability condit ions of sl(2, R)-valued

linear problems.

The second point above appears to be of some interest because earlier works on

equat ions descri bing pseudo-spherical surfaces d e d t almost esclusively wit h scalar

equations. The only mention of systems of equations having that structure appears

in a paper by Sasaki and Bullough [1981]. They remark that the Lund-Regge equation

describes pseudo-spherical surfaces, a fact already mentioned here as wellt and which

will be used in Chapter 4.

It is interesting that the existence of this nerv class of equations represents a

smootli passage frorn the intrinsic approach to Syrn's [1985] point of view. What one

does is to deform the one-forms wa appearing in Definition 1. The Gauss-Coddazi

equations to be satisfied by the deformed coframe if it is to sweep a surface immersed

in a flat 3-space, determine a system of equations which is, by c~ns t ruc t ion~ the

integrability condition of the Gauss-Weingarten linear problem. The deformation

chosen is patterned in the classical deformations of pseudo-spherical surfaces giving

rise to Calapso-Guichard equations (Eisenhart [1923], Gürses and Nutku [l98 11): and

therefore equations in this class will be called "equations describing Calapso-Guichard

surfaces". Of course, if the equation being deformed is kinematically integrable, this

construction gives new solutions of the generalized Lund-Regge problem.

This "immersion approach" is described in Chapter 6, the last chapter of this

Thesis in which new resuIts are presented.

It is time now to summarize this long introduction by stating explicitly the objec-

tives of this dissertation. They are these: to classify those et-olution equations of the

form ut = F ( x , t , u, uz, . . . , U=m) which are stnctly kinematically integrable; to sliow

that forma1 integrability and kinematic integrability are equivalent for second order

autonomous evolution equations and that this result cannot be estended as proven to

t hird order evolution equations; to characterize (local, nonlocal) conservation laws and

(generalized, nonlocal) symmetries of kinematically integrable equations; to remark

that Chern and Tenenblat's intrinsic point of view may be connected to Sym's theory

of solitoo surfaces; to show that this approach ailows one to obtain (in principle) yet

another family of conservation laws for the complete class of equations which describe

pseudo-spherical surfaces; to introduce, again within Sym's framework, new examples

of sys tems of equations describing pseudo-spherical surfaces which are interpret ables

as "long deformations" of scdar equations in this class.

The reader will find several esarnples scattered throughout. Two equations which

will appear often are the generalized Burgers' equation,

and the sineGordon equation, urritten either in the form

Finally, one must mention that open problems and possible directions of research

will suggest themselves in almost every chapter of the Thesis. They will be collected

in Chapter 7, the final part of this long report. This is a most satisfactory state of

affairs: the end of this Thesis is also a beginning.

Conventions: The following notation will be used for u and its derivatives if con-

sidered as jet coordinates:

20 = U, Z~~rnlra = 3m+n~ df u dxmdtn ' and Zj = -

dxj ' for j,m,n >_ 1.

The sets of natural, integer, real, and complex numbers wilI be denoted by N, 2,

R, and C respectivel. Equations d l be referred to as "Equation (mm)", where n

indicates the chapter number. E.xterna1 references will appear as Author(s) Cyearl.

Montréal and Logan, October 1998, 03:OO ,431.

Chapter 1

Prelude

This is an introductory chapter in which the basic definitions of the forma1 theory of

differential equations are presented (Section 1.1): and equations describing pseudo-

sphericaI surfaces are re-defined within this framework (Section 1.2).

1.1 A geometric framework for scalar equations

Fi s once and for al1 a trivial fiber bundle T;. : E + 1CI of base the space IV of

independent variables (x, t ) and typical fibre the space of the dependent variable u?

and consider the fiber bundles JkE of k-jets of local sections of E.

The bundles JkE determine three families of projection maps, s;f : J k ~ + JIE:

: J% E E , and &: JkE -+ 12.1, given by ', E

: jk(s)(z. t ) * jL(s)(x, t ) .

TL : jk(s)(x: t) l+ ( x , t , u ( ~ , t ) ) , k L 1 , and

respective13 in which s : (x, t ) I+ (x, t: u(xl t ) ) is a local section of the bundle i;- :

E hl, and ja(s) denotes the a t h jet of s, CY = 1 , 2 , . . .. The local section jk(s)(z' t )

is called the kth prolongation of the local section s : (x, t ) t, (x, t: u(x, t ) ) .

The injinite jet bundle of E, JmE + -44, is the inverse limit of the tower of

jet bundles 12.I t E . - c J ~ E t P+'E + . . . obtained by means of the projec-

tion maps ir:. As such, it cornes equipped with canonical sets of projection maps

~; r : J"E + J k E , : JmE + El and x z : J k E -t M , defined by expressions

analogous to the ones above. The local section j 3"fs ) (x , t) of JmE is called the in-

finite prolongation of the local section s : (x, t ) H (x, t: u(x, t ) ) . The bundle J"E

niay be described locally by adapted (with respect to the projection mappings defined

above) coordinates (x: t , u, u,, ut , ..., u,it', ...), in wbich ( x o t l u t u,: ut, ...: u,kti) are the

standard coordinates on the finite-order jet bundle Jki'E given by

for O 5 iy j 5 k + 1, i + j 5 k + 1. -4 function f : J"E + R is smooth if and

only if it factors through a finiteorder jet bundle, f = fk o TF, for some function

fi. : JkE + R. A vector field on JmE is a derivation on the ring of smooth

functions on J w E , that is,

in which A, B, Cmln are smooth functions on JmE. A generalized vector field on E is

a vector field on J"E along E, namely, a derivation of the form

in which il: B, C are smooth functions on JmE. Vector fields A- on M (and also

generalized vector fields on E) can be prolonged to vector fields prmX on P E (See

Olver [1993]). In particularo vector fields X on M can be canonically prolonged to

vector fields on J m E by

(p."x)(j-(s) (x, t ) 1 . f = X(=,t> (f O j W (1.2;

in which s is a local section of E, and f is a smooth function on JooE. This pro-

longation operation defines a connection on J"E called the Cartan connection (See

Krasil'shchik and Verbovetsky [1998] and references therein): the horizontal subspace

a t jm(s)(x7 t) in JooE is

cjw(,)(,,t1 = {@+mX)(jm(s)(x, t)) : X is a vector field on M } . (1-3)

The Cartan connection satisfies

prm [ X , Y] = hmx, p r m Y ]

for al1 vector fields X and Y on M , and is therefore flat. Locally, one easily sees from

(1.2) that the infinite prolongations of the coordinate vector fields a / a x and d / a t are

the total derivatives D, and Dt given by

6 d D, = - + C u,m+itn - a , and Dt = - + uZmtn+~

d

a~ m,n duZmtm a nLvn duZm t n ?

(1 -4)

and one also sees that horizontal vector fields on JwE are linear combinations (with

coefficients in the set of smooth functions on P E ) of the derivations D, and DL.

A vector field X on J"E is called vertical if X(f) = O for et-ery smooth function

f E M. The sets of horizontal and vertical vector fields will be denoted by C and V

respectively.

Differential forms on JaE are the pull-backs of differential forms on JkE by the

projection mappings rr, k 2 1. The space of differential one-forms on J"E d l be

denoted by R1 (JmE). The Cartan connection on Ja E induces a splitting of R ' ( J D E )

into contact and horizontal one-forms which is dual to the splitting of the space of

vector fields on JwE deterrnined by C and V. More generally, any differential form

w on P E may be written in the canonical coordinates defined above as

in which i, = 1,2, x' = x, x2 = t, g 2 0, 1, are multi-indices of the form 1, = x k ~ t f l

1 5 j 5 q, and the one-forms O1 are the basic contact one-fonns

00 = du - urdx - utdt , and 8+,r = du,ttt - ~ , k + ~ ~ i d x - z ~ ~ I ; ~ ~ + - ~ d t , k, 1 2 1.

One then says that a differential form w is horizontal if no forms BIJ appear in

local expansion (1.5). Xote that the pull-backs of the one-forms O1 by sections

15

its

joc

of JmE are identically zero. One says that the coordinates (x, t , u: u,: ut, ...: u,kt~ ...)

are adapted to the contact structure of J"E.

The Cartan connection induces a splitting of the exterior derivative operator. The

horizontal exterior derivative dH on Q L ( J m E ) is the operator uniquely determined in

local coordinates by the forrnulae

d H f = D, f d3: t Dt fdt, d H ( d z ) = d H ( d t ) = O , d ~ ( e I ) = dx A Br,dx i- d t A Brtdt.

( 1-61

where I is a muIti-indes of the form I = xktL, and which can be estended to arbitrary

differential forms (1.5) in an obvious way. It satisfies d ~ o d ~ = 0, so that it determines

a family of complexes on Jw E indexed by the "vertical degree" q appearing in formula

(1.5). Only the first such complex will be used in this work. It is given by

in which Rk*O(JmE), k = 0 , l : 2, is the space of horizontal k-forrns on JmE. Over

suitable domains M , or under appropriate restrictions on the trivial bundle E, this

comples is exact (Olver [1993]), a fact which is not necessarily tme, not even local15

for submanifolds of J"E, as it will be seen presently

-4 partial differential equation

defines a locus in the jet bundle Jm+" E of E. This locus will not be necessarily a

submanifold of Jm+"E, of course (an obvious example is the parabolic second order

equation u,,utt - u2t = O , lvhich determines a cone in J 2 E ) . It will be assumed that

Sm+" is an open, connected and contractible subset of the locus determined by (1.8):

tha t the function i is smooth in a neighbourhood of Sm'", and that the condition

holds on Sm+", so that Sm+" is a submanifold of the jet bundle Jm+"E. I t will be

further assumed that this submanifold is a sub-bundle of JminE, so that it fibers

over the space of independent variables.

The infinite prolongation Sm L, P E of Sm+" is the inverse limit of the finite-

order prolongations of Sm+": Sm+" c Srni"+' + - - S . In other words, it is the subset

of J*E described locally as the set of infinite jets in JmE satisfying (1.8) and al1 its

(total) differential consequences. The tower Sm+" + Sm+"+' +- a-- -- is well defined,

and each Sk+' is a submanifold of J k + ' ~ which fibers over Sk, because of the rank

assumption on E stated above. Thus, Sm is a well defined sub-bundle of JmE which

also fibers over M . I t is called the equation manifold of Equation (1.8).

-4 local solution of Equation (1.8) is a local holonomic cross-section of Sm, namelp

a local section of Sm v JaOE which is the infinite prolongation of a local section

s : (x, t) I+ (x, t, u(x, t ) ) of E. Sometimes the phrase "u(x, t ) is a local solution of

Equation (1.8)'' will be used instead of the more forma1 sentence "the local section

jOO(s): in which s : (x, t) t, (x, t , u(x , t ) ) is a local section of E, is a local solution of

Equation (1.8)".

One can prove (See for example Krasil'shchik and Verbovetsky [1998]) that if

0 = jm(s)(x, t) is a point in Sm, the horizontal subspace Ce given by (1.3) is contained

in the tangent space T O P . One obtains, therefore, a connection on Sm, called again

the Cartan connection of Sm, by restricting the Cartan connection on JOcE. This

connection is also flat, and one can see that locally, the horizontal vector fields on

S" are linear combinations (with coefficients in the space of smooth functions of

Sm) of the restriction to this manifold of the total derivatives LIz and Dt defined in

( 1 . ) For instance, for evolution equations of the form zoVt = F ( z , t , ZO, ZL? - . . ~ m )

(note that the convention for the derivatives of the dependent variable u stated in

the Introduction has been used here for the first time) they become

The space of diferential forrns on Soc is determined by pull-bach. Thus, for

instance, the space of one-forms on Sm, $2' (Sm), is given by RL (Sm) = L ~ R ' ( .JD E) .

Horizontal differential forms and the splitting of the exterior derivative can then be

defined as before.

A local conservation law of Equation (1.8) is an horizontal one-form u = Xdx+Tdt

on S* such that

dHw = ( -DtX + D,T)dx A dt = 0, (1.11)

and the conservation law is called trivial if w is dH-exact. These definitions can be

understood in ternis of the pull-back by the inclusion L : Sm L, JXE of the comples

(1.7). By defining conservation laws, one is saying that this new comples is not

necessarily esact. In other words, a non-trivial local conservation law of Equation

(1.8) is a non-trivial element of the first cohomology group of the comples

in which RkyO (Sm), k = 0 ,1 ,2 , is the space of horizontal k-forms on Sm. Esplicitly.

the non-trivial local conservation laws of Equation (1.8) are the non-zero elements

of Hi*', in which

Sometimes the lenguage will be relased in the test, and the espression "w is a non-

trivial conservation law" will be used, meaning that w is a representative of a nonzero

element of the cohomology group Hl*?

Kote that the derivative operator used in (1.11) is dH and not d. These two

operators coincide ''on solutions" in the following sense:

d((j"(s))*w) = ( j m ( s ) ) ' ( d ~ 4 (1.14)

for every differential one-form w = X d x f T d t in R(Sm), and every holonomic section

j" (s) of Sm. In particular, (1.14) holds if Equation (1.8) is the "empty" equation:

narnely: it holds for holonomic sections of JmE. Equation (1.14) will be used very

often in this Thesis.

-4 generalized synzmetry of Eqdation (1.8) is a generalized vector field (1.1) on E

whose infinite prolongation is tangent to Sm. Olver [1993] shows that one can assume

a priori that generalized symmetries are of the forrn

for a smooth function G on Sm. The generalized vector field uc is said to be evolution-

UT, and the function G is called its characteristic. If Equation (1.8) is an evolution

equation giveii in jets coordinates by z o , ~ = F(x2 t , Q, z l , . . . , h), one can check that

vc is a generalized symmetry of (1.8) if the equation

in which F, is the operator

and D, and Dt are given by (1.9) and (1.10) respectively, holds whenever z0 (1: t ) is

a solution of ~ 0 , ~ = F - Details are given in Chapter 5 .

Example: The Burgers' equation

determines the locus

in J 2 E . Since E = -zo,, + 22 + rozl is an smooth function on J2 E and jL O on

J2E1 the locus L: itself is a submanifold S' of J ~ E . The total derivative operators

D, and DL are given by (1.9) and (1.10) in which F = 22 + zozl. The equation

manifold SM v JmE of Equation (1.17) is obtained by successive prolongations of

S2. It follows that natural coordinates on Sm are of the form (x: t , 20, zl, ...: zk; ...),

as derivatives of ro with respect to t may be replaced by expressions containing only

x-derivatives by means of (1.10) and (1.17).

Equation (1.17) itself determines a conservation law, of course. If one sets

1 w = q d z + ( I l + 5*;) dt,

then d H w = - Z O , ~ +z2 +zOzl = O on Soo. This is the only non-trivial local conservation

law of Equation (1.17) (See Olver [1993] p.330).

Burgers' equation possesses also an infinite nurnber of generalized symmetries

(Olver [1977]). For example, the infinite prolongation of the evolutionary generalized

vector field with characteristic

is tangent to Sm.

Burgers' equation possesses also nonlocal conservation laws and symmetries. One

can check that if +y) is given by

then there esists a function O\') a h depending on integrations of functions on Sa:

such that the equation

~ t ( 4 r ) ) = D~(@\')) (1.19)

holds whenever zo (z, t ) is a solution of (1.17). Thus, 4\') should be considered as the

conserved density of a conservation law for Burgers' equation. In the same vein, the

"infinite prolongation" of the "evolutionary vector field" with cbaracteristic

is formally tangent to Sm: one can easily check that Equation (1.16) holds when-

ever zo(x, t ) is a solution of Burgers' equation. I t niIl be seen in Chapters 4 and

5 how one can understand Equations (1.18) and (1.20) in the contest of the forrnal

geometry of differential equations following Khor'kova [1988]. and Krasil'shchik and

L-inogradov [1984, 19891, and also how one can generalize them to the mliole class

of equations describing pseudo-spherical surfaces to be rigorously introduced in the

following section.

This geometric approach to the classical themes of differential equations can be

certainly pursued further. The reader is referred to Olver's [1993. 19951 treatises, and

to Ian Anderson's forthcoming book, Anderson [199?]. What is crucial is the fact

that the equation manifold SS is a genuine non-trivial geometric object determined

by Equation (1.8) which carries in itself important information on the equation under

study. Some recent papers in which this is made apparent are the detailed studies of

the variational bi-cornples for hyperbolic second-order scalar equations in the plane

made by Anderson and Kamran [199?] and Anderson and JuraS [1997]. the work on

characteristic cohomology of differential systerns and integrable estensions by Bryant

and Griffiths (1995, 1995a], and the realization by Krasil'shchik (19921 that generalized

and forrnal syrnmetries have cohomologicaI interpretations.

1.2 Kinematically integrable equations

The class of equations describing pseudo-spherical surfaces d l be now re-introduced

in the context of the forma1 theory of differential equations reviewed above.

Definition 3 Consider the differential equation

i n which S be a smooth function on P E , and let S.;r: be its equation manifold.

Equation (1.21) 2s said to describe pseudo-spherical surfaces if there ezist srnooth

furtctions fO4 (a = 1 2: 3; ,O = 1 ,2 ) O R Sw, for which the pull-back of the one-forms

b y local holonomic sections jS(s) of Sm such that

satisfy the structure equations of a surface of constant Gaussian cvruature equal to

-1 with metric ( w ' ) ~ t (w2)? and connection one-fom w3, namely?

3 d U L = w A u 2 , du2=w1hw3, and d u 3 = ~ ' h u 2 . (1 -23)

The dependent variable zo does not have an immediate geometric interpretation

in this set-up. However, from the point of view of SO" one can say the following.

Suppose that Equation (1.21) describes pseudo-spherical surfaces, and let s : ( x , t ) e

(x! t , zo (x: t ) ) be a section of E such that its infinite prolongation jw (s) is contained

in Sm. If one considers the space of independent variables (x, t ) as the domain of a

local chart on the graph Q of the section j m ( s ) , then Q itself is a submanifold of of

S" which possesses the structure of a pseudo-spherical surface, and this structure is

determined in the coordinates (x, t) by the oneforms (j"(s))'wa, a = 1 ,2 ,3 .

I t should be pointed out that if an equation describes a one-parameter family

of pseudo-spherical surfaces (kinematic integrability, see Definition 2 and subsequent

discussion) the dependence of the associated one-forms { w l , w2, w3} on the param-

eter may bel in principle, completeIy arbitrary. However, one of the strengths

of this theory is that one can effectively characterize large classes of kinematically

integrable equations for which the corresponding linear problems present a fixed a

pn'ori dependence on 77. Thus, for example, a complete characterization of evolution

equations of the forrn = F(x , t , za, ..., z,) describing pseudo-spherical surfaces for

which the associated one-forms satisS w2 = qdx + f22dt (SO that the x-part of the

associated linear problern (4) is an eigenvalue problem) has been performed by Reyes

[1998], generalizing earlier work by Chern and Tenenblat [1986], and will be reviewed

in Chapter 2. .&O, in a related vein, Rabelo and Tenenblat [1992] have proven that

there exist only four equations (up to changes t+ f (z0) of the dependent variable) of

the form ZO,, = +G(zO> ZL, z2) (no explkit x, t dependence) which are kinematically

integrable and such that the associated one-forms satisfy w2 = qdx i f22dt.

Finally, one should also stress that, as said in the Introduction, if the equation - c = O describes pseudo-spherical surfaces, this equation is the integrability condition

of the sl(2, R)-valued linear problem

and conversely, if an equation E = O is the integrability condition of an sl(2, R)-valued

linear problem, one easily sees that it describes pseudo-spherical surfaces.

The following proposition presents two esamples (one of them well bomn) of kine-

matically integrable equations which are not of evolutionary type, and therefore are

not covered by the characterization results cited above. This proposition also illus-

trates hom involved the dependence of the functions fap on higher order derivatives

of the dependent variable may be.

Proposition 1 T h e sine-Gordon and Mange-Ampère equations (giuen by (1.25) and

(1.26) below) are kinematically integrab le.

Proof: SineGordon is the equation which was used classicaly to describe pseudo-

spherical surfaces (See for example Eisenhart [1909]). In appropriate light-cone co-

ordinates reads

= sin 4, (1 -33)

in which 4(<, 77) is the angle between the asymptotic directions of the surface. It is

kinemat ically integrable, since the one-parameter family of one-forrns

satisfy the structure equations (1.23) whenever d(<, 7) is a solution of (1.25)-

The fact that the Monge.4mpère equation

is kinernatically integrable follows from the observation that the graphs z = u(z, y)

which are solutions of the Monge-Ampère equation (1.26) define surfaces of constant

curvature equal to -1 in R3, and that therefore one may hope t o find a relation

between the angle d(<, 7) satisfymg (1.25) and the function u(x, y) satisfying (1.26).

