29 January 1996

PHYSICS LETTERS A

EIXiVIER Physics Letters A 211(1996) 19-24

Reductions of the Benney equations

John Gibbons a*1, Serguei P. Tsarev b,2*3 a Department of Mathematics, Imperial College, London SW7 2BZ UK

b Steklov Mathematical Institute, Moscow, Russian Federation

Received 25 October 1995; revised manuscript received 30 November 1995; accepted for publication 1 December 1995 Communicated by A.P. Fordy

Abstract

The reductions of the Benney moment equations to systems of finitely many partial differential equations are discussed. Several families of these are constructed explicitly. These must all satisfy a compatibility condition, and the reduced equations are diagonalisable and semi-Hamiltonian. By imposing a further constraint, of scaling and Galilean invariance, the compatibility condition is reduced to a system of algebraic equations, whose solutions are described. A more general family of reductions is then constructed explicitly.

1. Introduction

The Benney moment equations [ 1,2] are

n *aA0 +nA - -=O. ax (1)

They admit many different reductions, in which the infinitely many moments A” are expressible in terms

of only finitely many variables. These are quite diverse in their properties. One of the best understood is the

Zakharov reduction [ 31, which can be derived as the dispersionless limit of the vector NLS. It is

N

A” = c

hiul, (2) i=l

for if the variables hi, Ui satisfy the equations of motion

’ E-mail: j.gibbons@ma.ic.ac.uk. 2 Present address: Department of Mathematics, KGPU, Lebedevoi

89, 660049 Krasnoyarsk, Russian Federation. 3 E-mail: tsarev@class.mian.su.

ahi a(h) = o

-Lg+- , ax (3)

(4)

then all A” satisfy ( 1). Here the A” are all express- ible in terms of 2N independent dynamical variables, and further, the reduced system is Galilean invariant

and scale invariant. It is also Hamiltonian with three different structures. Most important, it is diagonalis- able. If, following Ref. [ 21, we define the generating

function A(p) by

A=p+~A”,p’+‘=p++- n=o i4 P-h

(5)

then each turning point

an ap (Pi) = 07 A(Pi) = Ai

of this function corresponds to a Riemann invariant A’ with characteristic speed pi* That is, Eqs. (3), (4) are

0375-9601/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0375-9601(95)00954-X

20 J. Gibbons, S.P. Tsareu /Physics Letters A 211 (I 996) 19-24

equivalent to the system

a/v x +pi$ =O. (6)

In Ref. [4] it was shown how any diagonalisable Hamiltonian or semi-Hamiltonian system may be

solved exactly by the hodograph transformation; this

has been carried out in detail here by Geogdzhaev

[5]. A degenerate case of the Zakharov reduction can be derived by letting some of the Ui approach the same limit, so that A(p) has multiple poles.

Another important class of reductions consists of the dispersionless Lax equations, which were solved

in Ref. [ 61; here the A” are given by

p+eA”lp”+‘= N-2 IIN

pN + c unpN-“-2 . (7) n=o ( n=o >

These are also Hamiltonian and diagonalisable, but are

not Galilean invariant. A further class of reductions,

important in the numerical investigation of the Vlasov equations, is the so-called “waterbag” model, where

the variables A” are the moments of a piecewise con-

stant distribution function f( x, p) . This gives

p + 2 A”,p”+’ n=a

(8)

so that the A” depend on 2N dynamical variables UT and ~(7, as well as the N parameters pi. The resulting

system is Galilean invariant but not invariant under the scaling

A” -+ (ynf2An. (9)

Studying these different reductions individually can- not answer the obvious general questions which arise, such as whether the reduced equations are always diag- onalisable or Hamiltonian, how many such reductions

exist, or which of them are invariant under Galilean or scaling transformations. It makes more sense to study such reductions directly. This may be done in two ways; either by trying to construct much more general classes of reductions, or in general terms, by considering the consistency condition which all such reductions must satisfy.