Ferapontov and Nutku [1994] show that this is indeed the case. They set

and consider the Pfaffian system

de = A(u,,dx + (u, + H ) d y )

dg = B((u, + H)dx + u,dy),

in which

they then check that the Pfaffian system (1.27) and (1.28) is completely integrable

on solutions P ( { , v ) of (1.23) and u(x, y) of (1.26), so that i t determines a change

of independent variables between (c, g) and (z, y) whenever q5(<, g) is a solution of

(1.25) and u(x, y) is a solution of (1.26). Finally they show (see their Equation (10))

that this fact implies that the dependent variables # and u satisfy the interesting

equation

Here, & denotes variational derivative with respect to u. It is well known (Eisenhart

[1909]) that k a is exactly the mean curvature of the surface with graph z = u(z, y):

so that (1.31) has a readily geometric interpretation.

Now, since the sineGordon equation (1.25) is kinematically integrable: Equations

(1.27), (1.28), and (1.31) imply that so is the Monge-.Ampère equation (1.26). It

describes a one-parameter farnily of pseudo-spherical surfaces with associated one-

forrns

in which the differentials dq and d< are given by Equations (1.27) and (1.25). This

finishes the proof. 0

Chapter 2

Evolution equat ions describing

pseudo-spherical surfaces

Large classes of equations which describe pseudo-spherical surfaces have been char-

acterized (Chern and Tenenblat [1986], Jorge and Tenenblat [1958] : Rabelo [l989],

Rabelo and Tenenblat [1990], and Kamran and Tenenblat [1995]), by using the fol-

lowing approach: One chooses coordinates (x, t , ZO, ... , zk) as in (5) ' and realize that

the structure equations (1.23) give enough constraints on the functions faa and F

so that they can be effectively computed. This method is used here to perform a

complete classification of partial differential equations of the form

mhich describe pseudo-spherical surfaces: under the assumption t hat f21 equals 77: the

spectral parameter. This chapter consists of two sections, Section 2.1 (characteriza-

tion theorems) and Section 2.2 (examples). They correspond, essentially: to Sections

2 and 5 of Reyes [1998].

2.1 Characterization results

The following Lemma Nil1 be used to find F in terms of the functions fao7 and to give

tvorkable necessary and sufficient conditions on these functionsso that the oneforms

w Q sat isk the structure equations (1.23).

Lemma 1 Let 20,~ = F ( x , t , q, ..., zk) be a n euolution equation, and let SIX; be its

equation manifold. This equation describes pseudo-spherical surfaces with associated

o n e - f o m s u" = f a l d x i fo2dt, a = 1,2,3, in which fap are smooth functions o n Sr'

and f21 = q, if, and only if, these functions sntisfy the conjunction of

f 2 2 , ~ ~ = 0, and

and moreover the following equations hold:

Proof: Work in Sm coordinatized by the variables x: t , zi7 i 2 0: and suppose that

the functions fap depend at most on x, t , a, ..., zktl. The structure equations (1.23)

are satisfied if and only if

k+l

C fm.z,d+i A + ( - / i l f 3 2 + f n f x + f * ~ , + ) d ~ A d t = O> and (2.9) i = O

k i l k f 1

f31,;id~i Adz + f32,z;dzi A dt f ( 9 f 1 2 - fllf22 + f32,z - f31,t)dx A dt = 0- ( 2 - I o ) i=O i=O

Sow, clearly the differentials dzi , dx and d t satisfy the relations

dz, /\ dt = z i + , d x A d t , i > O and ( 2 - 1 1 )

on SOC. These constraints irnply that Equations (2.8): (2 .9) and (2 .10) above hold if

and only if

= f31,zi = O: iz l and (2 .13)

One obtains Equations ( 2 . 2 ) , (2 .5 ) and (2.7) by differentiating Equations (2.11). (2.15)

and (2 .16) with respect to z ~ + [ + ~ , ..., and using (2.13): while (2.3) and ( 2 . 6 ) are

found by taking the zk derivative of Equation (2 .15) , and using Equations (2.1) and

(2.2). Finally, (2.4) must be satisfied, othenvise Equations ( 2 . 5 ) 1 ( 2 . 6 ) : and (2.7)

could hold independently of F, and therefore the partial differential equation would

not describe pseudo-spherical surfaces. The converse is obsious: Equations (2.1) to

(2.7), (2 .11) and (2 .12) imply that Equations (2.8), (2 .9) and (2 .10) hold. 0

The following notation wilI be used in what follows:

In addition, whenever L # O one defines Aj recursively as foliows:

~ k - - ' .- -- O,

and, for O 5 j 5 k - 2,

Theorem 1 Let f a p : 1 5 a 5 3, 1 5 0 5 2 be dzflerentiable functions of x, t :

20, ..., z k S U C ~ thut Equations (2.11, (2.21, (2.3) and (2.4) of Lemrna 1 hold, and

assume that f21 = 7, a non-zero real parameter. Suppose that H L # O. The equation

zo,t = F ( x , t? Q, ..., zk) describes pseudo-sphen'cal surfaces zuith associated one-forms

wQ = f Q l d x + foîdt if, und only if, the functions faB and F satisfy:

1. The function F is given in terms of fa@:

2- T h e z, -derivatives of f and f32, 1 5 j 5 k - 1, are given in t e n s of f and

fS1 by the formulas

-3 . T I L ~ zo-derivatives of f12 and f32 sut* the equation

4. The functions f12 and f33 satiSfY the constraints

Proof: -4ssume that the equation q,, = F ( x , t , zo, ..., zk) describes pseudo-spherical

surfaces. Since by hypothesis the derivatives fil,, and f31,.o do not vanish sirnulta-

neously. one can re-mite equations (2 .5 ) ; ( 2 . 6 ) : and ( 2 . 7 ) of Lemma 1 as follows:

These equations allow one to compute f12,, and f3*,+ for 1 5 j 5 k - 1. Indeed,

taking the zk-derivative of Equation (2 .25) and using Equations (2.1), ( 2 . 2 ) and ( 2 . 3 )

of Lemma 1 one obtains

and, on the other hand, taking

and (2.3) again, one finds

(2.26) and using (2.1): (2.2)

= O . ( 2 . 2 9 )

The system (3.28) and (2.29) yields

and one can continue recursively. One obtains that the derivatives fL2,, and f3?,=,

satisfy the system

from which Equations (2.20) and (2.21) follow. Of course, one cannot say that the

same result holds for j = O. One only obtains the equality

which is Equation (2.22). Now one can find constraints on f12: f3?, and their x-

derivatives, solely in terms of fil, f31: fn, and their zj-derivatives: Substitute the

espressions for f 12,=, and /32,zj , Equations (2.20) and (2.211, into Equation (2 -25) and

use 'quation (2.22) to obtain

The conjunction of this equation and Equation (2.26) gives a linear system in f i2 and

f32 which, in turn, yields the constraints (2.23) and (2.24).

Final15 one solves for F out of the equation obtained by substituting Equations

(2.20): (2.21), (2.23) and (2.24) into (2.27). One finds Equation (2.19) as clairned.

Conversely, one shows that if the functions F and fOD satisfy conditions (2.19) to

(2.24), the oneforms w" = faldx + fa2dt sat isb the structure equations (1 -23) when-

ever zO(x , t ) is a solution of the equation za,t = F ( x , t , zo, ..., zk), and therefore that

this equation describes pseudo-spherical surfaces. It is enough to see that Equations

( 2 . 5 ) , (2.6) and (2.7) -or equivalentiy Equations (2.25), (2.26) and (2.27)- of Lemma

I hold, and this is a straightforward computation. O

This theorem characterizes the local structure of equations which describe pseudo-

spherical surfaces under the generic assumption H L f O. AS such: it is analogous

to the first classification theorem (Theorem 2.2) appearing in Chern and Tenenblat

[1986]. However, Theorem 1 has a different character. In their wcrk, the functions F ,

f 12 and f32 are expressed in terms of 77: f i l , f31 and f22r and differential equations to

be satisfied by these last three functions are found. On the contrary. one sees from

Theorem 1 and its proof that only constraints on f12 and f32 are to be found should F

depend esplicitly on x and t. Moreover, no differential conditions on fil: f31 and f2?

appear in this theorem, although one can certainly find equations satisfied by these

functions by simply taking the rj derivatives ( j = 1,2, ..., k - 1) of the constraints

(2.23) and (2.24), and replacing into Equations (2.20) and (2.21).

It turns out that Chem and Tenenblat's [1986] Theorem 2.2 can be directly gen-

eralized if oniy explicit t-dependence is allowed. Indeed, the following holds.

Corollary 1 Let faa, 1 5 a < 3, 1 5 P 5 2 be dgerent iable functions of t. zot . . . y such that Equat ions (2.1), 2 . 2 , (2.3) and (2.4) of Lernma 1 hold, and assume

that = q, a non-zero real parameter. Suppose H L # O. The equation Z O , ~ =

F ( t , zo: ..., zk) describes pseudo-spherical surfaces with associated one- forns w" =

fn1dx + jo2dt if: and oniy i f , the functions foa

1. The funct ion F is given in terms of faB:

and F satisfy:

2. The funct ions f l2 and f32 are given in t e m s of fil: f31, h2 and derivatiues:

3. The functions fil, f31, and f2? satisfy the equations

for j = 0 , l : ..., k - 1, where sj0 = 1 fo r j = 0, and zero othenuise.

Proof: The equations for f12 and f3Z follow immediately from the constraints (2.23)

and (2.24) of Theorem 1. Now, Equation (2.22) (Part (3) of Theorem 1) and the zo-

derivative of Equation (2.26) yieid

One obtains Equation (2.30) by solving this system for f 12,L0 and f32,ro, subst i tu thg

the resulting equations into the expression for F appearing in Part (1) of Theorem 1,

and using equations (2.31) and (2.32). The differential equations (2.33) are found by

replacing formulas (2.31) and (2.32) into the constraints for the derivatives of f12 and

f3? appearing in Parts (2) and (3) of Theorem 1. The converse is a straightforward

computation. O

Attention is now turned to the study of the non-generic cases L = O and H = O

which are, of course, not covered by the preceding results.

The condition L = O is equivalent to f31 = X(z,t) fil. One can naturalIy divide

this case into two subcases, namely f31 = O or fil = 0, and f3i = A(x, t) fil, A # O-

These subcases are considered in the following three results, which can be more clearly

stated if one defines the functions

JOB := + f22,s fus

whenever faB # 0-

..., zk such that Equations (2.1), (2.2), (2.3), and (2.4) of Lemma 1 hold, and assume

thaf fil = q, a non-zero parameter- Suppose that either f J 1 = O or f i l = O . The

equation zos = F ( x , t , zo' ..., z r ) describes pseudo-spherical surfaces with associated

one-forms wQ = f a l d x + fa2dt if, and only if,

1. Either fil = 0, the functions f12 and f2* satiSfy

2. The functions f12, f32 and F are given by

3. Or = O , the functions f32 and fn satisfy

4. The functions f I 2 , f32 and F are given b y

Proof: This theorem is a direct generalization of Theorem 2.3 of Chern and Tenen-

blat [1986]: and the proof is therefore omitted. O

Consider now the subcase L = O, f31 = X(x, t) fil, X # O. Equations (2.5), ( 2 . 6 ) :

and (2.7) of Lemma 1 become

Equations (2.42) k-1

and therefore two new subcases arise, depending on whether X 2 - 1 = O or not. These

are considered in the following two theorems:

Theorem 3 Let faB, 1 5 a! 5 3, 1 5 /3 5 2 be diflerentiable functions of x: t , z0 ...:

z, such that Equations (2.11, (2.2), (2.3), and (2.4) of Lemma 1 hoid. and assume

that f21 = q, a non-zero parameter. Suppose f31 = X fil, fil # O and X2 = 1. The

equation ro,t = F ( x , t , zo, -.-, Y) descn'bes pseudo-spherical surfaces with associated

one-forms wa = f,ldx + fU2dt if, and only if,

1. The function f22 satisfies

f22,= = O and fï2,Lj = O for O 5 j 5 k,

2. The functions f12 and f3* are constrained by

3. The function F is given by

Proof: .r\ssume that the equation = F describes pseudo-spherical surfaces.

Equations (2.42), (2.43), and (2.45) are equivalent to

Saking successive derivatives of (3.51) with respect to zk: ~ k - ~ , ..., zli one obtains

and therefore Equation (2.51) becomes

It follows frorn the last two equations that the functions f l2 and f32 satis-

for some function Q(t) . Replacing this equation into (2.50) and taking successive

derivatives of this equation with respect to ~ k - 1 ~ zk-2: - . - ? z1, one finds

/22,=, = O for O 5 j 5 k. (2.53)

It follows immediately from (2.54), (2.55) and Equation (2.50) that

Differentiating this equation with respect to zo and using (2.55): one Ends 4 = O and

therefore f22,= = O. These computations give (2.46) and (2.47). The formula (2.48)

for F is found by substituting Equations (2.46) and (2.41) into (2.49).

The converse is a straightfonvard computation. O

It remaios to study the subcase L = 0, jli # 0, X2 - 1 # O. Let I be the function

One obtains the following result:

Theorem 4 Let fQp, 1 5 cr 5 3, 1 5 ,L? 5 2 be differentiable functions of x, t, zol ...:

zk such that Equations (2.1), (2.2), (2.3); and (2.4) cf Lemma 1 hold. and assume

that f21 = 7, a non-zero parameter. Suppose f3i = X f 1 1; f # O and X2 - 1 # 0- and

set A = qX2 - 17 - A,. The equation z0,, = F ( x , t , ro? . .., zk) describes pseudo-spherical

surfaces with associated one-forms wQ = fa1dx + fez& i f , and only i f ,

1. Either A # O and

(a ) The function f22 satisfies

f22,=k-2 = 0:

(b ) The functions f 12, f32 are given by

(c) The function F is given by

X V + X Z -1 - - A A

JIl: and

2. Or A = O, A, # 0 , and

( a ) The function f2? satisfies

fZ2,., = O for O 5 j 5 ki

( b The functions fin and 132 satisfy the constraint

f22J Af12 - f 3 2 = --

fil

(c) T h e functions f i l and f22 satisfy the dgerent ia l equation

f22 . t v2X- f i l = X r f l l f 2 2 + ~1 (2) - f 1 , and (2.63)

(d) The function F is given by

Proof: Assume that = F describes pseudo-spherical surfaces, and suppose first

that A # O. One easily sees that the equations

are equivalent to Equations (2.42), (2.43): and (3.44). Taking the zk-derivat ive of

(2.67) one obtains

O f 1 2 - f32)r i - l = 01

and therefore by taking the zk-,-denvative of (2.66) and using Equation (2.68) one

obtains f22,Zk-? = O, as claimed in Part ( l a ) of the theorem. New: Equation (2.66)

can be considered as an equation for X f12 - f32. It yields

Replacing this equation into (2.67) and using (2.68) one finds

-(v + &r)f 12 + ~ X f 3 2 = 1-

Equations (2.58) and (2.59) follow a t once from this last equality and (2.69). Finally,

formula (2.60) for F is found by substituting (2.58) and (2.59) into (2.65).

-4ssume nom that zoYt = F describes pseudo-spherical surfaces, A = O? and A, # 0.

Equation (2.67) becomes

Taking successive derivatives of both this equation and (2.66) with respect to z k ? zk-l :

...: zl one obtains

This gives Part (2a) of the theorem. Notv, these equations imply B = 0, and therefore

Equation (2.69) becomes

which is the constraint (2.62). One finds the differential equation (2.63) for fli and

fi? by substituting this constraint back into Equation (2.71). Finally, formula (2.64)

for F follows from Equations (2.63) and Parts ( 2 4 and (2b).

The converse is a straightfonvard computation. O

One considers next the last non-generic case, H = O. Clearly, H = O is equivalent

to f:, - f& = c ( x , t ) ; and one can certainly assume that c # O for, otherwise? L = 0.

The following notation needs to be introduced:

In addition, one defines E' recursively as follows:

and for O < j < k - 2,

k-1 L v E' := - zi-+iE!CL + ( - z l - + 7 + -) (B + f22.r)=,+l -

CI; -a c c dil x + -,@M. (2.73)

i = O C

The characterization result one obtains is the following:

Theorem 5 Let fa3, 1 5 CY 5 3, 1 5 0 < 2 be differentiable functions of L, t ,

20, ..., zk such that Equations (2.1): (SA?), (2.3), and (2.4) of Lemrna 1 hold. and

assume that f 2 L = q, a non-zero parameter. Suppose that f,?, - f:, = c, in which

c = c(x, t ) # O. The equation zoTt = F ( x , t, 20, ..., zk) describes pseudo-spherical

surfaces with associated one-forms wa = faldx t fa2dt i f . and only iJ

1. The function F is given in terms of fao:

2. The zj-derivatives of f and f32: 1 5 j 5 k - 1 are given i n terms of fil: f 31

and f22 bg the formulas

3. The z,-, -derivatives of f l Z and f32 are related by the equation

4. The functions f 1 2 and f32 are related b y the equation

5. The functions f12 and f32 satisfy the constraints

Proof: This theorem may be proven by working along the lines of Theorem 1 and

Theorem 2.5 of Chern and Tenenblat [1986]. The proof is therefore omitted.

2.2 Examples

This section contains some

Section 2.1. In particular,

applications of the classification theorems just proven in

some new families of equations which are kinematically

integrable in the strict sense, are introduced.

1. The linear equation. Consider the general kth order linear partial differential

in mhich the coefficients are functions of x. That this equation describes a

one-parameter family of pseudo-spherical surfaces is a consequence of Theorem

3. Indeed, it is enough to take X = 1 and consider

f l l = f31 = I ( x ) ~ o , f21 = 71: f22 = 0: and k

fi2 = f32 = C k - j ~ k - j + E, j= i

mhere the functions Ck-, are determined by the equations

Ck = 0: and

the function -; # O is a solution of the linear 0.d.e. qCo + Cos = ?a0 and E is

determined by the equation 77E + Ez = a-11.

Remark: Some linear equatioas with t-dependent coefficients appear as spe-

cial cases of Theorem 4. Indeed, if one determines X by the equations

& = O and -Xt=A(t )A,

wliere h = qX2 - 11 - X I and A(t) is a rb i t rap then the family of equations

wit h B(t) also arbitrary: describes pseudo-spherical surfaces with associated

functions Ill = a ( x , t , q ) z o , fz l = 7, f31 = X f i l : f22 = 0: f i 2 = aAq, and

h2 = XaAto, where the function a(x, t, q ) is determined by the linear partial

differential equation

at - a,,4 - qXaA = -aB(t).

Ko other esplicitly x/t-dependent examples in which Theorem 4.1 can be ap-

plied are known to this author.

2. The equation = z3 +z&+czl, c E R. This formally integrable equation (See

Mikhailov, Sabat and Sokolov [1991] and Chapter 3 of this Thesis) describes

pseudo-spherical surfaces with associated functions faD given by Theorem 2: and

is therefore strictly kinematically integrable. Indeed, one easily finds fl1 = 0:

fi ( i 2 + $2; + (C + i12) 20). (See also Chern and Tenenblat [1986]).

3. An x-dependent famiIy. A new family of equations describing a one-parameter

family of pseudo-spherical surfaces can be constmcted by applying Theorem 3.

-- Indeed, the members of the family of equations

where b(x , Zr) is arbitrary, describe pseudo-spherical surfaces with associated

functions faD deterrnined by

fil = f31 = Q(Z)ZO, f22 = 0, f * ~ = q, and fl:! = f32 = 6ûz2,

in which 7 7 a + a, = O. It may be remarked that even if b, = 0, the associated

functions fa8 are ezplicitly x-dependent .

4. -4 non-generic esample. Let h(t) # O and consider the equation

This equation describes pseudo-spherical surfaces ni th associated functions sat-

isfying the non-generic condition f:l - f:, = c considered in Theorem 5 . Indeed.

it is a simple variation of an example due to Cavalcante and Tenenblat [1988].

One finds = q sinh(ehu), f21 = 7. fZ2 = -q2e-h/2v;1/23 fsl = q cosh(ehv).

f 12 = qe-h/2(v;1/2)1 cosh(ehv) + [eh/2(t.z)1/2 - 17e-h/2 sinh(ehz), and

f32 = ve-h/2(z;1/2)z sinh(ehv) + [ehj2(v=) - qe-h/2(vz)-1/2]il cosh(eht.).

5. An explicitly t-dependent KdV equation. A family of esplicitl? t-dependent

evolution equations describing a one-parameter family of pseudo-spherical sur-

faces to which the KdV belongs, is found by considering Corollary 1 and making

the estra a priori assumption fil,,, = f3i,zo # O. Indeed, the family

in which the functions A(t), C(t) , and K(t) satisfy the condition

is strictly kinematically integrable, with associated functions f i l = a.zO + j3. f 2 1 = 77, f . 3 ~ = QZO + P + a, F22 = - ( A / ~ ) z I + (A71/3)zo + and fi2, f3*

43

determined by formulas (2.31) and (2.32) of Corollary 1. Here, a and a are

functions such that

y is given by

and /3 satisfies the equation

6. The Calogero-Degasperis family. The family of equations

has been shown to be integrable by inverse scattering techniques by Calogero

and Degasperis [1978]. A one-parameter family of Iinear problems for which

Equation (2.84) is the integrability condition is determined by Theorem 1 and

the a priori assumption fll,,, = f31,za. One obtains that the functions La are

given by the forrnulae

f i 2 = ~ a 1 ~ 2 - rlcra~zi - 2a<rlr: + a(ao - a1[4z + a2 - q2])zo + - and a

in which

(a2 t a a = - 2 , a,=O and - 4

+ 2ula2 - 2qa1 = 0.

7. Equations related to KdV by Fuchssteiner transformations. In an interesting

paper, Fuchssteiner [1993] shows how to generate integrable evolution equations

with tirnedependent coefficients ("integrable" here is taken in the sense of pos-

sessing an infinite number of tirne-dependent generalized symmetries): starting

from equations knomn to be formally integrable. In particular, one learns from

that work how to find transformations between the KdV equation and its time-

dependent generaiizations. These transformations allow one to show that these

equations describe one-parameter families of pseudo-spherical surfaces. Two

esamples are given below.

The equation ut = (l/t)(u,, + 6uzu - (1/3)v - (2/3)xv,) . The invertible

point transformation

takes this equation to u:, = US,, + 6ufu', . Therefore, it follows frorn

Example 5 that it is kinematicaily integrable.

1 The cylindrical KdV equation u, = -vcc( -vu< - -u. The invertible point

20 transformation

takes this equation to ut = u,,, + uu,. This rneans (again by Esample

5) that the cylindrical KdV describes a one-parameter family of pseudo-

spherical surfaces.

This esample is interesting as not only illustrates t h e results obtained in Section

2.1, but also their limitations. Indeed, one can see: for example from (2.86):

that the functions fOa determining

some reasonable cases. It appears,

theorems proven in this Chapter if

kinematically integrable equations.

the one-forms biQ do not satisfy fZ1 = 77 in

therefore, that one needs to generalize the

one aims to a complete characterization of

8. The KdV hierarchy (-4fter Chern and Peng [1979]).

In an interesting paper, Chern and Peng [1979] study the connection between the

KdV hierarchy and the Maurer-Cartan structure equations of SL(2 , R). Their

results quickIy impIy that every equation in this hierarchy describes pseudo-

spherical surfaces. Indeed, for each n 2 1 one considers the one-forms

in which C, is a polynomial in q given by

and the functions cj , j = 1,2, . . .: are defined recursively as follo~vs:

2cr = -zo 7 and

One obtains that the one-forms wn', i = 1: 2,3: satisfy the structure equations

(1 -23) whenever zo(x, t ) is a solution of the nth-order KdV equation

in which I(, is the q-independent function given in terms of C, by the formula

1 Kn = Z I C ~ + 2~0Cn,z + 2v2Cn,z - 5G".xzz- (2 -93)

In particular, Ci = coq2 + ciqO = 82 - (1/2)zo, and (2.93) gives

3

the ubiquitous KdV.

Chern and Peng [1979] also study the modified KdV hierarchy, and their results

also show that every equation in this hierarchy describes- pseudo-spherical sur-

faces. This fact does not appear to illustrate further the issues discussed in this

chapter and will not be pursued here.

Chapter 3

Forma1 and kinematic integrability

Formal integrability is introduced in this chapter in the contevt of the theory of forma1

symmetries, and the implication "formal integrability =+ kinematic integrability" is

studied. There are three sections, "Formal symmetries and formal integrability"

(Section 3.1): "On the integrability of second order equations" (Section 3.2) and

''On the integrability of third order equations" (Section 3.3). The analysis of the

aforementioned implication appears in Sections 3.2 and 3.3, both of which are based

on a previous paper by the author (Reyes [1998]).

3.1 Formal symmetries and formal integrability

Forma! symmetries have been studied by Svinolupov and Sokolov [1982]. Mikhailov.

Shabat and Yamilov [198S], Mikhailov, Shabat and Sokolov (19911, and Shabat and

klikhailov [1993]. Exhaustive references may be found in the last three works cited.

A fine introduction t o this subject appears in Olver [1993]. The general notion mil1 be

neittier reviewed nor used here, the interested reader is referred to the bibliography.

What is of interest in this chapter is the particular case of a forma1 symrnetv of

infinite rank:

Definition 4 Let = F(Z,ZO, ..., zk) be an autonomous evolution equation with

equation manifold Sa. Let (See Equation (1 -9))

be the operator of total derivative with respect to x, and let

be the f o m a l Zznearization of F . A forma1 syrnmetry of infinite rank is a formal

pseudo-dzfferential operator N

where the coeficients fk (-00 < k 5 N ) are smooth functions o n Soc, such that the

Lux-type equation

a, = [F*, Q]

holds on solutions of = F .

In Equation (3.11, the derivative ikt is defined as

in which Dt is the total derivative with

(1.10)):

respect to t restricted to Sm (See Equation

Equation (3.1) is the equation characterizing the important "recursion operators"

first defined by Olver [1977]. It yarantees that if G is a (generalized) symmetry of

the equation zoPt = F, then so is @(G) . Note however that this notion includes the

assumption that ik(G) is a well defined local function on Sm, a non-trivial hypoth-

esis. Xonetheless: in principle, a forma1 s y m m e t l of infinite rank of the equation

~ 0 , ~ = F ( x : 20: .. ., zk) generates a sequence of generalized symmetries of this equation

depending on arbitrarily large numbers of x-derivatives of the dependent variable zo.

One can also prove (Mikhailov, Shabat and Sokolov (19911. Olver [1994]) tlie

following two important facts- First, if O satisfies (3.1), then the coefficient f-i is

a conserved density for the equation zogt = F. Second, if i3r satisfies (3.1): then

so dues Q ( ~ / " ) for every rn and n > O. Thus, if an equation posseses a forma1

syrnmetry of infinite rank, one can find an infinite sequence of conservation laws

for this equation. I t does not follow, however, that these conservation laws are not

trivial: Burgers' equation possesses a formal sqnimetry of infinite rank (Olver [1977!,

SIikhailov, Shabat and Sokolov [1991]) but. as mentioned in Section 1.1. it has no

non-trivial local consenation laws other than itself.

The last two paragraphs mean that the existence of a non-trivial (in the sense

of not being simply multiplication by a constant) forma1 pseudo-differential operator

satisfying (3.1) may be advocated as an indicator of what "integrability" should

rnean. Indeed: the existence of such a Q has been proposed by Fokas [1987] and Olver

[1993] as the defining property of integrable equations. An interesting discussion on

this point appears in Shabat and Mikhailov's [1993] review paper. They point out

tlie intriguing fact that even if the construction of a sequence of local generalized

symrnetries by means of \k fails, or if the conserved densities one finds by computing

fractional powers of \i yields only trivial conservation Law, the fact that a non-

constant \k satisfying (3.1) exists, appears to indicate that the equation in question

can be eit,her linearized or integrated by means of inverse scattering, and should

trlerefore be considered "integrable". They present two esamples:

1. The Burgers' equation Z O , ~ = 22 + qq- AS pointed out above, this equation

possesses a recursion operator ik and a sequence of generalized symrnetries (See

Olver [1977] and [1993] p.315 for esplicit computations) but the conservation

laws one obtains by taking fractional powers of \k are al1 trivial. However,

Burgers' equation can be linearized bÿ means of the Cole-Hopf transformation,

2. The Svinolupov equation

2 Zo,t = ZOz2 + x2zL - 3 1 ~ ~ .

Svinolupov (unpublished? quoted by Sliabat and Mikhailov [1993]) showed t hat

this equation possesses a recursion operator Q and it is therefore (See Mikhailov.

Shabat and Yarnilov [1987], Theorem 5.1) equivalent, via a contact transforma-

tion, to oue of the four equations appearing below (Equations (3.3)-(3.3)). But_

Shabat and Mikhailov [1993] point out that al1 of the symmetries one obtains by

applying iI are nonlocal functions on Sm: they depend on x, y, and DiL (z/zo).

It turns out, however, that Equation (3.2) can be linearized by means of the

transformation

In view of the comments above, and following the Russian school, it appears

natural to make the following definition.

Definition 5 A n autonomous euolution equation zoz = F ( z , ZO? ...: zk) zs said to be

formally integrable if i t possesses a non-trivial forma2 s ymme t ry of infinite rank.

3.2 On the integrability of second order equations

This section consists only of the theorem below, stating that every second order evo-

lution equation which is formally integrable, is kinematically integrable. -4s pointed

out in the Introduction, the proof consists of checking that every formally integrable

second order equation appearing in the exhaustive (up to contact equivalence) list

provided by Mikhailov, Shabat and Sokolov [1991] is kinernatically integrable in the

strict sense. Of course, it does not follow that e u e q formally integrable second order

equation is strictlg kinematically integrable, as the property "The equation zo, = F

describes pseudo-spherical surfaces" is invariant under contact transformations, but

on the other hand, the a priori assumption fil = 17 is not.

Theorem 6 Every second order evofution equation of the form 20, = F ( x : zo: zl: z2)

which possesses a fonnal symmetry of infinite rank describes a one-parameter family

of pseudo-spherical surfaces. Moreover, there ezists an exhaustive list of represen-

tatives (up to contact transformations) of fomal ly integrable second order equations

which are kinematically integrable i n the strict sense.

Proof: hlikhailov, Shabat and Sokolov [1991] provide an exhaustive list of second

order equations which possess a forma1 s p m e t r y of infinite rank. Up to contact

transformations they are (Mikhailov, Shabat and Yamilov [1987], Theorem 5.1: and

OLver (19941, Theorem 5-48) :

The fact that linear equations describe pseudo-spherical surfaces with = 71 \vas

easily checked in Section 2.2. On the contrary, the computations needed for proving

that the non-linear equations (3.4). (3.5) and (3.6) are kinematically integrable are

not straightfonvard. They will be tvorked out in detail.

1. The equation zont = 22 i zOzl + h(x). Theorem 1 is the result reIevant here.

Equate the right hand side of the aforementioned equation with F given by (2.19)

and k = 2. Set F;ll = 22 + z0zl + h(x) , and assume that the functions fao do not

depend explicitly on t.

- The coefficient of z2 in F(Il is one, so that fn,, = L: B = and

This means that F equals

Kow; Equations (2.20) and (2.21) of Theorem 1 imply that = fl l ,ro and f32.-i =

f31,=0- Thus, the constraints on f12 and fS2 appearing in Theorem 1. Equations (2.23)

and (2.24), yield the following equations for the zo-derivatives of fl1 and f31:

The zl-derivatives of these equations yield P = 0, and it follows that the second

Since the second zl-derivative of F(l1 is O, L., = 0, and therefore f22,zozo = O.

The equations P = O, Lzo = 0, and f2zlzozo = O are enough to determine the

functions f i l , f& and f2* up to functions of x. Indeed, one easily finds

fll = a w o + @)? (3.1 1)

f31 = X(x)a(z)ro + A ( x ) b ( x ) + v ( x ) , and (3.12)

f22 = -u(z)a(x)zo + P ( x ) - (3.13)

Som one uses the fact that the zl-derivative of F(1) is ro. One finds the equation

1 Fr, = ~ ( v H + f11/32.:0 - f31f 12.~0 - R) = 20- (3.14)

Since Equation (2.22) (Theorem 1, Part (3)) must hold, one obtains

The last two equations imply that

f 1 2 . q = azo -a, -Aar) , and

f32,ro = - (Aa)r -

Aa 2 f32 = Aazl f -q-, - ((Xa), + ai)) zo + 6.

2

These computations imply that the formula for F- Equation (3.7), becomes

Cornparison of this formula with F(i) yields an intricate under-determined system of

ordinary differential equations for the x-dependent coefficients of the functions fa$.

One then supplements it with equations coming from the constraints on f12 and fm

Equations (2.23) and (2.24), which have not been used so far. (Indeed, only their

zl-derivatives were.) The result is an intricate ouer-determined system of 0.d.e.k

for the aforementioned functions. The obvious way to proceed then, is to impose an

ansatz on some of them. It is enough to assume that X = O. The functions 6! u, 0,

a , 7, and b appearing in Equations (3.11), (3.12), (3.13), (3.18) and (3.19) are then

determined by the following ten equations:

a& -- 2a + a,, = O

The first four equations corne from comparing F with F(l). Equatioos (3 .25) : ( 3 .26 ) :

and (3 .27) appear after comparing the constraint (2 .23) \vit h Equation (3.18)- Finall-.

t h e last three equations can be deduced from comparing (2.24) with (3.19). This

system can be solved without trouble. One obtains

1 1 -0 a = - 7 = - / h ( x ) d x , b = -

2 ' 2 , v = -77, and 6 = -p?

77

in which 0 is determined by the Riccati equation

For the convenience of the reader: the final result is stated here: The equation

describes pseudo-spherical surfaces with associated one-forms

w2 = vdz + (5% + 0) d t ,

u3 = - ~ d x - ( zz0 +@) d t ,

in which 0 a solution of Equation (3 .32) . These one-forms reduce to the ones found

by Chern and Tenenblat (19861 for the Burgers' equation if fl = h = 0.

-2 2. The equation z0.t = D,(Z~Z;~ + axzo + P z O ) Set F(21 = zo z2 - 2zt3z? + (ax i

/3)rl + uq. The fact that the equation a,c = F(2) describes pseudo-sphericai surfaces

is a consequence of Theorem 3. Indeed, assume that L = 0: f Z 1 = Xf l t : and X2 = 1.

Suppose, moreover: that the functions fOy do not depend explicitly on t. Equation

(3.43) implies that F is given by

Comparing the z2-coefficients in F and q 2 ) , one sees that the functions f 12 and f i l

must satisfy

The function b(x, zo) is found by comparing the z:-coefficients in F and F(*). One

obtains b(x, zo) = 7(x)zc2, and therefore

Since by Equation (2.46), Part (1) of Theorem 3, /22 = k< the function F becomes

X ~ C Xkd C, - Xkz, + - - - + -.

Y 7 I

Compare the zl-coefficients of F and F(21. One finds that c must be given by

for a function d(x) to be determined. Replace this equation back into the expression

(3.38) for F. It follows that the functions F and F(zl are equal if, and only if.

Choose the ansatz 6 = d = k = O. Equation (3.42) is then vacuous, and Equa-

tions (3.40) and (3.41) become an over-determined system for y(z) which admits the

immediate solution

-2 To summarize, the equation zoYt = zo 22 - 2zr3zf + (OZ + 4)z1 + azo describes

pseudo-spherical surfaces with associated functions f = esp(-qAx) zo : f21 = qr

fzz = O, fSi = Aexp(-qAx)ro, f12 = e ~ p ( - ~ ~ z ) z ~ ~ z l + exp(-qXz)(ax i P ) i o , and

f3? = X e ~ ~ ( - q X x ) z ~ ~ z ~ + X esp(-~Ax) ( ax + 8 ) a 7 in which X2 = 1.

-3 2 3 . The equation zoBt = ~ , ( z ~ z ~ ~ + 2) = zo2z2 - 2z0 z1 + 1. That this equation

describes pseudo-spherical surfaces is also a consequence of Theorem 3. One easily

finds, by using (3.38), (3.39): (3.40) and (3.41): that the aforementioned equation

describes pseudo-spherical surfaces with associated functions f = esp( -Xqz)zo.

f21 = q, f22 = 0- f31 = Xexp(-Xqx)zo, f12 = e ~ ~ ( - A ~ z ) z ~ ~ z ~ f b(x): and f3? =

X e s p ( - ~ v x ) ~ 2 z l + A6(x), in which the function b(x) is determined by the equation

and .A2 = 1.

This finishes the proof that Equations (3.4), (3.0) and (3.6) are kinematically

integrable in the strict sense, and therefore the proof of Theorem 6 is complete. O

I t may be worth noticing that even though the last part of the proof deals with

an equation without eq l ic i t x dependence, the associated functions Lfl are esplicitlÿ

x-dependent, and therefore could not be found by using only the results appearing

in Chern and Tenenblat [1986].

3.3 On the integrability of third order equations

Attention is now turned to third order equations. I t will be shown that one may not

tiope for exhaustive lists of formally integrable third order equations formed by only

strictly kinematically integrable p.d.e.'s:

Abellanas and Galindo [1985] have proven that an equation of Harry-Dym type.

= f ( t , zo: z1)z3, is formally integrable if, and only if, either f = ( a z i + pro + 1)3/2

or f = ( z L + 6)3, a, p, y r 6 in R, up to reparametrizations of the t variable. One

lias the following theorem.

Theorem 7 1. Every equation of the form

where a(zo) 2s a n arbi t raq snzooth function, describes a one-parameter family

of pseudo-spherical surfaces.

2. There are no functions faO depending o n x, t and a finite number of variables

zi, i > 0, with f21 = 7, such that the formaily integrable equation

describes pseudo-spherical surfaces m'th associated functions faB.

Proof: Assume that a # O. The members of the family of equations z0,t = az3

describe pseudo-spherical surfaces with associated functions fao determined by The-

orem 3: Assume from the onset that the functions fas do not depend explicitly on t .

The function F: the right hand side of the differential equation, is given by Equation

(2.48)

1 F = - ('3f 12,iZ + ~ 2 f 1 2 . a + if 12.z0 + X v f 1 2 - X f l l f 2 2 + f12.1) r

f 1 L a

and therefore one must have f12 = a f ll,zoz2 + cr(x, zo: z l ) . Since no Z* term must

appear in F? one finds the equations

Reemplacing these equations into the expression for F, one sees that no z: terms

appear and that fil must also satisfy

for functions v # O and 6 to be determined. Since no zl terms must appear in F ,

one must have oz,, = 0, and so one may choose = O. Taking f2* = = 0: one finds

F = a(zo)zs, as long as satisfies (3.44) and (3.43). The explicit formulae for the

functions f a p are

in which X~ = 1.

Xow set Fp) = (z1 + 6 ) 3 ~ 3 . Comparison of F as given by formula (2.19) (Theorem

1) with F(3), yields f22,z, = L(z1 +6)3 and this fact, together with equations (2.20) and

(2.2 1): yields in turn FL2=? = 6(z1 + 6)2, so that F # FL3). The function F(3) cannot be

given by formulae (2.37) (Theorem 2) or (2.60) (Theorem 4.1): it ~vould follow that

the coefficient of z3 in F(3) depends at most on zo. It is not @\*en by formulae (2.48)

(Theorem 3) or (2.64) (Theorem 4.2) either: Comparison of F as given by them and

4 3 ) implies f lZ,z2 = fll,zo(zl + 6)3, and it would follows that F(3)ii2z2 = 6(zl + d)2

instead of zero. -4 similar argument, using Equations (2.75) and (2.76); shows that

F(3) cannot be given by formula (2.74) of Theorem 5.

One may stilI wonder if a result analogous to Theorem 6 holds for a different

subclass of formdly integrable third order equations. Indeed this is so:

Theorem 8 Every third order evolution equation of the form

- 3 2o.t = 20 23 f 4 2 , ZO, f l ) ~ ; + ai@, f o i f&2 + ao(G 20: zi) (3.46)

which possesses a formal symmetry of infinite rank describes a one-parameter family

of pseudo-spherical surfaces. Moreover, there exists a n exhaustive list of representa-

tives (up to contact transformations) of fonnally integrable third order equations of

the form (3.46) which are kinematically integrable in the strict sense.

Proof: An eshaustive list of representatives of formally integrable third order equa-

tions of the form (3.46) appears in Mihailov, Shabat and Sokolov [1991] pp. 162-163.