In Sections 2-4 we study the consistency condition, and hence show that the reductions to N variables

are parameterised by N functions of one variable; in Section 5 such a family of reductions is constructed

explicitly, by deforming the Lax reduction.

2. The compatibility condition

Let us consider the most general case in which all the moments A’ are functions of only N independent variables ~j. If the A’ satisfy (1) it is straightfor-

ward to show that the mapping (u’, . . . ,uN) -+ (A’, . . . , AN-‘) is nondegenerate. Hence we may without loss of generality set u1 = A’, , . . , u” = AN-‘. The first N moments are the independent variables, while the higher moments are functions of them,

Ak = ak( A0 ,..., AN-‘), k> N. (10)

The equations of motion for (A’, . . . , AN-‘) then be-

come

3Aj JAi+l --=

at ax + jAj-lg, j<N-2, (11)

aAN- C3 A0 -- dt

=$z+(N-1)AN-2an_, (12)

while each higher moment (aN, . . .) must satisfy the overdetermined system,

dak -- =

dt

ask N-1 aaN aAi dA0 +-

aAN- * c :- + (N - l)AN-‘= i=. aAl ax

N-1 aak+l aAj 0 =

c --

j=. aAj ax + kak-‘!!-

ax . (13)

Hence we find, on comparing the coefficients of aAj/ax, the system

auk+ * ask aaN ask -=-- aAj aAN-’ aAj

+- a&i ’ l<j<N-1,

iA i-l aak ; aak aaN ,&-I. (14) aA’ aAN- ’ dAO

The compatibility condition for these with k = N, gives a system S of i N( N - 1) nonlinear second order

J. Gibbons, S.P. Tsarev / Physics Letters A 21 I (i996) 19-24 21

equations for the single unknown aN( A’, . . . , A’-‘). It can be shown by induction that if S is satisfied then the analogous compatibility conditions for uk with k > N, are satisfied too. Taking N = 2 we get the simplest such system; on denoting x = Ao, y = At and z =

a2 - ia& we get

ZXX - z”vzxy + zxzyq’ - 1 = 0. (1%

If the variables u and u are defined as $( zY &

F z, - 4zX), then this equation becomes

1 1 u* = uuy - -

u-v’ V.X =uuy+----.

U-V

This has one obvious hydrodyn~ic type conserved density (U + u), one involving first derivatives (U - u) (u; - u_;), together with several involving X, y and z explicitly. A simple form of S for general N will be given in the next section.

Haantjes [ 71 derived the condition that a system of N partial differential equations of the form

ad i ad ~+u.--=o J ax (17)

should possess Riemann invariants. That is, that there should exist N functions A”, depending on the vari- ables ui, in which these equations are diagonal&d,

ah” $+vndt=o, where the V” are the eigenvalues of the matrix U$ called characteristic speeds. The condition is the fol- lowing. Define N;k as

and then define Hi, as

System ( 17) can be diagonalised whenever H$ van- ishes identically, and the V” are all real and distinct. In the case of system ( 11 f , ( 12), the reduced Benney equations, the former condition holds precisely when S is satisfied. Hence any consistent reduction of the Benney equations is diagonalisable. Thus we can use

the Riemann invariants and their characteristic speeds as the natural variables in which to discuss the prob- lem further.

3. The characteristic form of the consistency condition

Suppose that each moment Ak is expressible as a function of N Riemann invariants A’, which satisfy the equation

aA\’ j ah” -$-v-=0.

dX

The moment equations are then satistied if

(21)

.i?A” aA”+’ V’z=----

i?A0

d/i’ -t- nAn-lT.

aA’

Following Ref. [ 21, we consider the generating func- tion A(p) defined by

A=p + ~A”/$‘+‘. n=o

This satisfies the equation, equivalent to (22),

or, rehanging,

a/1 dA” an 1 (3A’=dl\‘dpp--

(23)

f24)

(25)