Each member of the aforementioned list is of the form

in which A = A(x, a, ..., z2). The resuit then follows from the rather general propo-

sition proven next.

Proposition 2 Equations in the fornt of conservation laws,

in which -4 = A(x, 20: . . . zk), are kinematically integrable.

Proof: Indeed, associated functions fap are determined by Theorem 4.2. It is

enough to set f i l = a(x)zo, fil = q: fn = O, fJ1 = X(x)a(x)zo, fil = a(4.4,

and 132 = X(x)a(x)A, where the functions X(x) and o ( x ) are determined by the

equations

r l ~ 2 - q~ - Al = 0, and

qX(z)û + a, = 0. El

What about quasi-linear formally integrable third order equations of the form

20,~ = z3 + G(to, ri, t2)? It appears that no result as Theorem 6 may be proven in

this case. For esample, one has the following proposition.

Proposition 3 There are no functions depending on a finite nurnlier of variables

zi, i > O, with faBF = fafigr = O and f21 = T), and satisfying the concLusions of Lemma

1 '(Chapter 2) such that the fonnally integrable equation

describes pseudo-spherical surfaces with associated functions fa8.

Proof: It is enough to use the results proven by Chern and Tenenblat [1986]. Taking

F(41 = z3 + z: + C Z L + CI> one sees that if F(4) were given by their Theorem 2.2. one

would have f21.z1 = L and L2, = PZ, = 0, and then the differential equation

which FL2: f and f3L must satisfy would imply L = O, a contradiction. The function

F(4) is not given by Chern and Tenenblat's [1986] Theorems 2.3 or 2.4 either: One

easily sees that the fact that the coefficient of z3 be 1 precludes the term z: from

appearing. Finally, it is proven in Rabelo and Tenenblat [1992] that no q-independent

equation of the form q t = z3 i G(zO, 21: Q) is given by Chern and Tenenblatk [19S6]

Theorem 2.5. a Long computations suggest that this proposition holds without the assumption

fû3,= = = 0. but no forma1 result in this direction wi11 be reported here.

Chapter 4

Conservation laws and kinemat ic

int egrability

This chapter is about conservation laws of equations describing (a one-parameter

family of) pseudo-spherical surfaces. There are three sections: Section 4.1 considers

the classical method (Wadati, Sanulci- and Konno [1975]) for finding conservation laws

of kinematically integrable equatious of AIWS type, and extends it to the complete

class of strictly kinematically integrable equations. Section 4.2 studies the Chern-

Tenenblat method and its relation with the material esposed in Section 4.1. Section

4.3 is a short introduction to the theory of coverings developed by Krasil'shchik and

ITinogradov [1983, l989]. Some of its applications to the geometric understanding

of the conservation laws studied in Sections 4.1 and 4.2 are presented here as well.

Section 4.1 represents an improved version of some results appearing in Section 4

of Reyes [1998]. The results of Section 4.3 will be also used in the next chapter on

(generalized? nonlocal) symmetries of kinematically integrable equations.

4.1 Conservation laws via Riccati equat ions

-4 classical discussion on conservation laws of AKRI'S equations based on the "Riccati

form of the linear problem" appears in Wadati, Sanuki and Konno [1978]. The formal

content of this method generalizes to e v e l strictly kinematically integrable equation,

and is reviewed in the following theorem.

Theorem 9 Suppose that Z(x7 t , a,. . . , z ~ , ~ ~ , ~ ) = O is a dzfferentzai equation describ-

ing pseudo-spherical surfaces wi th associated functions

manifold Sm. The systems

and

1 Dt42 = Dz (?f2..+ f12 + f32

f i l + hl in vlhich D, and Dt are the total derivative operators o n Sw, are integrable whenever

- z0 (2: t ) is a solution of the equation c = 0.

Proof: It has been pointed out in the Introduction and Section 1.2 tha t the equation - = = O is the necessary and sufficient condition for the integrability of the sZ(2, R)

linear problem (l.24), namely,

Equivalently, by (1.23), the functions faa satisfy the equations

wlienever zo (x , t ) is a solution of E = O. Set v = ($1: & ) t and define functions dl and

& by

41 = hl - 2 h (2) 7 and

Straightforward computations using Equations (4.5): (4.6) and (4.7), allow one to

check that, whenever zo(x: t ) is a solution of E = O, if v = q2)' is a non-trivial

solution of the linear system dv = RI;, then given by (4.8) is a solution of the

systern ( 4 (4.2), and d2 given by (4.9) is a solution of the system of equations

(4.3): (4.4). a

This theorem provides one with at Ieast one 7-dependent consemation law of the -

equation z = O if this equation is strictly kinematically integrable, to wit? Equations

(4.2) and/or (4.4). One obtains a sequence of 77-independent conservation laws by

espanding @1 and/or d2 in (inverse) powers of q. Thus, for instance, a short corn-

putation shows that if one assumes that S = O is an evolution equation, that is, - - - - - ro,, - F ( x , t , ro, . . . , zr ) , the functions fil , f31 are independent of 7 and satisfjr

fi 1 $ f3 1 , and qj1 is expanded in inverse ponrers of 77,

then consideration of Equation (4.1) yields the recursion relation

which determines c $ ~

Ddfll - f31)&) n-1

(i) ( n - i ) - D - 1 41 , n 3 1: (4.11) fll - f31 i= l

whenever .Q (x, t ) is a solution of = F. The function $1 : in

turn, yields, by replacing into (4.2), the sequence of conservation laws of evolution

equations integrable by AKNS inverse scattering found by Wadati, Sanuki and Konno

[1973]. Note? however, that the present result is in a sense stronger, as it holds for

cquations with expiicit x and/or t dependence.

Example: The members of the t-dependent KdV family (2.83) possess an infinite

sequence of (esplicitly t-dependent) local conservation laws. Indeed? Equations (4.10)

and (4.11) imply that the functions

are conserved densities for these equations. As usual, one sees that for odd numbers n,

the conserved density $1) contains a pure power of t$I1) and is, therefore, non-trivial.

Further examples of sequences of conservation laws obtained from Theorem 9

are provided by the nest two technical, and straightforward, propositions. The idea

behind them is simply to assume that the functions fil and f31 depend on I] in a

predetermined fashion:

First: suppose that fll and f31 are poIynomials or power series in 77. One obtains

the following result.

Proposition 4 Let Z = O be an equation describing pseudo-spherical surfaces with

assaciated functions defined o n the equation manifold o j = O . Suppose that

f i l # f31, and let f21 = v, a parameter. Set 2a = fil - f3i, 2 b = fil + fS1 and assume

that the functions ab and (D,(a)/a) - 7 can be expanded as formal power series in I].

The system of equations (4.1)) and (4.2) detennined by the functions fQa i s integrable

whenever zo(x, t) is a solution of the equation f = O . Moreover, the fvnction

is a solution of Equations (4.1) and (4.2), if and only i f the coeficients satisfv

the equations

n n ( i ) ( n - i ) D,~P' = A, + C B~@P-') - C 4, #1 , n L O,

whenever q(x, t ) is a solution of E = O , in which the functions A, and Bn are

de temined by

Xow suppose that the functions fil and f31 are given by formal Laurent series in 7.

Of course, one stil1 has (Notation as in Proposition 4) :

for some functions Bi- One obtains the following.

Proposition 5 Let E = O be an equation describing pseudo-spherical surfaces with

associated functions fa* defined o n the equation manifold of 5 = O . Suppose that

that the function ab can be written as a fonnal Laurent series in q of the form

The system (4.1), (4.2) determined by the functions faa is integrable whenever z o ( x , t )

i s Q solution of the equation E = O. Moreover, the function

i s a solution of Equations (4.1) and (4.2) i f and only if the coeficients &\l'l satisfy

the equations

- whenever q ( x , t ) is a solution of the equation r = 0.

Proposition 4 yields conservation laws which are in general of a "nonlocal" nature:

as the coefficients 4y) are to be found (whenever zo(z7 t) is a solution of the equation - z. = O) by solving ordinary differential equations. On the other hand, it is not

hard to convince oneself that the consenation taws obtained from Proposition 5 are

local. Certain13 there are instances when the hypotheses of Propositions 4 and 5 are

satisfied, and tmo examples illustrating these results are collected in Subsection 4.1.1

belon-. Thus, in particular, it is of interest to have a geometric theory in which local

and non1ocaI phenomena are treated on the sarne footing. This is what Krasil'shchik

and Vinogradov% [1983, 19891 t h e o - of covenngs accomplishes, as it will be seen in

Section 4.3.

4.1.1 Example: Burgers' equation

Theorems 6 and 9 allow one to find, in principle, conservation laws for ewry formally

integrable second order equation. In particular, Propositions 4 alid 5 allow one to

prove t hat the Burgers' and generalized B urgers' equations, Equations (1.17) and

(3.33), possess an infinite number of conservation laws. These computations are

summarized here.

1. The generalized Burgers' equation describes pseudo-spherical surfaces with as-

sociated functions determined by Equations (3.35): (3.35), and (3.36). For the

convenience of the reader, they are collected here again: fL l = (1/2)zo - ( ,d /q) .

fi2 = (1/2)2~ + (1/4)z: + (1/2) Jh(z )dx , fii = r ) , f22 = (f1/2)~0 + 0, f i l = -I),

and fnz = - ( 1 7 / 2 ) z ~ - ~ ~ in which 0 is deterrnined by the Riccati equation (3.32).

68

Assume that the solution P(x,v) of (3.32) can be espanded in pomers of 77 of

the form

for functions Pi (x) independent of 7. It follows that (Notation as in Proposition

- q2) . and

so that

in which the coefficients n/i depend polynomially on ro, and the coefficients Si

depend polynomially on zo and zl. Proposition 5 nonr implies that the following

are conserved densities for the generalized Burgers' equation:

and

in which Bo = f l o , z / f l o , and Bi = &, i > 0.

Sote that: in agreement with the rernark after Proposition 5, al1 the consm-ation

laws obtained from the conserved quantities c $ l ) are local.

2. Tt has been already pointed out in the Introduction that the mual Burgers'

equation (Equation (1.17)) does not possess a sequence of (non-trivial) local

conservation laws. Conservation laws of "nonlocal" type for this equation can

be found by using Proposition 4. Indeed, Burgers' equation describes pseudo-

spherical surfaces with associated functions fil = (1/2)z0, fi? = (1/2)zi + (1/4)*:, /21 = q7 = (q/2)z0, f31 = -q, and f3* = -(77/2)z0 I t follows that

(Notation as in Proposition 4)

Equation (4.14) now gives

so that q5t) = &(1/4)zo, and one recovers from (4.2) the obvious fact that

the Burgers' equation itself is a conse-nation lam. On the other hand: the

functions d l 1 , n 2 1, are determined by integrations, and correspond, therefore,

to 'honlocal" conservation laws, a concept to be formally introduced in Section

4.3 in the context of the theory of coverings. Interestingl~ not a11 of them

are non-trivial. For esample, if one chooses = -(1/4)zo, Equation (4.14)

irnplies that QI') is a solution of the ordinar). differential equation

mhenever zo(x, t ) is a solution of Burgers' equation zo,, = z2+zozi , and one finds,

after using the elementary formula for first order linear differential equations

and integrating by parts, that

On the other hand, also assurning &) = -(1/4)zo, one easily finds from Equa-

tions (4.14) and (4.18), that #) is determined by the linear equation

whenever zo(x' t ) is a solution of Burgers' equation a,t = Z* + ZOG. -4 straight-

fonvard calculat ion gives 1 J a,& &' = zOez 7 (4.19)

so that 4$) is a genuine "nonlocal" conserved density of Burgers' equation. A

more complicated example is provided by di1) if one chooses = (1/4)z0.

One finds from (1.14) that 4\') is determined by the equation

whenever zo (2, t ) is a solution of Burgers' equation. A computation analogous

to the one yielding (4.18) gives

Xote that dl'), as given by (4.20) above, does not depend sirnply on integrations

of smooth functions on the equation manifold Sm of Burgers' equation, but on

int egratioos of functions which in turn depend on indefinite integrals of functions

on Sn". One says that d(,') (and in general gr)) depends on nested integrations

of smooth functions on Sm.

4.2 Conservation laws: The Chern - Tenenblat

method

It was seen in Section 4.1 how to find conservation Iaws for equations describing a

one-parameter family of pseudo-spherical surfaces with f21 = 7 by using the basic

equivalence between this notion and strict kinematic integrability. On the other hand,

conservation laws of equations describing pseudo-spherical surfaces (with no a priori

conditions on the functions jaB) have been obtained in a purely geometric fashion

by Chern and Tenenblat [1986]. This construction, and its relationship with the

computations carried out in the Iast section, are considered here.

The followïng fact from the classical geometry of pseudo-spherical surfaces is a t

the base of Chern and Tenenblat's (19861 method:

Given a coframe {cl, W? } and corresponding connection one-form c3 on a smooth

surface S equipped with the Riemannian metric ( ~ l ) ~ + ( z ~ ) ~ , one can find, by means -1 -2

of a rotation in an angle p ( x , t ) , a new coframe {O , B } and a new connection one-form -3 8 satisfying the structure equations

-3 d ë l = O d @ = P 2 ~ P L , and 9 + 8 2 = 0 , (4-2 1)

if and only if the surface S is pseudo-spherical.

Following Chem and Tenenblat [1986], one now gives an analytical interpretation

of this fact as follows. The "old:' and "new" one-forms, w<L and T (a = 1,2,3)

respectic-eiy, are connected by means of the rotation

ëL = Z' cosP + c 2 s i n p -9

0- = -G' sin p + g2 COS P,

and the gauge transformation

e3 = d + dp.

Thus, if the differentiai equation

with equation manifold Sm describes pseudo-spherical surfaces with associated one-

forms wa, a = 1,2,3, Equations (4.21)-(4.24) imply that the Pfaffian system

on the space with local coordinates (x, t: p), in which P = ( jo"(s)) 'w": a = 1 ,2 ,3 ,

is completely integrable for p whenever s is a local section of E, s : (x,t) H

(x, t , zo(x , t)) , such that jm(s) is contained in Sm, and that moreover, for each solution

z i ( x , t ) of (4.25) and p(x, t) of (4.26), the one-form

is closed.

If Equation (1.25) is assumed to be kinematicdly integrable, and the associated

functions can be formally expanded as a power series in a parameter 7, the

function p can be also expanded in powers of 77. The one-form 8 given by (4.27) then

determines a sequence of one-forms which are closed whenever zO(z, t ) is a solution of

(CE), and dH-closed on Sm if they belong to the space of differential forrns on Ss.

An esplicit algorithm to construct these conservation Iaws appears in Cavalcante and

Tenenblat [1988], toget her wit h several examples.

Further applications of the method reviewed above have been made by Wyller

[1989] and Alekseev and Kurdryashov [1992]. Wyller [1989], in particular, shows that

for the derivative nonlinear Schrodinger equation, the equation he is interested in,

Cavalcante and Tenenblat's [1985] algorithm produces nonlocal conservation laivs.

He conjectures that his conservation laws should be related to the ones obtained from

the corresponding linear system with spectral parameter by the classical method of

Riccati-type substitutions reviewed in Section 4.1. Indeed this is the case. It d l

be shown that for the whole class of equations describing a one-parameter family of

pseudo-spherical surfaces, Cavalcante and Tenenblat's [19S8] conservation laws are

related (in a very precise fashion) to the ones obtained by Riccati-tvpe substitutions

from the corresponding one-parameter family of linear problems. The proof of this

daim is based on the following technical result.

Lemma 2 Let E = O be a diflerential equation describing pseudo-spherical surfaces

with associated one-fonns wu, a = 1 , 2 , 3 , and consider the one-form (4.27) de-

termined by the cornpletely integrable Pfa f ian system (4.26). Under the changes of

uan'ables r = tan(p/2) and f = cot(p/2), the one - fom (4.27) becomes, respectively,

and

Proof: Recall that the following notation is being used. If s : (x, t ) H (x: t , zo(x, t ) )

is a local section such that z0 (2, t ) is a solution of the equation E = 0, the one-forms

LF are given by P = ( j " ( s ) ) * w Q , 0 = 1,2,3.

The substitutions of variables p c, r and p c, r introduced in the staternent

of the lemma appear in (or can be easily deduced from) the classical literature on

pseudo-spherical surfaces. In particular, the transformation

is in Eisenhart [1909]. Equation (4.28) is obtained as follows:

First, one substitutes the transformation (4.30) into the Pfaffian system (4.26).

One finds

(WJ + w*) - 2 E 1 + rZ(W3 - z2) = -2dr, (4.31)

while the closed one-form 8 considered in (4.27) now reads

Of course one can check by direct computation that Or is closed if I' satisfies (4.31)

and the forrns w", a = 1,2,3, satisfy the structure equations of a pseudo-spherical

surface whenever zo(x,t) is a solution of Equation (4.23). Kow one co~siders the

one-form O given by

8 = Br + d ln(1 + r2). (4.33)

-4 straightfoward compu tation using Equation (4.31) yields

which is (4.28). If instead of (4.30) one novv uses the change of variables

one obtains (4.29). Indeed, the Pfaffian system (4.26) now becomes

(W3 - w2) - 2îW1 + F 2 ( d + z2) = -2&, (4.35)

and one obtains (4.29) by considering the one-form (4.217)' using the change of vari-

ables p r, l? t o obtain a one-form Bi. analogous to (4.32): and taking

O

The one-forms 8 and E ) given by (4.28) and (4.29) are very similar to the ones

appearing in Sasaki's [1979] seminal paper on equations which are the integrability

condition of Iinear problems of AKNS t-vpe. One obtains his formulae (up to iinessen-

tial constants) if one rotates the coframe {wL,w2} by an angle p~ = ~ / 2 . Indeed, the

Pfaffian system (4.31) and the closed one-form (4.34) become, respectively,

while the Pfaffian system (4.35) and one-form (4.36) now read

= ( w J - ~ ~ ) + 2 F ~ 3 + F i ( d + w ' ) , and (4.39)

6 = i Z 2 + p o ( d + ~ 1 ) . (4.40)

Equation (4.35) is exactly formula (4.1) appearing in Sasaki's [1979] paper, and Equa-

tion (4.38) is -2 times Sasaki's [1979] formula (4.5), if one replaces Sasaki's rl by

-To. On the other hand, Equation (4.39) is precisely Sasaki's [1979] Pfaffian systern

(4.2) if one replaces Fo by -f and the one-form (4.40) is 2 times Sasaki's [1979]

formula (4.6), if one replaces Sasaki's r2 by 40.

Suppose now that the equation E = O is kinematicaIly integrable. Since Sasakits

one-form (- l/2)@ = (1/2)$ + ( l /2)ro(z3 - d) is closed whenever 20 (x, t ) is a

solution of (4.25) (it was obtained by rotating the one-form (4.34): mhich is closed

whenever zo(x, t ) is a solution of (4.25)) one finds, by &ting (4.38) explicitly in

terms of the functions faB y the following parameter-dependent conservation law for

Equation (4.25):

The change of variables i

yields the Riccati equation

and also the equation

to be satisfied whenever zofx, t ) is a solution of the equation

(4.42)

- = = O. They are

precisely (if f21 = 77) formulae (4.1) and (4.2) for conservation laws obtained from

the well-known systems of equations of Riccati type which appears in Theorem 9

of Section 4.1. In the same way, equations analogous to (4.3) and (4.4) are found

if one now considers Equations (4.39) and (4.40). Summarizing, these comments

mean that Chern and Tenenblat's [1986] construction of conservation laws: reviewed

at the beginning of this section, provides an exact geometric interpretation for (and

generalization of) the forma1 computations of (not necessarily local) conservation laws

of Section 4.1 so familiar from the inverse scattering literature.

4.2.1 Example: The Lund-Regge equation

The Lund-Regge equation \vas mentioned in the Introduction as an important kine-

matically integrable equation which was obtained via geometric considerat ions by

using the "immersion" approach to be discussed in Chapter 6. Conservation laws for

this equation wi1I be computed here, as they provide a non-trivial example of the use

of Equations (4.41) and (4.42) found above from the Chern and Tenenblat one-form

(4.27). I t should be remarked that a sequence of conservation laws for the Lund-

Regge equation has been found already by Lund hirnself (Lund [1978]). His analysis

holvever, was based on scattering techniques, so that the calculations presented below

do not duplicate his work.

The Lund-Regge equation reads

1 cos e O,, - - sin 28 + -

2 &At = 0,

sin3 8 and

This equation describes a one-parameter family of (complex) pseudo-spherical sur-

faces with associated one-forms

w1 = (2iX,cot 0)dx + -sin28 dt, c r i )

and therefore Equation (4.41) now reads

and (4.16)

(4.47)

in which the functions

.cos 28 R = D, log(iX, cot 8 - 28,) - z-

sin2 O (4.50)

do not depend on the spectral parameter q. -4ssume that $l can be espanded in a

power series of the form

Equation (4.48) implies that is determined by the recunion relation

whenever (B(x, t ) , X(x, t ) ) is a solution

yields, via Equation (4.42), a sequence

of the power series in

of Equations (4.43) and (4.44), urhich in turn

of conserved densities given by the coefficients

cos 28 00

-2iq + i - ~ ~ + 1 $ l " ) ~ - ~ . sin2 0 .=,

The first tivo conserved densities (corresponding to the coefficients of q0 and g- ' ) are

esactly Lund's consex-ved densities Co and Cl given by Equations (3.50) and (3.5 1) of

Lund [1978], if one sets the constant A' appearing in those equations equai to zero.

4.3 Coverings and conservation laws

The analysis carried out in Section 4.2 uncovers two characteristics of the Chern and

Tenenblat [1986] procedure. First (See Reyes [1998], and the enamples of Subsection

4.1.1) the conservation laws one finds from (4.42) by espanding the functions dl and

fOa in forma1 power series of q, are not always local, and therefore the correspond-

ing one-forms, although closed whenever q(x, t ) is a solution of the kinematically

integrable equation E = O, do not necessarily determine elements of the cohomology

group (1.13). Second (See Reyes [1998], and the comments after the proof of Theorem

9), even if these conservation laws are local, the function dl given by Equation (4.41)

will not be smooth as a function on Sm: It will depend on an infinite number of

variables zo,zm,n. Thus, as it has been anticipated already, if one is interested in the

forma1 geometric aspects of the theory, one needs to entend the setting being used

until nom. A very convenient geometrical framework encompassing local and nonlo-

cal aspects of the geornetry of differential equations is Krasil'shchik and Vinogradov's

[1984, 19891 theory of coverings, which will be now introduced.

The exposition which follows is based on Khor'kova [1988], and Krasil'shchik and

Vinogradov 11984, 19891. It has also profited from a recent survey of the theory a p

pearing in Krasil'shchik and Verbovetsky's [1998] lecture notes on homological meth-

ods in the geometric theory of differential equations.

Consider the differential equation

-, = L X , t , 20, . - - , ~ g ~ r n p ) = O

with equation manifold Sm. Let 7iz be the projection map from Sm to the space

of independent variables, and let

be a fibre bundle over Sm. Following Krasil'shchik and Verbovetsky [1998], one defines

the notion of coverings of Equation (4.53) as follows.

Definition 6 A covering structure is determined b y iïr (or, the bundle 9 is a covering

of S") if and only i f

- 1. There ezists a Jat connection C on the bundle i r z o n : S -t M , and

2. T h connection C is in agreement with the Cartan connection C (defined in

(1.3)) on Sm, that ts, for any vector field X on M,

in which the vector field X on is the unique horizontal iij? of X induced by

the connection C.

- Two coverings ai : Si + Sm and 7r2 : S2 + Sa with flat connections CI and C2

- respectively, are said to be equivalent if there exists a diffeomorphism rp : Si + S2 such that

- TI = 7~~ O q, and y.(X(l)) = X(2)

for al1 vector fields X on M , in which the vector field X(,, on SI, and X(2) on 3 2 : are

the unique horizontal lifts of X induced by the connections CI and C2 respectivel-

' Suppose now that the bundle ?r : + Sm is a covering of Sm, that this space is

equipped with canonical coordinates (z, t , zo, . . . , ~ o , , - ~ n , . . .), and that (wl, . . . , w N ) ,

1 5 N < oc, are vertical coordinates on 3. Let Dr and Dt be the total derivatives

on S" defined by Equation (1.4), or by (1.9) and (1.10) if (4.53) is of evolutionary

type. Then, the covering 3 is determined locaiiy by the triple

-- 1. D,, DL are diiferential operators on 3 (which will be called total derivatives on

3) of the form

- D, = D, + X , and - Dt = D,+&

n-here Xi, i = 1,2, are vertical vector fields on given by

and X{ are functions on 3, and rnoreover

2. and satisfy the integrability condition

In Equation (4.57), D,(Xi) is the vertical vector field

and D t ( X i ) is defined analogously. The number N appearing above is called the -

dimension of the covering n : S -t Sm. Note that N may well be equal to infinity.

For example, one may wish to study coverings which are themselves inverse limits of

finite dimensional coverings. The fiber coordinats wi, 1 5 i 5 N, are called nonlocal

variables (with respect to Sm) in the covering S.

Also from a local point of view, the notion of equivalence of coverings given above

corresponds to changes of coordinates on these spaces. Consider two coverings 3, and - S? of Sm with vertical coordinates (w:, . . . , w?): 1 5 NI 5 I, and (w:: . . . , wp).

1 5 iV2 5 m equipped with total derivatives DI, and p, i = 1,2. Then, the coverings - SI and SÎ are equivalent if iVl = Nz = N and there exists a map p transforming the

vertical coordinates of into the vertical coordinates of SÎ, namely,

such that

-2 ~ L D : = D,: and

or! in other words, using the notations of (4.54): (4.55) and (4.56);

The Rat connection C on the covering S determines a splitting of the space of

vector fields on S into horizontal and vertical vector fields. In the canonical local

coordinates considered above, horizontal vector fields are simply linear combinations

of the total derivatives D, and Et. The dual splitting of differential one-forms on

gives rise to the notion of horizontal exterior derivative, as in (1.6). The horizontal

exterior derivative operator dH on R(S) , the space of diEerentia1 forms on S, is defined

on one-forms by a "bar" version of (1.6), namely, by the formulae

in which XP, i = 1,2, are the components of the vector fields Xi defined in (4.56)

determining the vertical parts (with respect to Sm) of the total derivatives D, and

a -

Since the total derivatives D, and Et satisfy the integrability condition (4.57): the

operator dH satisfies d H o z H = O. -4 non20cai conseruation law of type 7i for Equation -

(4.53) is a oneform w = X d x + Tdt in R(S) such that

- -- -- d H w = ( - D t x + f = T ) d z A d t = O ,

and the conservation law is called trivial, if it is dH-exact.

The following e~amples show that coverings do arise naturdly in the geonietric

theory of differential equations. -4 classical construction of "new" coverings from

"old" : is also presented.

1. The most elementary, yet non-trivial: esample of a covering of a differential

equation is the following. Let

be a nonlinear equation, not necessarily describing pseudo-spherical surfaces.

and assume that the oneform K = crdx + Bdt is a non-trivial IocaI consema-

tion law (ie., a representative of a non-zero element of the first cohomology

group HL.', cfr. the discussion following (1.13) in Chapter 1). This one-form

determines a onedimensional covering

(Sc, (0:' D;): x,) (4.60)

of Equation (4.59) (-4 more general result, not to be used here, appears in

Krasil'shchik and Vinogradov [1989]). Indeed, it is enough to define SK locally

b y

SIC= { ( x : ~ , z ~ , . . . , ~ o , ~ m p , - -..IL'))-

where (x, t , 20' zl, ...) are coordinates on Sm, and set

Thus, in other words, one introduces a potential w for the conservation law

a d x + pdt. Indeed, w satisfies &W = a d x + Bdt. It is trivial to check that :

since K is a local conservation lan- of Equation (4.59), the integrability condition

(4.57) for DE and Dr holds.

2. The nonlocal transformations of Mikhailov, Shabat and Sokolov [EN 11.

In their work on the classification of forrnalty integrable evoiution equations,

hlikhailoi-, Shabat, and Sokolov [1991] point out that useful transformations

can be defined with the aid of a non-trivial local conservation law (again, a

non-trivial element of the cohomology group Hl.') as follows.

If K = a(to, zl)dx + P(zo, 21, . . .)dt is a non-trivial conservation law of the

evolution equation Z O , ~ = F ( x , t , zo, ..., a), one defines new independent and

dependent variables by

This change of variables can be readily understood in the context of the the-

ory of coverings. It is simply an invertible transformation of the covering SK

defined above which prolongs the function f : E + S" given by f : (x, t , zo) t,

(w t , $+O)).

3. Suppose that SI and S2 are two coverings of Sm. One constructs the fibered

product of S1 and S2, S1 @ S2, as follows. Coordinates along the fibers of SI @ S2 1 Ni 1 are simply w,, . . . , w, ,w2, . . . in which w:, . . . , w? and wi , . . . ,urp are

vertical coordinates on Sr and S2 respectively, while the total derivatives are:

using an obvious notation adapted frorn (4.56),

Further facts of the theory of coverings will be prsented in the next chapter, in

which generalized and nonlocal symmetries of kinematically integrable equations are

discussed. Non; one would like to apply this theory to understand the results obtained

in Sections 4.1 and 4.2 in terms of coverings. First of all? one would like to consider

functions depending on nested integrals (in the sense of Esample 2, Subsection 4.1.1)

of functions on S" as functions defined on adequate coverings of the equation manifold

S". The following proposition, and its obvious generalizations, is ail one needs for

t his.

Proposition 6 Consider the ring 3 of smooth functions o n the equation manifold - -

Sm of Equation (4.53), and let f E F. There ezists a covering (Si, {Dr, D l ) , îr) of

Sm, and a function wf o n SI such that

so that one can always adjoin the indefinite integrals off E 3 to Sm as new nonlocal

variables.

Proof: Let S 3 be the set of infinite forma1 sums of elements of 3? and let /3 be the

operator

p : 3 + s 3

given by

The covering SI is then given by the product

equipped with local coordinates (x, t , 4 , . . - , zopmr~ , . . . , wf) , and with total deriva-

tives (4.54) and (4.55) defined as

One easily sees that the integrability condition (4.57) for D, and D, is satisfied.

O

Clearly. this proposition allows one to interpret functions depending on nested

integrals, as functions defined on coverings of the form Si, in which 7 is a function on

some covering of Sm. Thus, for exarnple, if Equation (4.53) is Burgers' equation,

the function 41) considered in (4.19) can be now understood as a function on the

covering S7 of the equation manifold Sm, in which

is, in turn, a function defined on the covering ~r : S" + Sm given by (4.60) for

K = zodx + (zl + ( 1 / 2 ) r i ) d t .

These comments imply that the conservation iaws for kinernatically integrable

equations depending on a finite number of nested integrations which were obtained in

Sections 4.1 and 4.2, can be considered as genuine nonlocal conservation laws. Indeed?

generalizing what was just said for Burgers' equation and its conserved density

given by (4.19), it is enough to "iterate" exarnples (4.60) and (4.64) a finite number

of times. Since these results have a very formal flavour, they will not be pursued

here. It is worth pointing out, however, that one can prove, for example, tha t for any - equation = = 0, not necessarily kinematically integrable, there esists a covering of

its equation manifold Sm such that every local conservation law of E = O possesses

a potential defined on 5 (Khor'kova [1988]), and that i t appears that the same is

true for nonlocal conservation laws, although a forma1 result d o n g these lines is.

to the best of this author's knowledge, an open problem. These studies are a t the

base of the so-calied "reconstruction results for nonlocal symmetries" summarized by

Krasil'shchik and Verbovetsky in their lecture notes (Krasil'shchik and Verbovetsky

[1998]) on homoIogica1 methods in the theory of differential equations.

Chapter 5

Generalized symmetries and

kinemat ic integrability

This chapter is about the geometry underlying another fundamental theme of the

theory of equations describing pseudo-spherical surfaces name13 the existence of gen-

eralized symmetries for equations in tbis class.

There are three sections and one appendix. -4 geometric characterization of (gen-

eralized) symmetries is made in Section 5.1. This characterization can be used also

for symmetries of nonlocal type: the theory of nonlocal symrnetries is rigorously intro-

duced in Section 5.2 in the context of coverings, and an existence theorem is proven.

-An interesting application of this theorem to the Burgers' equation is also presented.

Section 5.3 contains a construction of nonlocal conservation laws of kinematically

integrable equations: they are obtained by "deforrnations" (in a sense to be made

precise in Section 5.1) of the conservation latvs constmcted in Chapter 4. Finally, an

appendix on Harniltonian structures built out of conservation laws and symmet ries is

also included.

It \ d l be assumed henceforth that the equation 5 = O describing pseudo-spherical

surfaces is of evolutionary type.

5.1 Generalized

87

symmet ries and infinitesimal de-

format ions

Consider the evolution equation

du - = F(x , ~ , u , u ~ , . .. , u z m ) at (5.1)

with associated equation manifold Sm c, .JWE equipped with local coordinates

(z? t7ro , . . . , t k , . . .): and total derivatives D, and Dt given by Equations (1.9) and

(1.10). The following two notions are central to the analysis to be carried out here:

If f (x: t , 20: ..., zk) is a smooth function on S", the forma1 Zznean'zation of 1 at zo

is the differential operator f, whose action on functions g on S" is given by

namely,

One says that

the dependent

in which T is a

the order of f. is k. Also, an infinitesimal deformation of order m of

variable zo is a change of coordinates on J" E of the form

and

small parameter and G is a smooth function on Sm such that the order

of the operator G. is m. Generalized symmetries of Equation (5.1) are introduced

t hus:

Definition 7 A generalized symmetry of order m of Equation (5.1) is an infinitesi-

m a l defornation (5.2), (5.3), (5.4), of the dependent variable zo of order m such that

for any local section s : (x, t ) I+ (x, t , zO(x, t ) ) of E for which joo(s) is contained in

s ~ , the local section ( x , t ) rt (x' , t', zh, D,zh, D~z;,.. .) of JmE satisfies (5.1) up to

terms of order r2.

In other mords, G is a generalized symmetry of (5.1) if and only if the equation

DtG = F.G (5.5)

holds on solutions of (5.1): or still, if and only if Z o , = G and the mixed partial

derivatives qtT and ~ 0 , ~ ~ are equal on solutions of (5.1). This last characterization

can be also expressed in the following form: A function G is a generalized symmetry

of Equation (5.1) if and only if the differentiai operators D,: Dt2 and L,, in which

commute on solutions of (5.1).

-4ssume now that Equation (5.1) descnbes pseudo-spherical surfaces wit h associ-

ated functions fa8 (2, t, zo, ..., z ~ - ~ ) . In this case, the generalized symmetries of (5.1)

may be characterized in a very appealing way.

Fis an infinitesimal deformation of ZQ, z0 * a+rG, of arbitrary order, and denote

the linearizations of the functions at zo acting on G, fnB.(G), by gao- Lemma 1

of Section 2.1 implies ttiat these functions are given by the formulae

and

as a straightfonvard computation shows. Now consider the "formal linearization" of

the one-forms wQ = faldx + fa2dt at zo acting on G,

and also the deforrned one-forrns

~ h e following straightfomard Lemma belongs to the classical theory of infinitesimal

deformations of surfaces, although the emphasis here is different. Classically (See for

esample Eisenhart [1909] Chapter XI), the interest was in infinitesimal deformations

of surfaces S immersed in R3 of the form (S, ds2) ct (S,, ds:), in which r is some

small parameter, such that So = S and ds2 = dsf up to terms of order greater than: or

equal to, r2. Here, one is interested in infinitesimal deformations (S, ds2) c, (S, ds;)

of metrics ds2 on S (considered intrinsically) such that the Gaussian cunature of

(S: ds:) is preserved up to terms of order r2 or higher. C

Lemma 3 Suppose that three one-forms P, a! = 1,2,3 , s a t i s h the stmcture equo-

tions o/ a pseudo-spherical surface of metric ( z ' )~ + ( w ~ ) ~ and connection fonn g3.

The deformed o n e - f o m i P + r X a satisfy the structure equations of a pseudo-spherical

surface up to t e m of order r2 i f and only if the formai linearizatiorts h, satisfy the

stmcture equations

The analytical content of this Lemma is the following:

Proposition 7 Suppose that Equation (5.1), zoV1 = F , describes pseudo-spherical

surfaces with associated one-forms wQ, cr = 1 ,2 ,3 . T h e deformed one-fonns wa+rAa:

in .which the o n e - f o n s A, are given by (5.8), satisfy the structure equations of a

pseudo-spherical surface -up t o tenns of order r2- whenever zo (x , t ) zs a solution of

Equation (5.1), i f and only i f G is a generalized symmetry of (5.1) .

Proof: A trivial generalization of the basic Lemma used in the classification of

strictly kinematically integrable equations (See Lemma 1 of Chapter 2) shows that the

equation zo,r = F ( x , t , zo, ..., zk) describes pseudo-spherical surfaces with associated

functions fap if and only if the function F and the functions fas satisfy the equations

faiyZi = O, i 2 1, and fo2,=>. = 0, O = 1,2,3(5.12)

and moreover

Evaluate the linearization of Equations (5.13), (5.14) and (5.15) a t G. One obtains,

where F. is the formal linearization of F. Now, if one takes P = ( jm( .s) ) 'wa, and - ila = (ja(s))*A,, in which s is a local section s : (x, t ) * (x. t , zo(x, t ) ) such that

j " ( s ) is contained in the equation manifold Sm of Equation (5. l ) , then one can use

Lemma 3. Equations (5.9), (5.10) and (5.11) will hold if and only if

nyhenever zo (2, t) is a solution of Equation (5.1).

Substituting Equations (5.16), (5.17) and (5.18) into the last three equations, one

finds that Equations (5.9), (5.10) and (5.11) hold if and only if

whenever ~ ( x , t ) is a solution of Equation (5.1). Since f:,,=,' t f&,, + f&,, # O,

Equations (5.9), (5.10) and (5.11) are equivalent to

whenever zo(x7 t) is a soiution of (5. l) , and the result follows. 0

This proposition is interesting because it relates the (generalized) symmetries of

equations describing pseudo-spherical surfaces to the geometry of the pseudo-spherical

surfaces themselws. It can be also understood at the level of linear problems as

foIlows:

-4 smooth function G on Sm is a (generalized) symmetry of Equation (5.1) if

and only if the formal linearizations (5.8) of the one-forms wa satisfy the compatible

4 2 : 1)-linear problem

One can check that this linear problem is compatible mhenever zo(x7 t ) is a solution

of Equation (5.1) precisely because this equation describes pseudo-spherical surfaces

with associated one-forms wa.

Proposition 7 wiIl be used in the following section to prove the existence of (gen-

eralized, nonlocal) symmetries of kinematically integrable equations.

5 -2 Symmetries and coverings

The result just proven (Proposition 7) does not depend on the face that G is a local

function. Indeed, only the fact that G satisfies Equation (5.5) whenever zo(x7 t ) is

a solution of Equation (5.1) was used. This elementary observation is important

because it allows one to show that if Equation (5.1) is kinematically integrable, it

possesses a large number of generalized and/or "nonlocal" symmetries. -4s in the last

Chapter, the adequate geometrical setting for this discussion is the theory of coverings.

The relevant constructions are as follon-s (the notations and notions introduced in

Subsection 4.3.2 wiil be used throughout).

-- Definition 8 Let (3, (D,, Dt) , rr) be an N-dimensional c o ~ e n n g of Equation (5.1)

equipped with coordinates (x, t, r o , 21, . . . , wg), 1 5 /3 5 N . A nonlocal symmetry of

type n of (5.1) is a generalized vector field Dr tangent to 3 of the f o n n

in which G and

equutions hold:

ID, 1 5 P 5 N are smooth functions on such t h a t the follouBng

- k a F . DtG = 1-0, (G),

i=o ' z i

und

The first condition says that the "t-fl OW?' generated by the evoiution equation (5.1)

in cornmutes with the "7-flow" generated by the vector field D, in 3: namelx ./

20,tr = Z0,7t.