Cross-differentiating, we derive consistency condi- tions, equivalent to S,

a2Ao dA” dA” 1 ----:-- c?JVa& a# d,Q (vi - vj)2

and

avj dA” 1 -- - ahi - j$F Vi _ Vj ’

(26)

(27)

for all i # j. The higher moments A”, with II > 0 are then given recursively by solving (22). These equa- tions are compatible, and their solutions are param- eterised by 2N functions of a single variable. IIalf of these are inessential, corresponding to the freedom

22 J. e~bbo~, S.P. Tsareu /Physics Letters A 211 (1996) 19-24

to reparameterise each /\’ separately without chang- ing the form of (21), but the other N functions dis- tinguish essentially different reductions. If the A’ are replaced as independent variables by the moments A’, . . . I A”‘-‘, then (27) may be taken into a form analogous to ( 16).

A direct calculation confirms that the reduced equa- tions (21) are semi-Hamiltonian; that is, the charac- teristic speeds satisfy

(28)

for i, j, k ah distinct. The reduced equations are thus integrable by the generalised hodograph ~ansfo~a- tion [ 4 J . However, (28) does not imply that they pos- sess a Iocal Ham&on&n structure. Even if they do, it may be distinct from the reduction of the Hamiltonian structure of the full system ( 1).

4. Homogeneity and Galilean invariance

We may impose two further restrictions on these reduced systems. First, we require that the reduced system is Galilean invariant in the sense that

A’ -+ A’ + c rs v’ --+ vi + c, A0 --$ A0 (29)

or, ~uivaIently,

&Vi = 1, (30)

&A0 = 0, (31)

where 6 = Cz, d/dh’. Secondly, we require the func-

tions A0 and V” to be homogeneous in the A’; for bal- ance, A0 should be of weight 2, and the Vi of weight 1. Thus

RV’ = Vi, (32)

R$=$, (331

I I where R = Cz, A’~/~A’. We may now substitute in (26)) (27), eIiminating the second derivatives,

N

= xl j=l,j#i

hi - A’ i3Ao cTAo -- (Vi - vi)2 dAi iuj

Hence, either 8A”/3Ai = 0 or

N

z: 2 h’- A’ &A0

j=l .j#i (VJ _ vi)22=‘*

Similarly we may obtain

(34)

(35)

(36)

Together these are a system of 2N algebraic equations for 2N unknowns, the Vi and dAo/dA’. The solutions of this system are essentially unique. With two Rie- mann invariants, the only solution is the Zakharov re- duction with one pair (u, h), the classical shallow wa- ter equations. With 2N invariants, one solution is the Sakharov system with N pairs (u’, hi); we may dso obtain degenerations of these, with fewer independent variables, by letting ui --+ uj. All these reductions are tri-Hamiltonian, and their metrics are of Egorov type. It would be interesting to know whether any other such reductions exist.

5. A deformation of the Lax reductions

If the A” are the moments with respect to p of some distribution function f( n, p), then the generat- ing function A(p), analogous to (5) may be defined

by

(37)

Here, for definiteness, the principal value of the inte- gral has been taken. AIternatively, the contour can be indented so that p’ passes below p, giving a function A.+ analytic in the upper half p-plane; its boundary

J. Gibbons, SP. Tsoreu / Physics Letters A 211 (1996) 29-24 23

value on the axis is then A - i7rf. It is straightforward to show that if f(x,p) is advected along the charac- teristics

dx z =P*

dp dA“ x=--z

(39)

then the moments A” satisfy ( l), and A is advected along the same characteristics. It follows that any rela- tion such as f = F(A) is preserved by the dynamics. The generating function A will then satisfy the non- linear singular integral equation,

(40)

-CO

If F < 0 then we may describe the solutions of this equation in terms of a conformal mapping. The con- struction is to take the upper half of the complex A+-plane, and to cut it, along the curve Im(A+) = +rF(Re(A+)), from the real axis as far as A - irrF( A), where the parameter A of the branch point depends on x and t.