The second and third equations Say that the flows generated by and commute

with the 7-flow in the vertical directions, namely,

Kote that Equation (5.26) depends only on G and F . The vector field

(compare with L, defined by (5.6)) can be interpreted as a vector field on along

the equation manifold Sm. This vector field, or just G, the "nonlocal function" (from

the point of view of Sm) which generates the operator (5.29), is called the shadow of

the nonlocal symmetq- Dr. In general, vector fields on 3 along SOE satisfying (5.26)

are called n -shadows. --

Xow, let (3, {Dr, D l } , ii) be a covering of Equation (5.1): and assume that this

equat ion describes pseudo-sp herical surfaces. It has been seen already t hat t hen the

Pfaffian systern defined in Equation (4.26), narnely

in which P = (jm(s))'wQ, CY = 1,2,3 and s : (x:t) c, (x, t _ zo (x , t ) ) , is completely

integrable for p whenever jS (s ) is a local solution of Equation (5.1). One would like

to consider this completely integrable Pfaffian system as the pull-back of a system

defined on 3. One proceeds thus:

- Lemma 4 Consider the covering a : S -t Sm of Equation (5.1) equipped with a fiat

connection C. An hoionornzc section jm(s) of Sm c m be lzfted to an horizontal local

section on 3, that is, to a local section s for which the tangent space at each point i n

its image is contained in the horizontal distribution o f S determined by the connection

Proof: Let s : (x, t ) (x, t , r o ( x , t)) be a local section of E such that j S ( s ) is

contained in Sm. The graph Q of jm(s) is a subrnanifold of Sm, and one can therefore

lift its tangent bundle TQ to by means of the Bat connection C. This lift determines -

a distribution TQ contained in the horizontal distribution of 3 and defined on every

point p E 3 with n(p) E Q. But one of the characteristics of a covering is that C

is integrable (see Equation (4.57)). Thus, there exists locally a submanifold of

about every point p E 3 Mth x ( p ) E Q, such that its tangent bundle TQ is m. This

submanifold projects in a one-to-one fashion to Q, and therefore it determines a lift

5 of j" ( s ) . U

Consider now the following "lifted" Pfaffian system on 3 (recall that the esterior

derivative dx was defined in (4.58))

and let s : (x, t ) i-, (x, t , zo(x, t ) ) be a local section of E such that the holonomic

section j w ( s ) is contained in Sm. By the last lemma, this section can be lifted to

a local section 3 of the covering 3. One can use this lifted section to pull Equation

(5.31) back to the space of independent variables JI. Since by construction the one-

forms w" do not depend on nonIocd variables, one obtains s'us = 3, as before.

Since the total derivatives and satis@ the compatibility condition [z7 = O

(Equation (4.57)), the operator dH pulls-back to the exterior derivative operator on

the space with coordinates (x, t , p) . Thus, the completely integrable Pfaffian system

(5.30) is the pull-back of the Pfaffian system (5.31) defined on 3. Kow let G be a (for now arbitra-) function on 5: consider the differential operator

- - L, given by (5.29) determined by G, and set o = L,p. The Pfaffian system (5.31)

can be "linearized a t t.o in the direction G that is, one can compute its r-derivative.

One obtains a new Pfafnan system on 3, -

i l3 - d H o + COS p)wl + (sin p)Al - o(sin p)w2 + (COS p)A2 = 0- (5.32)

One would like to study the complete integrability of this system whenever zO(x , t )

is a solution of Equation (3.1), and p is a corresponding solution of the cornpletely

integrable Pfaffian system (5.30). First of all, one needs to compute its &-esterior

derivat ive:

Lemma 5 The horizontal exterior derivative of (5.32) is given b y

wheneuer zo(x, t ) is a solution of Equation (5.1) and p is the corresponding solution

of system (5.30).

Proof: -4 direct computation. One only needs to use the structure equations (1 -23)

of a pseudo-spherical surface: and Equation (5.31). 0

The analytical interpretation of this Lemma is the following.

Proposition 8 Assume that Equation (5.1) i s kinematically integrable. With the

foregoing notations, the Pfaf ian sgstern (5.32) is completely integrable for a whenever

20 (x, t ) is a solution of Equation (5.1) and p 1, 0 is a corresponding solution of the

cornpletely integrable Pfafian system (5.30), if and only if G is a shadow of a nonlocal

symmetry of type n of Eqvation (5.1).

Proof: The if part follows from Lemma 3 and Proposition 7. On the other hand, - -

it follows from the proof of Proposition 7 (with D,, Dt replaced by D,, Dt) that

Equation (5.33) is equivalent to

where

If G were not a generalized symmetry of Equation (1.8), it wvould t,hen follows that

but this equation says that p depends at most on 20, a contradiction.

The Pfaffian system (0.32) is, explicitlv.

Introduce some notation:

A straightforward computation shows t hat Equations (5.34) and (5 -35) are e q u i d e n t

31 a t ters being so, Proposition 5 and the preceding computations yield the following

result:

-- Proposition 9 Let (3, {D,, D,), i;) be a couering of Equation (5. l ) , and assvme that

this equation is kinematically integrable. Suppose that G is a shadow of a nonlocal

s g m m e t q of type n of Equation (5.1). Then, the PfaBian sys t en (5.32) is completely

integrable for o whenever zo(x, t ) is a solution of Equation (5.1) and p is a correspond-

ing solution of the cornpletely P fa f ian systern (5.30). Moreover, e v e q a satisfying

this system solves equations (5.38) ~ n d (5.39). Conuersely, i f a is a solution of (5.39)

whenever zo(x, t ) zs a solution of (5.1) and p i s a corresponding solution of (5.30):

and G is defined by (5.35), then the Pfaf ian system (5.32) is completely integrable

whenever zo (x, t ) is a solution of Equation (5.1) and p zs a corresponding solution of

(5.30). Moreover, G is a shadow of a nonlocal symmetry of type 7; of Equation (5.1).

Xow, from the point of view of 3, Equation (5.39) is simply a Iinear equation

whenever ~ ( x , t ) is a solution of Equation (3.1). Thus, one can, in principle, find a

a5 a forma1 power series in 77. In particular, standard facts about esistence of solutions

of linear equations yield the following result on the existence of shadows of nonlocal

syrnmetries for kinematically integrable equations:

Theorem 10 Let zoVt = F be a kznematically integrable equation with equation man-

ifold Sm, and let (5, (z, n), ir) be a couering of Sm. If the associated functions

fa8(q) are analytic functions of the parameter 7 wl~enever ~ ( x , t ) is a solution of

z g , ~ = F , then so are p, the solution a of Equation (5.39) and the solution G of

(5.38). Thus, in particular, a solution a of (5.39) giues rise to an infinite number of

shadows of nonlocal syrnmetries of the equation = F by means of (5.38).

This result is analogous to Cavalcante and Tenenblat's [1988] result on conservation

laws for kinematically integrable equations. Note, however, that analyticity is not

strict ly necessary (Cavalcante and Tenenblat [19S8] make the same comment with

respect to their theorem on conservation Zaivs). What really is of importance, is the

fact that one can expand the functions fPa, p, and a as fonnal power series in q.

Then, one obtains a formal power expansion for G from (0.38), and therefore an

infinite number of shadows of nonlocal symmetries of Equation (5.1).

Xote aIso that in pnnciple, the solutions p of (5.31) and a of (0.39) may be

themselves nonlocal n i th respect to 3. One then obviously wonders if there esists

a covering SL on which the coefficients of the formal series expansion of G are well

defined local (with respect to 3') functions. I t appears that the answer is in the

affirmative. However, no forma1 generd result along these lines will be reported here.

Another related question is the following. Can one obtain genuine nonlocal symme-

tries of zoIt = F from the shadows determined by G ? This problem is called "the

reconstruction problern" by Krasil'shchik and Vinogradov [1989]. Important "recon-

struction theorems" for nonlocal symrnetries have been proven by X n a Khor'kova

[1958]. Kiso [1989], Krasil'shchik and Vinogradov [1989], and Krasi1:shchik and Ver-

bovetsky [1998]. One conjectures t hat the following holds:

Conjecture: Let zoIt = F be a kinematically integrable equation w-ith equation

manifold Sm. There exists a covering n* : S* + Sm of Soo and an infinite number of

nonlocal symmetries of type .rr' of the equation ~ 0 , ~ = F.

5 -2.1 Example: Nonlocal symmetries of Burgers' equation

The constructions leading to Theorem 10 will be now illustrated with the simple case

of Burgers7 equation. Before performing the necessary calculat ions: however: it wiIl

be shown that one can rewrite the linear system (5.38) and (5.39) in a way more

amiable for cornputations.

One considers again the change of coordinates (4.30) used in Section 4.2 on con-

servation laws, namely, l? = tan(p/2) and replaces into (5.38) and (5.39). Then, one

easily obtains, whenever zo (x, t ) is a solution of Equation (f i l ) , the formulae

in nhich l? satisfies Equation (4.31). The functions b and A~ defined in (5.36) and

(5.37) become

and finaily, it follows easily that the linear system (5.38) and (0.39) nonT reads

whenever zo (z, t ) is a solution of Equation (5.1).

Xow one can specialize the foregoing discussion to the Burgers' equation case. Re-

cal1 that this equation is kinematically integrable, and that associated one-forms are

determined by the functions faa given in Subsection 4.1.1. Take = S". Equations

(4.31) for l?, and (5.41) and (5.45) for G and C become, whenever zo(z, t) is a solution

of Burgers' equation,

and

respectively. The function

pomers of 77, ofie finds that

l? can be easily computed from (5.46). Espanding r in

is determined by the recursion relation

zOr0 = 2ro,=, and k-1

whenever to (x, t ) is a solution of Burgers' equation. Now write

Since Equation (5.49) is linear in C, one of course obtains linear equations for each

coefficient function Sn. In particular, for So one has the equation

whenever zo(x, t ) is a solution of Burgers' equation. It is straightforward to check

that the Iast equation can be written in the f o m

This equation, in turn, can be re-written in a very suggestive way as follows:

Kow: expand G, as given by Equation (5.48), in powers of 71,

.At zero order in q, Equation (5.50) i m p k s that

Equations (5.54) and (5.56) contains al1 the known integrability properties of Burgers'

equation. Indeed, if, following Krasi17shchik and Vinogradov [1989]: one now asks

for solutions zo(x, t ) of Burgers' equation which are "invariant" under the (still not

completely deterrnined) nonlocal symmetry Go, these solutions should also satisfy

or in other words,

But if this is so: Equation (5.54) implies that So must be a solution of the heat

equation! Thus, Equation (0.57) is a transformation between solutions of Burgers'

equation and the heat equation. It is, of course, the classical Cole-Hopf transforma-

tion. It follows then that if So satisfies So,r = s,,, Go given by (5.56) is a nonlocal

symmetry of Burgers' equation, and solutions zo of this equation, may be transformed

into solutions of the heat equation by means of (5.57).

This is an interesting and intriguing observation, even more so if one recalls that

only the zero order term of the expansion (5.52) has been computed. A better un-

derstanding of this phenomenon, however, must be left for the future.

5.3 S ymmetries and nonlocal conservation laws

The geometric characterization of (generalized? nonlocal) symmetries of kinemati-

cally integrable equations of evolutionary tevpe studied in Section 5-1: and the anal-

ysis carried out in Section 5.2, allows one to construct infinite families of nonlocal

conservation laws for these equations. One proceeds as follows.

Assume thai Equation (5.1): namely zoVt = F(x, t , zoo . . . , zk) is kinematically in-

tegrable with equation manifold Sm and associated one-forms wa = faldz + fa2dt. Let G be an arbitrary (generalized, nonlocal) symmetry of Equation (5.1) with asso-

ciated evolution equation q,, = G. One can interpret the total derivative operator - D, defined by (5.29), as an operator measuring the "r-evolution" of tensors on S*

which are defined along the space A I of independent variables.

In particular, if the operator D, is applied to the Chern-Tenenblat one-form

(4.27): namely to 0 = w 1 cos p - w2 sin p, in which p is a solution of the Pfaffian

system

w 3 + d H p - w ' s i n P + w 2 c o s P = ~

whenever %(s, t ) is a solution of Equation (5.1), one obtains the following:

Proposition 10 T h e one-form = D,8 is giuen by

Oc = -o(wl sin p + w2 COS p) + Ai cos p - h2 sin p. (5.58)

Moreover, whenever % (2, t ) is a solution of Equation (5. l ) , p(x, t ) is a corresponding

solution of the completely integrable Pfa f ian system (4.261, and a ( x , t ) is a solution

of the Pfa f ian systern (5.32) corresponding tu the solutions zo ( x , t ) and p(x, t ) the

pull-back of the one-form Oc b y the solution .q ( x , t ) i s closed.

Proof: The proof is a direct computation using the structure equations (1.23) for

pseudo-spherical surfaces, the linearized structure equations (5.9) : (5.10) and (5.11):

and Equations (4.26) and (5.32) for p and a respectively. 0

Thus, if G is a (generalized, nonlocal) symmetry of Equation (5.1) and this equation

is kinematically integrable, the one-form Oc given by (3.58) generates an infinite

number of conservation laws of this equation. These conservation laws are in principle

nonlocal, as the following cornputation shows:

Consider the representative (4.35) of the Chern and Tenenblat one-form,

The deformed one-form Oc becomes

- in which C = D,ro is a solution (whenever zo(x, t ) is a solution of Equation (5.1)) of

the Pfaffian system

obtained by deforming the Pfaffian system (4.37) along ET. One can of course check

directly that QG is closed on solutions of (5.1) if To is a solution of (4.37) and E is a

solution of (5.60) whenever zO(x, t ) is a solution of Equation (5.1).

Define the functions of 77

The function T(q) is, of course, the conserved density corresponding to the conserva-

tion law dHO = O. One can find C by integrating the x-part of (5.60). -4n elementary

cornputation yields the formula

In'terms of flows; this formula is saying that C, the infinitesimal variation of To in

the direction a/&, is nonlocal with respect to x. The exact dependence of C on G

can be found by recalling that Lemma I of Section 2.1 implies that the linearizations

gii are multiples of G, see Equations (5.7). Expansion of the functions fal,, and To

in powers of 77 yields an expression of the following form for U ( q ) :

in which J is a subset of 2. Thus, the formula (5.63) for C becomes

E.spansion of T in powers of the parameter q determines C as a power series in 77,

and then the one-form ec given by (5.59) determines a one-parameter family of

conservation laws of Equation (5.1) which are, of course, nonlocal in principle.

5.3.1 Exarnple: The KdV hierarchy

The above schemme will be used here to indicate how to find nonlocal conservation

laws for al1 the equations in the KdV hierarchy studied in Section 2.2. -4n interesting

feature of this example is that, as it has been known since Olver's [19TiI work on

recursion operators, one has a t one's disposai an infinite number of symmetries G for

each equation in the hierarchy, namely, the higher order KdV equations themselves.

Recall from Section 2.2 (Equations (2.87), (2.88): and (2.89)): that the functions

for the KdV hierarchy are given by fll = 1 + ZO, f21 = 271: and f3t = 1 - ZO. The conservation laws contructed above for the n th equation of the hierarchy! say?

are obtained as follows.

First, the function ro is determined, whenever zO(x,t) is a solution of the nth

KdV equation, by the Pfaffian system

which one obtains frorn (4.37). Thus, in particular, ro satisfies the Riccati equation

whenever ~ ( x , t ) is a solution of the nth-KdV equation. Espand To in inverse pomers

and substitute into (5.66). One finds that whenever zo(x, t ) is a solution of the nth-

KdV equation, the coefficients rp) of r0 are determined by the recursion relation

1 = -- 2 y

and (5.61)

On the other hand, the conserved density T(77) given by (5.61) becomes:

and moreover, for an arbitrary (generalized, nonlocal) symmetry G of the nth-KdV

equation, the function U ( 7 ) given by (5.62) now reads

so that its power espansion (5.64) becomes

In order to find C as a forma1 power series in powers of 77: one only needs to expand

appearing in the formula (5.65) for C , as forma1 power series in 77. This espansion

can be done with the help of the classical Faa de Bruno polynomials. Recall that if

is a parameter, and f , g are functions of u such that

t hen

in which the functions h, are given by

and the sum is taken over the set II, of al1 the partitions a = {al,. . . , a,} of n, that is.

on the sets of numbers a, satisfying al+2crz+. . .+na, = n, and k = al +az+ - - tct,.

In the case of interest here, one notes that

so that one needs to take

and

I t follows that f (g(l/q)) is given by

i r i which the coefficients h, are given by (5.73) with functions gn appearing in that

formula found from (5.74) and f k = (llk!). Thus, the function E(q) is given by

and one finds an analogous expression for F(q). Thus, one can find explicit expressions

for the conservation laws of the nth-order KdV equation induced by al1 the equations

of the hierarchy.

Appendix: Nonlocal Hamiltonians for evolution equat ions

The modern theory of Hamiltonian methods for partial differential equations is a t

least three decades old: it seems reasonable to consider Gardner [1971] and Zakharov

and Faddeev's [1971] papers on the Hamiltonian structure of the Kd\- equation as

the foundational Iandmarks. Hoivever: it is, still now, a many Iayered theory. Fi-

ni te dimensional canonical met hods extend to partial differential equations onIy in

i l restricted fashion: It does not appear to be always possible to interpret (a system

of) equations as Hamilton equations 0x1 a symplectic manifold. although work in this

direction lias certainly proved worthwliile: one only needs to think. for example. of

.\dler's [19T9] work on the symplectic interpretation of KdX' t>-pe eqiiations (See also

Faddeev and Takhtajan [1957], and Adams: M.R.. J-Harnad. and J-Hurtubise ( l ! N O ] ) .

?;or it scems possible to base one's studies on Lagrangian field-theoretical consider-

atioris: For esample, one may not espect to find Lagrangians for one-dimensional

cl-olution equations (unless one allows, for instance, integrating factors or potential

formuiations): their existence is forbidden by the Helmholtz conditions characterizing

the image of the Euler-Lagrange operator (Theorem 5.92 and Esercise 5-46 of Olver

[1994]).

This short appendis has been inspired by two interesting papers by Hojnian [19961.

and Gornberoff and Hojman [1997]. in which they point out a natural way of providing

evolution equations with Hamiltonian structures. Their approach is phrased in terrns

of Poisson brackets, and therefore it is naturczl to attempt to formalize it in the szme

terms. Olver's [1994] point of view (Hamiltonian meclianics r ia Poisson geornfq-)

will be taken throughout.

Let ~ 0 , ~ = F be an autonomous et-olution equation, not necessarily describixig

pseudo-spherical surfaces, and let SM be its equation manifold. The Hamiltonian

interpretation of this equation (if it e-uists) has three main ingredients: a forma1

notion of Hamiltonian functionals, t h e construction of an appropriate differentiation

on them, and the forma1 notion of a Hamiltonian operator. These two aspects are

considered in the following paragraphs.

Let 3(SDO) be the algebra of smooth functions on S". natural equivalence

relation on F(S*) is defined as folloivs. Two functions f: g E 3 ( S a ) are equivalent

if and only if

for some h F(S").

Definition 9 A linear functional on Soc zs an eqviualence class of the equiualence

relation (5.75) defined aboue.

The equivalence class 3 of a smooth function f E 3(Sm): ivill be derioted by

r = / fdz .

while C ( F X ) will denote the rector space of linear functionals on S".

The variational derivative is the operator

It acts on linear functionals F by

in which 3 = J f dx. This differentiation can be interpreted geometrically as defining

a natural Lie derivative operator on the space of linear functionals. Indeed. if q E

3(Sm). one can consider the evolutionary vector field .Yq = q(a/i3zo) and set

in which 3 is the linear functional

The last equality appearing in (5.45) is, of course, the classical definition of the

operator d / d q .

This finishes the discussion of the first two aspects of the Harniltonian formal-

km. Hamiltonian operators wi11 be introduced nest as a special case of the following

concept, which appears nat urally as the formal counterpart of the classical integral

operators of analusis:

Definition 10 A /ormal inteyu1 operator with kernel (Ki; ...: I<,), h; E 3(SX). is

a finite formai sum

in which gi! f j E F(SE ) .

These operators do not necessarily transform the set 3(Sm) into itself. but one can

prove the follo~ving proposition:

Proposition 11 Each fonnal integral operator K given by (5.79) determines a R-

iinear map (aiso denoted bg K) transforming snzooth f~rnctions on SX into smooth

[unctions on in which zs a svitable couering of SX.

Proof: Let be the coverinp in n-hich al1 indefinite integrals of functions on SX

appear as nonlocal variables, n a m e l ~

n-here SI is the covering of S" on which the indefined integral of f is a ive11 defined

nonlocal variable on S". This covering was constructed in (4.64). For each g E

F(S*) set rn

Ttien. clearly K ( g ) is an element of S, as by construction, al1 the "indefinite integrals''

of smooth functions on S" are elernents of 3. 0

Below, forma1 integral operators with coefficients in wi11 be also used. Hamil-

tonian operators are now defined thus.

Definition 11 A forma1 integral operator K wyith kernel (Ici, .... K.). Ii, E 3(Srj.

is called Harniltonian if

1. The map KI : 3 ( S r ) -> 3(3) defined by (5.80) satisfies

and moreover,

2. The bracket

zs a Poisson bracket on the space of junctionals L(Fm)? namely: it zs R-linear.

skew-symmetric, and satisfies the Jacobi identitg

for all functionals P, Q. R E L(FS) .