We may now define a function p( A+) uniquely by three properties:

(i) p is real and continuous on the real A+-axis and on the cut;

(ii) it is analytic in the cut half A+-plane; (iii) as /A+/ --+ 00, with Im(A+) > 0, then p =

A+ + G( l/A,.). The moments A” then are obtained from the asymptotics of A+(p) as p -+ oo,

A+ = p + 5 A”/$‘+‘. n=Q

(41)

We may similarly construct other functions H, with asymptotics H, = A$ /n + 0( 1 /A+). The equation of motion then takes the simple form

JP aH2 x+x=0. (42)

This equation may be expanded near infinity, to give the conservation laws, or near the branch point, to see that A is a Riemann invariant, with characteristic speed P(A).

This construction may be generalised to the case of N nonintersecting cuts given by Im(A+) =

-rFj (Re( A,) ), each starting on the real A+-axis and ending in a branch point Re(A+) = A’; these Aj are the N Riemann invariants of the system. If the cuts are taken along the N rays through the origin

a%(A+) = jr/ (N + 1 ), this construction gives the dispersionless Lax reduction. We have shown above that the most general reduction with N Riemann in- variants is similarly parameterised by N functions of a single variable, but it is easy to see that the waterbag and Zakharov reductions do not fit directly into the scheme described here, which forces the even num- bered moments AZ,, to be negative. It seems likely that the Riemann surface used by Geogdzhaev in Ref. [ 51 can be deformed to give a similar large family of reductions.

6. Conclusions

The most important open question concerning these systems is whether the compatibility conditions (3 1 ), (32), and their analogues for other integrable moment hierarchies, can themselves be regarded as integrable systems. This is related to the problem of finding the most general reduction of the type described in Section 5; of the reductions described in the introduction, only the dispersionless Lax equations belong to this family. Other problems include classifying those reductions of the Benney hierarchy which are Galilean or scaling invariant, but not both. For n = 2, the only Gal&an invariant solutions are the waterbag r~uctions, with their limit, the Zakharov reduction. Looking for homo- geneous solutions of ( 15)) however, leads to a nonlin- ear ODE which we have not been able to solve; it does not have the PainlevC property. One intriguing connec- tion has recently been discovered by Ferapontov and Fordy [ 8,9] who have related the system (26)) (27) to isometries of Stackel metrics, and have hence found some solutions. This approach needs further study.

Acknowledgement

S.P. Tsarev is supported in part by grants No. RKR 300 of the International Science Foundation, and No. 95-01 l-168 of the Russian ~undation for Scientific Research. This work was begun during his visit to the UK, supported by the Royai Society.

24 J. Gibbons, SP. Tsareu /Physics Letters A 211 (D96) 19-24

References

] I] D.J. Benney, Stud. Appl. Math. 52 ( 1973) 45. [2] B.A. Kupershmidt and Yu.1. Manin, Funct. Anal. App. 11

(1978) 188; 12 (1978) 20. ]3] V.E. Zakharov, Funct. Anal. Appl. 14 ( 1980) 89. [4] S.P l&rev, Sov. Math. Dokl. 31 (1985) 488. [5] V.V. Geogdzhaev, Tear. Mat. Fiz. 73 (1987) 255. [6] J. Gibbons and Y. Kodama, in: Proc. Singular limits of

dispersive waves, eds. N. Ercolani et al. (Plenum, New York, 1994).

171 J. Haantjes, hoc. K. Ned. Akad. Wet. 58 No. 2 (1955); fndag. Math. 17 No. 2 pp. 158-162.

[S] E. Ferapontov and AP Fordy, Separable Hamihonians and integrable systems of hydr~yn~c type, preprint ( 1995).

[ 9] E. Ferapontov and AI? Fordy, Nonhomogen~us systems of hydrodynamic type related to velocity dependent quadratic Hamiltonians, preprint ( 1995).