.An autonomous evolution equation t ~ , ~ = F ( x y zo, ...: zk ) is said to lie Hamiltonirrn if

there esists a Hamiltoriian operator K and a linear functional 3C suc11 that

Many esamples of Hamiltonian operators mith trivial kernel appear in Chapter 7

of Olverk [1993] treatise. On the other hand, the existence of Hamiltonian operators

with non-trivial kernels is an important issue. It appears that the first esample of such

an operator was found by Sokolov [19S4]. He proved that the operator T = -zlD;'zi

is Hamiltonian, and used it to show that the Krichever-Piovikov equation can be

writ ten in Hamiltonian form. Ferapontov [1991] points out that another noteworthy

esample is due to Irina Dorfman. She proved in her Doctoral thesis that

is a Hamiltonian operator, and that it can be used to m i t e the modified KdV equation

in Hamiltonian form. More general exampIes have been obtained by 'ulokhov and

Ferapontov (19901 in their study of the theory of systerns of hydrodynarnical type.

and developed further by the same authors (See for example Ferapontov [1991] and

Slokhor [1905]).

Assume now that the autonomous evolution equation

possesses a nontrivial conservation law D,T = DZx, and a generalized synmctry G.

Consider the conserved quanti ty

The function H is a fuiiction on the covering 3 defined abote. Let li be the Lie

derivative of 3C along the evolutionary vector field deterrnined by G, nameIy.

in u-hicli Sc = G(B/Bzo). Again. fi is a well defined function defined on 3. Ir will

be assumed that fi is nonzero.

The Harniltonian operator constructeci by Hojman [1996] is the follou-ing forma1

integral operator with kernel (G: F) and coeficients in 3:

1 1 3 = F-D,-'G - G-D;'F.

K fi'

This clioice of coefficients appears to be one of the reasons wliy Hojmnii (19961 (~ills

3 a "non-standard" Hamiltonian. In his paper, Hojman proves the follou-ing crucial

lemrna:

Lemma 6 The equation zo,t = F is equivalent to

Moreover, the integral operator 3 is Hamiltonian if the Lie derivative

Ls, / Gdx

is zero.

Esamples are readily available within the class of kinematically integrable equa-

tions, since in many cases one has a t one's disposal a large nurnber of consenmion

l a w and generalized syrnrnetries. Gomberoff and Hojman [1997]. for instance. con-

sider the KdV and Harry-Dym equations. They also study Burgers' equation. for

which they consider the consen-ed density T = zo, and take as G an special case of

the nonlocal symmetry found in Subsection 5.2.1 of this Thesis. Esamples with non-

local G, however, are outside the scope of this short appendis, as the corresponding

functions Ii given by (5.83) mil1 not be neçessarily functions on 3. Its forma1 stiidy

presents an interesting problem.

Chapter 6

Immersed surfaces and kinematic

integrability

The pseudo-spherical structure determined by an equation whicli describes pserrdo-

çphericai surfaces is described intrinsically by the one-forms a = 1.2 .3: no

mention of an "ambient space" is made. Kow, as pointed out in the Introduction.

t here esist also geometrical investigations on the integrability of nonlinear equat ions

n-hich are in a sense inverse to kinernatic integrabiiit- and will be studied here.

Since three-dimensional spaces will appear, the one-form w3 i d 1 be denoted here-

after by ~ 1 2 . -41~0. extending what has been done already in Chapters 4 and S. a bar

(-) will be used henceforth to distinguish one-forms defined on finite-dimensional

manifolds from t hose defined on the infinite-dimensional spaces SOC and J" E.

This Chapter has three sections. In Section 6.1. sonie general comments on im-

mersed surfaces, mainly to fis notation, are made. In Section 6.2 , conservation lan-s

of equations describing pseudo-spherical surfaces are revisited. now from an estrinsic

point of vien.. The class of equations describing Calapso-Guidiard surfaces is intro-

diiced in Section 6.3, and finally, conservation laws for tliem are briefly considered in

Section 6.4.

6.1 Immersing pseudo-spherical surfaces

Suppose that P : M -i E3 is a (local) isometric immersion from a Riemannian

two-manifold i\f into a flat (pseudo-)Riemannian three-space E3 equipped with a

metric of signature (+: t. c), É = &l. Choose a point uo E II and consider a n open

neighbourhood C2 of uo. C-' C dl. such that Plrc is an embedding. .Us0 ctioose

a n open neighbourhood I - O 3 of P ( c ~ ' ) , 1'" E3, and consider a nioving orthonormal

frame {el, el, e s } in isP3 such that el,e2 restricted to P(CLL) are tangent to P ( L y 2 ) (so

t hat e3 restricted to P(Lj2) is normal to the immersed surface P(LT2)) . One says that

the mo\-ing frame {el e2. e, } is adapted to the immersed local surface P(L:').

Sow, the moving frame {el y e2, e l ) determines a moving coframe{&. 2'. c3 } on

1-3, and three connection one-forms w,?, w:? and @ satisfying the structure ecpa-

tions (hereafter indices arc raised and lon-ered with the metric tensor q,b = SZb.

77,3 = cdL3. i = 1.2,3 , b = 1.2).

CG' = A and

k ai = - A . i. j : k = 1.2,3 ,

and the compatibility condition

The immersion P : U 2 C :LI + 1'3 C E3 induces one-forms P'D., P ' j i j in Lq c -11.

These one-forms will satisfy Equations (6.1)-(6.3). and also the constraint

R-ith a slight abuse of notation, writing P ' S = 3: and P g Z I j = Zij. one obtains

t hat the structure equations of the immersed surface U2 are

and

and t h a t the compatibility condition (6.3) holds. Equations (6.8)-(6.10) are the

Gauss-Codazzi equations. It is well h o w n tha t the local geometry of embedded

in E3 is now determined by the first and second fundamentai forms

In particular, JL2 is the Levi-Civita connection one-form corresponding to ds'. and

equations (6.7) and (6.8) imply that

in u-hich K is the Gaussian curvature of the irnmersed surface Ci2.

Conversely one has the folloming classical result (Eisenliart [1926] p.198. Guggeen-

heinier [1963] p.229). which is a t the heart of the "estririsic" approach to integrability.

Theorem 11 Let 8' and zi1 (b = 1; 2, io j = 1,2 .3) be one-fonns o n LY2 c I -3 c E3

such thnt ~ ' ~ 3 are linearly independent! equation (6.3) holds, and the structure

equations (6.5)-(6.10) are satisfied. Let uo E 02 and p0 E 1 be given. and assume

thut { e y _ e3. e!} is a n o r thonomal basis of the tangent space of 1.'"~ po. Ther~. t f~er-e

erczsts a unique (up t o isometries of E" immersed surface P : C" + [-:% c E? aurd a

unique o r t h o n o m a l moving frarne { e ? e2 . ea ) o n P(L7') s u c f ~ that

P(~o) = PO:

ei(u0) = ep, i = l , 2 , 3 ,

dP = d e b , 6 = 1,3, and

dei = s ' e l i . j = 1 , 2 . 3 .

The map dP is the differential map of the immersion P, while the map dei is the

differential of the vector field ei considered as a map ei : Li2 -+ E? Equations (6.17)

are the Gauss- Weingarten equations.

Thus, if two vector fields w1 8 n1 + d & d and C(J' @ q3 + 3 @ q3) on Ccl are

giveri. they are the metric and second fundamental forrn of a local surface immersed

isometricallÿ in E3 if and only if (6.3) and the structure eqiiations (6.3)-(6.10) are

satisfied. \if iting

one can determine the one-form w , ~ from Equations (6.5) and (6.6). The Gaiiss-

Codazzi equations then becomes a system of nonlinear equations for the functions

and dQJ which is the integrability condition of the Gauss-Weingarten liriear problem

(6.1'7).

It follows that, as anticipated in the Introduction, if one h a some control on the

functions had and daB. one can find interesting esamples of nonlinear (systems of)

cquations which will be, by construction. the integrahility condition of a linear systeni.

One can then use this linear system !pravicled tiiat it admits a b-spe~tra l parameter" )

to study the analytic propertics of the Gauss-Codiizzi equations. It is tliis autlior's

opinion that a striking esample of this procedure is the discover? and analysis of the

Lund-Regge equation of classical field theory by Lund and Regge [19T6]. and Liirid

[197S].

Xow, it is ccrtainly natural to ask whether one can relate this "estrinsic approach"

to the original "intrinsic" point of view developed in the previous chapters, and if stich

an esercise is worthwhile. This will be esplored in the following tn-O sections, Bcfore

considering these questions, however, one must of course have a reasonable metliod to

immerse pseudo-spherical surfaces into an appropriate flat three-dimensional space.

Proposition 12 A (local) pseudo-spherical surface described by o n e - f o m s d: b =

1 ,2 , and W L 2 satisfying the structure equations (1 -23) can be imrnersed in to a three

dimensional Minkowski space E3 of signature (+? +: -). Moreouer. it can be identifieci

with (an open subset o f ) an orbit of the pseudo-orthogonal group SO(2.I).

Proof: By the theorem quoted above, in order to immerse a pseudo-splierical surface

into the Minkowski space E3, one only needs to specifi one-forms 2 and gil ( b = 1.2. . . z' 3 = 1: 2 3) satisfying (6.3) and (6.5)-(6.10) with E = - 1. Consider then oneforms

3. z,? satisfying (1.23)' and set (Eisenhart [1926] p. 201)

- 3 -1 wi = -/ Ir/ , and

- 3 -3 W2 = -w-.

It is easy to check that the structure equations (6.5)-(6.10) indeed liold. TIierefore.

the tensors

are the first and second fundamental forms of a space-like pseudo-spherical surface

of Gaussian cumature - 1 locally isometrically immersed in E". The connection with the orbits of SO(3: 1) can be made ver- esplicit. The last

eqiiation of the compatible linear system (6.17) is

t h is: using (6.18) and (6.19),

The right hand side is, by Equation (6.16), minus the differential of the immersion P

given by Theorem 11. Thus, one finds the equation

(derlc.~)u(t.) = -(dP),(,u) for al1 c E T.C~':

in which the same notation as in Theorem 1 has been used. This last equation has the

obvious solution P = -e3Ic;2, in which one is considering e~ as a map e3 : b-' - E3.

Therefore. if E3 h a coordinates zi (i = 1,2.3) , this computation shows that the

immersion P is of the form P(z: t ) = (-zl(x. t ) ) , where the map e31c= : b* i E"

is (x, t ) i+ (zi(x7 t ) ) . Since e3 is an unit time-like vector field. one obtains that the

points P ( x , t ) of the immersed surface satisfy

which shows that the immersed surface may indeed be identified with (an open subset

of) an orbit of SO(2,l). 0

6.2 Conservation laws induced by immersions

Suppose that E = O is an equation describing pseudo-spherical surfaces. It will be

siiown in this section that Chern and Tenenblat's [19SG] one-form (4.27) is a member

of a one-parameter family of one-forms which are closed whenever ro(x. t ) is a soli~tion

of the equation 2 = O. First of d l . one needs to set the stage (Notation as in Section

6.1):

Proposition 13 Suppose that A1 is a srnooth Riemannian surface locully isonletri-

cal19 immersed in E3 with normal vector field iV. Suppose rnoreover that this im-

mersion is descrïbed b y one-fonns id' and zii (6 = 1: 2 : i' j = 1: 2 ,3) satisfging

(6.3) and (G.5)-(6.10). Let S(<) and C(<) l e two real-ualued functions such that

C(<)' + ES(^)' = 1. T h e surface M is pseudo-spher-ical if and only if for any f i e d

number < with C(<) f O: there ezists an orthonormal local moving frame { e ; . el,. .jY} 4

on M, such that the dual one-fonns 0 and connection one-forms ëij ( b = 1 . 2 . . i: j = 1 ,2 .3 ) sa t i s f i the equation

In this case, -

el - o ~ ~ s ( E )

is a one-parameter farnit9 of closed one-farms.

-1 -2 Proof: Since {B , 8 ) is to be a new moving cofrarne on AI. if it esists it must be

related to the coframe {cL. 3) by a rotation

el = e; cos p - e; sin

e2 = e; sin p i e; cos p

connecting the moving frame {el, e 2 ) on M dual to {zL, c2}. with the new moving -1 -2

frame {e;: E ; ) on M dual to {B : O }. Explicitly. this rotation induces the change of

coframes given by formulas (4.22): (4.23): and the change of connection one-forms

given by (4.24) and the equations

- t931 = TZ31 cos p -+ z32 sinp, and - 03? = -Z3L sin p f E3? cos p.

Let Z be the differential ideal generated by the one-form

- 7 = 012C(<) + 8? + &LS(<) -

.A straightforward computation using the structure equations shows tha t

thus. 1 is closed under esterior differentiation if &I is a pseudo-spherical surface -4 -

immersed in E3. Conversely7 if d y = 0, there exist one-forms 0 , B i j ( b = 1.2.

i, j = 1: 2 , 3 ) satis@ing Equation (4.12) for < such that C(<) # 0. In particular. there -

are one-forms 06, 08, (b = 1.2. i, j = 1.2.3) on M satisfying t $ = 0. so that el.

k2: and 1 9 ; ~ satisfy the structure equations

dei = 0:

d@ = - 6 1 and

o;, ++ = 0.

This means that one can Iocally find functions U and b- such that

9 1 = du,

2 = = - u d ~ ,

but in these coordinates the metric of A l becomes

a standard form of the pseudo-spherical metric. Thus the first claim of the proposition

follows from the Frobenius theorem. The fact that the parameter dependent one-form -

ë1 - &S(<) is closed follo~vs from Equation (4.12) and the structure equations (6 .5 ) -

The analytical interpretation of this proposition is that zo = zO(l: t ) is a solution

of a given equation describing pseudo-spherical surfaces with associated one-forms

;*, sl> (6 = 1: '1: i, j = 1, 21 3) if and only if the Pfaffian system

is completely integrable for Q ( x . t ) for each satisfying C(<) f O. in which $ =

( j x ( . s ) ) * s b . w~' = ( j r ( . 5 ) ) ~ i i z j , and s is the local section .9 : (2. t ) rt (1. t . z o ( r . t ) ) .

Example: Consider the ubiquitous sine-Gordon ecluation in the fosni

d'O d28 -- - = sin 8 cos 8. as2 dt2

One readily sees that it describes pseudo-spherical surfaces immersed in a flat Eu-

clidenn threespace E3 with associated one-fornis

u1 = cosûdx.

W* = sin Bdt,

~ 1 2 = Otdx + Bzdt,

~ 1 3 = sin Odx,

q ? 3 = - COS edt.

and

Set 5 = T/:! - al C(<) = cos<: and S(<) = sine. Substituting into Equation (6.3L)

one finds the first order equations

sin a(& t Bt) = sin 4 cos 8 - cos a cos 6 sin 8, and

sin a(& + 8,) = - cos 4 sin 8 + cos a sin Q cos 8.

This is esactly the cornpletely integrable Pfaffian system appearing in Eisenhart [1909]

wliicli determines a pseudo-spherical surface !VI from the pseudo-spherical surface M.

by rneans of a Backlund transformation with paranieter o.

The geornetrical interpretation of the above proposition is precisely the one sug-

gested by this esample. Indeed, the first half of Proposition 2 (mith E = 1) is a

rnoving frames version of the integrability part of the Backlund theorem as proven b ~ .

Ctiern and Terng [1980] and generalized by Tenenblat and Terng [19SO]. They point

out that if (notation 'as in the hypotheses of Proposition 13) CO is a11y unit vector

r;iiigent to JI at xo E M satisfying co = clrA + c2r,2. in which CI. (2 f O and { V A . CG} is an orthonormal b a i s a t xo consisting of principal curl-atiire 1-ectors. ancl ,II is a

pseudo-spherical surface imrnersed in R3 (SO that (4.12) is integrable). then the first

I i d f of Proposition 13 implies that there esists a unique surface :Il1 satisfying the

foIlon-ing conditions:

1. d l l is pseudo-spherical: and the moving frame (e;: e;. iV} on JI satisfies e; (xo) =

'Co

2. AI! lies at a distance sin O from AI along the line aei. and moreover. x ~ ; =

sin avo, where xi is the point in Ml corresponding to xo.

3. -1 moving frame {e;, e;) and a normal vector field !Y' on JII are git-en by the

rotation

e; = ; e.: = cos ae; + sin oM

1V* = - sin oek + cos a:V.

The Pfaffian system (6.31) reduces to the one used by Chern and Tenenblat [L9S6]

(Equation (4.26)) if a = r / 2 : of course. -As in that case? it yields conservation Iaws

(Sotation as in Equation (6.31)):

Proposition 14 Let E = O be a n equation which describes pseudo-spherical surfuces

rn~mersed in u pst (pseudo)-Riemanniun three dimensional manifold (E? dds'). where

ds' = d iag ( l . 1. c) and c = &1. wzth associated o n e - f o m s i' = f t idx+ f&t and ~ $ 3 , =

htldx + h,-dt. Consider two functions S(<) and C(c) satzsjymg CI<)' +- ES(<)' = 1.

Tlren. for cach for which C(<) + 0 . the Pfaf lan sgstem (6.31). in co7nponents.

C(<) (4, + fil) = f sin q5 - cos 6 - S(<) (hl 1 cos ,j t sixi 6). (6.35)

C(<)(&+ f3?) = f12s ind - f 2 2 ~ ~ ~ 9 - S ( ~ ) ( h 1 2 ~ ~ ~ ~ t h 2 2 s i n o ) . (6.39)

i.i. cornplettilg integruble for ~ ( x , t ) .wheneuer zo(x , t ) is a solution of the eqvation , = 0 :

und nloreover. jor each solution of this equation and corresponding solution o of (6.38)

und (6.39). the one-form

L, = ~1 cos o i J' sin O - S(<) ( - z ~ ~ sin O t Z3? COS O ) . (6.40)

in uhich i' and Z,, denote the pull-backs of w' and rg,, b y zo (x. t ) . z s closed.

A change of variables analogous to (4.30). namely

a) = ta* (;) allows one to replace the one-form (6.40) by a simpler differential form n-hicli is also

closed on solutions z O ( x . t ) of E = O. The Pfaffiar, system (6.31) becomes

and the closed one-form w now reads

-4 simpler representative of the same conservation law is obtained by consideririg

One finds, using Equation (6.41),

Generalizing the Chern-Tenenblat case. if the equation S = O is kinematically

integrable, the proposition above can be used to find a two-parameter family of - consen-at ion lams for the equation = = O by espanding the one-forms ub: :i' (b = 1.2:

i: j = 1,2 .3) and the function in powers of the "spectral" parameter.

Example: The sineGordon equation (6.32) t?,, - Ott = sin 8 cos 8, is kinernatically

integrable. -1ssociated one-forms are, for esample,

Uj ' = 1 - siri 38(dx - d t ) . (6.43) 2r7

Using the immersion constructed in Section 6.1: one nou- sets

1 W I S = - sin 28(dx - d t ) ?

277 and

Set C(<) = cmhc and S(c) = sinhc. Since the Pfaffian system (6.41) is completeIy

integrable for @(x, t ) , one can compute this function from the Riccati equation

77 1 2 ~ 0 s h e @ ~ = -cosh<O, - - - 1 - cos 28 + +- sinh < sin 38 2 37 277

77 1 i + a2 (- eosh <B. + - + - COI 28 - - sinh <sin 20 2 2 q 2q

and the conservation law (6.44) non: reads

1 7 7 1 D,[- sin 20 + <P(, + - cos 28 - cosh <(8, + e t ) )

277 - 277 1 77 1

+sinh<(-@-sin28 + - + -cos28)] = 277 2 277

1 77 1 D,[-% sin 28 + a(? - -cos 28 - cosh <(O, + 8,)) - 27

I 17 1 + sinh <(a - sin 20 + - - - cos 29)]. (6.51) 277 2 277

This conservation law gives rise to a sequence of a-dependent conservation lan-s of

the sine-Gordon equation by espanding the solution cI, of Equation (6.50) as a formal

series in powers of 77.

6.3 Equations which describe Calapso - Guichard

surfaces

The main observation behind the work to be reported in this section, is that the

Gauss-Codazzi equations of Calapso-Guichard surfaces in isotliermal spherical coor-

dinares (See Eisenhart [1923j) generalize the sineGordon equation. and can be iiiter-

preted as the integrability condition of a tn-O-dimensional linear problem 11-ith spectral

parameter (Gürses and Kutku [1981]). It is natural to ask dietl ier this interpretatiori

may be used to generalize the whole class of equations describing pseudo-spherical

surfaces. This is certainly the case. as it wi11 be secn presently.

The classical Calapso-Guichard equations are obtained as follows. One corisiders

a pseudo-spherical surface immersed in R3 with metric

and second fundamental form

rp = sin 0 cos 8(dx2 - dt2).

The standard Calapso-Guichard surface is the surface determined by the met ric

ds;, = e2'ds' = (eC cos 8)2dx2 + (eC sin 0)dt2

and the second fundamental form

qcc = ec cos @(sin 0 + h cos 8) dx2 + e' sin 8(- cos O i h sin 0)df2.

The Gauss-Codazzi equations (6.8)-(6.10) imply three nonlinear eqitations for the

functions <, h, and B. These are the Calapso-Guichard equations. Esplicitly these

equat ions are

-(-et + 5 cot O ) , + (-8, - & tan O), = (- cos 0 + h sin 8) (sin 0 + h COS BX6.JZ)

h, = J , (h+tanB): and (6 .53)

h t = < t ( h - ~ ~ t t 3 ) . (6.54)

which clearly generalize the sineGordon equation ( 6 . 3 2 ) .

Sow one can find Calapso-Guichard equacions generalizing arbi t rary equar ions - describing pseudo-spherical surfaces. Let =(x. t . -0. .... z ~ . , ~ , ~ ) = O b~ a ( nc - n l r l i-

order equation describing pseudo-spherical surfaces with associated one-fornis

so that the structure equations

are satisfied whenever q(z, t ) is a solution of t = 0.

Consider a trivial fiber bundle Ë with base coordinatized by the independent \-xi-

ables x, t . and three-dimensional fiber mith coordinates zo: <, and h. The discussion

on jet bundles appearing at the beginning of Section 2 generalizes t r i~ial ly to this

case and wiIl be used here without fuxther ado. Define two one-forms ai and o2 on -

.Jm+'E by conformally deforming w1 and w2:

One can now determine uniquely a one-form

as the solution to the structure equations

Sote that no pull-backs are being taken in these equations. The idea is to find

a unique one-form a12 satisfying Equations (6.62) identically. One easily finds the

formulas

in which 6 = fll f2?, - f l l j L I .

Suppose chat tliere esist connection one-forms {w13' w," ssucli tliat the one-forms

w'? ~ 1 2 : u13? and -3 sütisfy the structure equations ( 6 . 5 ) - ( 1 0 of n surface

ininiersed in a Rat space E3 as in Proposition 1, mhenever zo (x . t ) is a solution of - r = 0.

Define one-forms oI3 and by the formulas

o13 = w 1 3 + h 1 , and

O: = u~~ + hw2.

Tlieorem 11 now implies that the one-forms o' and üil (2-e.' oi and Oij puiled back

by the infinite prolongation of a local section s : (x, t) rt (x, t, zg (x, t) , <(x, t) : h(x' t ) ) )

d l determine a surface immened in E3 if, and on- if, the following structure equa-

tions are satisfied (The signature of the flat metric on the ambient space E3 does not

need to be fised a priori):

Now, Equations (6.59)-(6.64) imply that Equations (6.67), (6.68): and (6.69) are

identities. On the other hand, Equations (6.70), (6.71) and (6.72) are equivalent to

a system of three nonlinear equations for zo(x , t) , E(x, t ) , and h(x, t). This motivates

the following definition.

Definition 12 Let f = O be a PSS equation with associated functions f i j : (i =

1' 2,3: j = 1, 2). Consider a system of differential equations

in which Zi are smooth functions on J ~ F , and let be its equation manifold.

The system (6.73) is said to describe Calapso-Guichard surfaces of type 5 if the

pull-back of the one-fonns 01, gji (i = 1,2,3, j = 1,2) dejîned by (6.59); (6.60),

(6.611, (6.651, and (6.66) by local ho~odrn i c sections joo (s) of y, s : (zl t) + (x: t: z0(x, t ) , <(x, t ) , h(x, t ) ) such that

satisfy the structure equations (6.70), (6.711, and (6.72).

EspIicit ly, the nonlinear equations (6.43) one obtains are the following:

in R-hich the functions hf are determined by wls = hi3dx + h;3dt, and ~ 2 3 = h l d r +- hZ3dt.

Xote that if is constant, h = 0: and a(z, t ) is a solution of Equation (6.73): the

structure equat ions (6.67)-(6.72) become simply

The first three equations Say that the coframe {w1,z2) spans a surface immersed in

E3 mith connection form alz. By uniqueness of the Levi-Civita connection, o12 = Z l ~ :

and therefore the conditions h = 0, constant, imply that the integrability condition

of Equations (6.67)-(6.72) is exactly E = O. Moreover, it folloms that the coframe

{ol' z2} spans a pseudo-spherical surface with connection form Z12 immersed in E3

which is liomothetic to the one spanned by {JI, s2}.

Exarnple: Gürses and Nutku's (19811 generalization of the sineGordon equation is

recovered from the foregoing computations. Indecd, the equation (6.32), O,, - 8, =

cos B sin O, describes pseudo-sphencal surfaces immersed in Euclidean three-space if

one now sets (compare wit h the associated one-forms introduced in (6.33)-(6 -37))

~ 1 3 = - COS Odt, and .

The one-forms oz, ai, defined above are given by

0 1 2 = (-8, + Ccot 6)dx + (-6, - <, tanO)dt,

0 1 3 = (- cos 0 + h sin 9)dt: and

0 2 3 = ( s i n e t hcosO)dz,

and therefore equations (6.74), (6.E), and (6.76) becorne exactly (6 .S) , (6.53). and

(6.54). As said at the beginning of this section' these equations are the classical

Calapso-Guichard equations. It should be noted that they differ from the equations

appearing in Eisenhart 119231 p.231 and Gürses and Nutku [1981] by some signs.

This s i p difference can be traced back to different conventions for the structure

equations (6.1) and (6.2). Gürses and Nutku [1981] use (see their Equation (2.8))

d 3 ; Po h w3 = O (summation on the second indes).

If the explicit immersion for pseudo-spherical surfaces developed in Section 3 is

used, one obtains deformations of the complete family of PSS equations. Indeed,

suppose that w l , w2 and w12 are the one-forms associated with a PSS equation E = 0,

and define the one-foms w13 and w~~ by w13 = w l , and w23 = u2. Equations (6-T-L) ,

(6.75) , and (6.76) become (6 = - 1)

in which 6 = f l l fZ2 - fi2 f Z L .

Example: The generalized Burgers' equation (3.33) admits a one-parameter family

of deformations of the form (6.82), (6.83), (6 -84). Parameter-dependent one-forms

wl, w2 are given by (3.34) and (3.35), so that the functions hS1 and h32 become

in which E = f:, + f&, F = fiif12 +fiifi2, and G = f:, + f& are the coefficients of

the tensor (d)* i- (w2)'.

It remains to be seen that Equations (6.74), (6.73) and (6.76) are the integrability

condition of a two dimensions1 linear problem. This is indeed elementary, since as i t

has been said before, the Gauss-Codazzi equations (6.70), (6.71) and (6.72) are the in-

tegrability condition of the Gauss-Weingarten Iinear system (6.17). Thus, Equations

(6.74); (6.75), and (6.76) are the integrability condition of the linear problem

One can nonr obtain a two-dimensional linear problem by standard methods, esploting

the local isomorphisms between SU(2) and SO(3) if E = 1, and between S L(2. R)

and S0(2,1) if E = -1. One finds that Equations (6.74): (6.75) and (6.76) are the

integrabilitÿ condition of the linear system

which in turn, says that the one-forms

satisfy formally the structure equations (1.23) of a pseudo-spherical surface on solu-

tions of (6.14), (6.75) and (6.76).

Of course, if the original PSS equation = O is kinematically integrable, the linear

problem (6.88) will depend on a "spectral" parameter. For esample, if the one-forms

associated to E = O satisfy u2 = qdx + f22dt, and one uses the esplicit immersion for

pseudo-spherical surfaces introduced in Section 2: the linear problem (6.88) becomes

In particular, the linear problem associated wit h the classical Calapso-Guichard equa-

tions (6.52), (6.53), and (6.52): is of the form (6.92). Gürses and Nutku [1981] have

shown that it can be written as a 2 x 2 AKNS linear problem.

6.4 Conservation laws and Calapso-Guichard sur-

faces

One may easily obtain conservation laws for equations describing Calapso-Guichard

surfaces. This follows, of course, from the esistence of a linear problern for which a

given equation ciescribing Calapso-Guichard surfaces is the integrability condition.

Indeed, consider a defonnation (6.74), (6.75): (6.76) of an equation describing

pseudo-spherical surfaces with associated one-forms w l , w 2 , and ~ 1 2 : and associated

estrinsic curvature one-forms wl j and ~ 2 3 . Define the one-foms a', Oij by means of

(6.59)-(6.66). Then, as it was said at the end of the last section: the one-forms al, a2 ,

and il12 given by (6.89), (6.90): and (6.91), satisfy formally the structure equations

of a pseudo-spherical surface on solutions of the s p t e m of Equations (6.74), (6.'is),

and (6.16). The constructions of Section 3 may be applied, then. One obtains that if

( z ~ ' ( x , t): <(x, t ) : h(x, t)) is a solution of (6.74), (6.75), and (6.76): the Pfaffian system

is cornpletely integrable for r c c ( x , t). Moreover, one aIso obtains that for each solu-

tion (z0 (x: t ) , <(x, t ) , h(x, t)) of (6.741, ( 6 . X ) , and (6.76), and corresponding solution

r c c ( x , t ) of (6.93), the one-form

is closed.

One can go one step further and immerse the formal pseudo-spherical surface de-

scribed by the one-forms a ' , (r2, and a12, into a three-dimensional space l? equipped

with a Aat rnetnc of signature (+, +, -) as in Section 2. One finds that the Pfaffian

system

(- sin oa12 - a2 + cos oal) + 2 9 c c ( a 1 t cos o a 2 ) + a&(- sin aat2a2 - cos oaL) = 2 sin o d ~ 9 ~ ~ . (6.94)

is completely integrable for acc(x, t ) for each solution (zo(x, t ) , <(xo t ) h(x, t ) ) of

Equations (6.74): (6.75), and (6.76). As before? for each solution of (6.74). (6.73)'

and (6.76) and corresponding solution 9 c c ( x : t ) of (6.94)? the one-form

6cc = a' + aCG(a2 - s inoai2) + cos o ( - < ~ ~ ~ c r l t a*)?

is closed.

The adjective "Formal" has been used in the preceding discussion because the

one-forms (6.89): (6.90), (6.91) may be cornplex-valued, depending on the signature

of the ambient space. However, even if this is so: one will find bona fide conservation

l a w ? as it was seen already for the case of the Lund-Regge equation, Subsection

42 .1 . Histoncally, the fact that one can formally estend the original Cavalcante

and Tenenblat [1988] algorithm for conservation laws to the case of cornplex-valued

associated functions fao: has been noticed for Wyller (19891. This is what he does

in his study of the derivative nonlinear Shrodinger equation already ment ion4 in

Section 4 .2 ,

Chapter 7

Summary, furt her comments and

open problems

This final chapter contains a very brief summaw some comments on the results

proven in this Thesis, points out some problems left open, and also some possible

directions of research.

Summary: The following have been obtained in this work: -4 classification of ar-

bitrary scalar evolution equations in two independent variables which are strictly

kinematically integrable (Theorems 1 to 5 and Corollaq- 1. Also in R e ~ e s (19981): a

proof of the fact that forma1 integrability implies kinematic integrability for general

second order autonomous evolution equations (Theorem 6. Also in Reyes [1998]); a

proof of the fact that forma1 integrability implies kinematic integrability for a spe-

cial case of third order autonornous evolution equations (Theorem 8. Also in Reyes

[1998]), and a counter-example for a generalization of Theorem 8 (Theorem 7. Also in

Reyes [1998]); an algorithm generating sequences of conservation laws for strictly kine-

matically integrable equations (Theorem 9. Also in Reyes [l998]) ; an exact relation

hetween the classical construction of conservation laws for kinematically integrable

equations and the Chern-Tenenblat method (Section 4.2); a geometric characteriza-

t ion and proof of existence of (generalized, nonlocal) syrnmetries for kinematically

integrable equations (Propositions 7, 8, 9: and Theorem 10): a construction of (in

principle nonlocal) conservation laws of kinematically integrable equations via de-

formations of the Chern-Tenenbiat family of conservation laws (Proposition 10): a

construction of consewation laws of kinematically integrable equations within Sym's

estrinsic approach (Propositions 13 and 14); a new class of kinematically integrable

systems of equations (Section 6.3).

Comrnents and open problems:

It \vas seen in Chapter 3 that the proof of the implication "formal integrabil-

ity kinematic integrability" cannot be extended from second to third order

equations, as the lists of formally integrable third order equations provided by

Mikhailov, Shabat and Sokolov [1991] are not composed of only strictly kinemat-

icaHy integrable equations. One immediately wonders if relaxing the a priori

condition f21 = 7 is enough to yield a proof of the implication in this case,

or if really forma1 integrability is a notion of integrability more general than

describing one-parameter families of pseudo-spherical surfaces.

Since the equations appearing in the blikhailov, Shabat and Sokolov's lists de-

pend esplicitly on the independent variable x, the first step in such a program

should be to extend the characterization results described in Chapter 2 for

evolution equations of the form Z O , ~ = F ( x , t , 20, . . . , zk ) to the general case of

arbitrary fZ2. This step is actually straightfonvard and has been done already

(Reyes [1997], unpublished) because one can use the results of Kamran and

Tenenblat [1995] on the classification of equations describing pseudo-spherical

surfaces of the form zovt = F(zO, . . - , z k ) (no explicit x, t-dependence) without

the a priori assumption f21 = q. What it is not clear to this author is how to

include a spectral parameter in a natural way, although one can easily include

one by hand, of course.

It appears that a more conceptual approach to this problem is needed. and

that one should aim to prove (or disprove) the implication above without using

lists of equations. This plan clearly passes by understanding more fully the

mathematics behind the notion of formal integrability, since one should be able

to esplain: a priori, for esample why the Lie algebra sZ(2,R) appears as a

common feature of every second order equation which is formally integrable.

'2. I t was proven in Theorem S that every formally integrable third order equation

of certain form is kinematically integrable. The result is, however, unsatisfactory

in the following sense (the author thanks Jacek Szmiagelski for conversations

on this point).

R e d 1 again that a kinematically integrable equation with associated one-forms

u" is the integrability condition of the si(2, R)-linear problem (1.24): name15

Following Crampin, Pirani, and Robinson [1977], one can now interpret this

linear problem in terms of a linear connection on a principal SL(2, R)-bundle.

Indeed, Equation (7.1) may be interpreted as saying that the section (x? t ) i+-

(u~(x, t ) , u2(x, t ) ) of a vector bundle i'1.f x V -+ M , in which V is some vector

space of dimension two, is covariantly constant. NOW, this vector bundle can be

considered to be associated with a principal fibre bundle ~'lf x SL(2: R) + M ,

and the matriv

can be considered (once the one-forms w" are pulled back to M by sections s of

E) as defining a connection on the principal bundle 1I.I x SL(2, R) -t M . The

basic observation of Crampin, Pirani, and Robinson [1977] is that the linear

system (7.1) is compatible if and only if the connection V is fiat.

Xow; once one takes this point of view is natural t o ask about gauge trans-

formations of the principal bundle M x SL(2, R) -t M. Since it is not clear

t hat arbitrary gauge transformations (depending on derivat ives of zO) change

the associated one-foms wa into another set of associated one-fonns: gauge

transformations are assumed here to depend only on the independent variables

and (rnaybe) the spectral parameter. Of course, application of a gauge trans-

formation wiil not alter the integrability condition of the linear problem (7.1):

but it will change the one-forms wa. For example, Sasaki [1979], and Gürses

and Xutku [1981] use this freedom to find convenient representatives for the

one-forms w". It is in this sense that Theorem 8 is unsatisfactory. One obtains

the one-forms of Theorem 8 by applying a gauge transformation depending

only on (x, t ) and the parameter q, to the trivial Iinear problem. Thus, even

though Theorem 8 does prove that a certain class of formally integrable third

order equations is kinematically integrable, the linear problem it provides is not

useful for, for esample, inverse scattering purposes.

These observations point out to an aspect of the theory presented in this Thesis,

the "gauge equivalence" of equations describing pseudo-spherical surfaces? which

still needs to be developed.

3. There are two important problems related to (generalized: nonlocal) synme-

tries of kinematically integrable equations which have not been addressed in

this Thesis. First of all, the computation of generalized and nonlocal symme-

tries bas been considered only in Theorern 12, which although not completelv

straightfonvard to apply in specific examples, appears to contain in itself impor-

tant characteristics of kinematically integrable equations. as the calculat ions of

nonlocal symrnetries of Burgers' equation performed in Subsection 5.2.1 show.

One needs to investigate whether this example is only an accident or it can be

generalized. This is an open problem which should be solved in the near future.

In the same vein, the construction of recursion operators for kinematically inte-

grable equations bas not been completed. This problem is important not only

for its otvn sake, but also because it is related to the construction of Hamilto-

nian operators for kinematically integrable equations which was outlined in the

Apendi~ to Chapter 5.

The second important and related problem is the study of hierarchies of equa-

tions (if any) determined by one equation describing pseudo-spherical surfaces.

Does every equation in the hierarchy describe pseudo-spherical surfaces as well?

The anstver is obviously in the affirmative if one considers hierarchies of the type

studied by Ablowitz, Kaup, Newell and Segur [1974], for esample; because in

this case each equation in it is obtained from the one before by application of a

two-dimensional recursion operator. This au t hor does not know to what estent

this observation can be generalized.

4. - in interesting probiem anses by considering the equations describing Calapso-

Guichard surfaces introduced in Section 6.3. These equations depend on second

derivatives of the deformation functions < and h. Can one generalize this? More

exactly, one could define the deformed coframe (6.59) and (6.60) as folloms:

(JI = egiwLT and o2 = e g 2 u , 2

and also set, instead of (6.65) and (6.66):

in which gl g2,913, and 923 are functions of c, hl and their derivatives. Then, one

may be able to check whether or not the Calapso-Guichard equations analogous

t o Equations (6.74), (6.?5), and (6.76) one would obtain from (6.70) (6.71): and

(6.72), impose restrictions in the order and/or the form of these functions. as

Chern and Tenenblat [1986] and several other researchers have done for the

class of equations describing pseudo-spherical surfaces. It appears tliat some

restrictions must indeed exist, since it was s h o m in the 1 s t section that the

one-forms oi: oil, determine one-forms (6.89), (6.90)' and (6.91) satisfying the

structure equations of a pseudo-spherical surface on solutions of the equation

at hand, but this needs to be better understood.

5. -4 problem in the same spirit as the one considered in Chapters 2 and 3 of this

Thesis presents itself once one recalls that some kinematically integrable equa-

tions can be interpreted as Hamiltonian systems on coadjoint orbits of a loop

group (See for example Adler [1979], Faddeev and Takhtajan [1987]' .Adams.

Harnad, and Hurtubise [1993], and references therein). Can one understand the

complete class of kinematically integrable equations in the same terms?

6. The following problem seems to indicate an interesting direction of research. in

the spirit of Bobenko's [1991] integrable surfaces theory. In the seminal mork by

Pinkall and Sterling [1989], compact tori of constant mean cumture were con-

structed from periodic solutions of the sinh-Gordon equation, ci kinernatically

integrable equation. Are surfaces of geornetric interest obtained from arbitrary

equations describing one-parameter families of pseudo-spherical surfaces? In

particular, Kapouleas (see Kapouleas [1995] and references therein) has shown.

by methods entireIy different from the one mentioned above, that compact mean

cun-ature surfaces of arbitrary genus exist. Can one generalize Pinkall and Ster-

ling's approach and obtain examples of these surfaces by studying solutions of

(some) equations describing pseudo-spherical surfaces?

- I . Final15 the issue of describing integrable equations on three or more indepen-

dent variables must be addressed. Are there classification results like the ones

obtained in Chapter 2 for the two independent variables case? In general, can

the analyses carried out in this Thesis be extended to equations with several

space variables? Can the observed cases of multidimensional integrable equa-

tions like the ubiquitous K P equation be recovered from such an approach'?

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