connections between free-surface systems of hydrodynamical type and generalized kinetic equations

Volume 144, number 4,5 PHYSICS LETTERS A 5 March 1990

CONNECTIONS BETWEEN FREE-SURFACE SYSTEMS OF HYDRODYNAMICAL TYPE AND GENERALIZED KINETIC EQUATIONS

John GIBBONS Department of Mathematics, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK

and

Boris A. KUPERSHMIDT The University of Tennessee Space Institute, Tullahoma, TN 37388, USA

Received 9 June 1989; revised manuscript received 4 December 1989; accepted for publication 20 December 1989 Communicated by A.P. Fordy

The dynamics of moments of basic integrable two-dimensional free-surface hydrodynamical systems, such as the Benney hier- archy, has been found in some cases to result also from Vlasov-type kinetic equations. We find a general hodograph transformation which directly relates systems of hydrodynamical type and generalized kinetic equations.

1. Introduction

The subject of this paper is the dual description - hydrodynamical and kinetic - of certain Lie algebras of infinite-dimensional dispersionless dynamical systems. We illustrate this first with the Benney system, origi- nally introduced as a description of long waves on shallow perfect fluid with a free surface. This system may be written in three very different forms. The first of these is the original (2 + 1 )-dimensional hydrodynamical form [1]

y t ~

ut=uux+hx-uy J uxdy, u=u(x,y, t), o

h

where u is the horizontal component of the fluid velocity, while h is the height of the free surface above the horizontal bottom, y=O. The subscripts t, x and y denote the partial derivatives with respect to t, x and y respectively. The second form is an infinite-component system for the moments An in ( 1 + 1 )-dimensional space [11,

An,t=An+l,x+rtan_lAo,x, n~71+ , A n = A n ( X , t). (1.2)

This system results from ( 1.1 ) upon the introduction of the moments of the velocity u with respect to y,

h

An= f Undy . ( 1 . 3 )

o

0375-9601/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland) 21 1


The same system (1.2) also results from a (2+ 1 )-dimensional kinetic (Vlasov) equation [2]

f=Pf~-Ao,x£, f=f(x,p, t), (1.4)

for the single-particle distribution function f if the moments are defined by the formula

A,= i p"fdp. (1.5)

Thus two different systems, the hydrodynamical ( 1.1 ), and the kinetic (1.4), are mapped into the same moment system (1.2), via the maps (1.3) and (1.5) respectively. Moreover, a similar diagram of systems and maps arises for the higher analogues of the systems (1.2), ( 1.1 ) [ 3 ], and (1.4) [ 4 ]. It is then natural to expect that this diagram can be completed into a commutative diagram, that is, that there exists a map from the hydrodynamical to the kinetic system (or vice versa), as indicated by

hydrodynamica[ equations

? or ? Moment (1.6) equations

J . / ' ~ . 5 )

kinetic equations

Indeed, a connection between (1. l ) and (1.4) has been exhibited by Zakharov [ 2 ]. In the more general case this result remains partially true, and our first result is the construction of a hodograph transformation which maps generalised kinetic equations in N+ 1 dimensions into (N+ 1 )-dimensional free-surface hydrodynamical systems; the qualification "partially" referring to the fact that these systems possess only a scalar velocity vari- able u, so that the moments are naturally indexed by 7/+ rather than (7/+)N with N> 1. This result is derived in section 2. The case of multicomponent, or vector, systems is discussed in section 4.

We now turn to Hamiltonian aspects of the diagram (1.6). It is easy to see that the systems (1.1), (1.2)

(1.7)

O,=O/Ox, (1.8)

(1.9)

(1.10)

[5], (1.8) [3] and (1.9) [4] are all Hamiltonian, and further, that the [ 3 ], and (1.5) [4] are canonical. In section 3 we show that the hodograph map constructed in

and (1.4) can be written, respectively, in the forms y

ut=(umHm)x__Uy f dy(mum_lHm)x ' Hm:= 5H S A m '

o

ht= (mAm_lHm)x

(we sum on repeated indices),

A n , t = B n m ( H m ) , B n m : = n A n + m _ l O + O m A n + m _ l ,

and

8H f \S f}p

where

H = ½ (A2 +AoZ),

and it is known that the forms (1.7) maps (1.3)

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section 2 is also canonical, between the Hamiltonian structures (1.7) and (1.9), as well as between other im- portant Hamiltonian structures. The commutative diagram (1.6) thus becomes Hamiltonian.

We conclude this introduction by demonstrating the hodograph transformation between the systems ( 1.1 ) and (1.4). Set

z=z(p,x , t ):= f f (x ,p ' , t ) dp' , O<~z<~Ao(x,t)= f ( x , t , p ' ) d p ' , (1.11) -- oz~ -- oo

and let

p=p(z, x, t)

be the inverse of the map p ~ z, for fixed x and t. Then P

Oz f P' Ot - f ( x , , t) dp' (by (1.11))

(1.12)

P P

f. d.-A0x - - o o m o o

(by (1.4)) (1.13)

which is (since, by ( 1.11 ), Oz/Op=f) P p

O Z f p , 0 2 Z . , , 07- ~.~OZ" , Ot - ~ dp -Aoxf=P ox - f v: aP -Ao,xf.

-- oo -- oo

(1.14)

Now, multiplying (1.14) by - 1/fwhich by (1.13) can be written as

1 Op Oz/Op Oz'

(i.15)

we get

( OzOp)+Op" Oz[(Ozl-l ] OzOp=p I dp' +Aox OtOz O x ~ ) ~ OxL\~p') Op 3 ' "

- -oo

(1.16)

Now, if

b=~(a,c)

and

(1.17a)

a=~,(b,c) (1.17b)

are a pair of mutually inverse maps, depending on the parameters c, then, on differentiating the identity

b=~(~(b,c) ,c)

with respect to % we obtain the standard result

Ob Oa Ob Oa Oc~ = - Oc-~ " (1.18)

Using this formula, and transforming the integral

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P P

Oz , dp') f ( ) ~ d p ( = f ( ) f ' (1.19a) - - o o - - ~

into

i ( ) dz ' , (1.19b) 0

eq. (1.16) becomes

Op Op Op i ( Op' ~ O~ =p -~x + Oz - ~x,I dz' +ao,x ; (1.20) 0

here Ao is given by

Ao= i f dp ,

and it satisfies

Ao,t = i f dp - - o o

AO

= ( ! p d Z ) x (by (1.19)). (1.21)

Evidently the system (1.20), (1.21) is identical with (1.1), provided we identify u with p, y with z, and h with Ao. Finally,

p'~fdp= pndz (by (1.19))

h

= fu"dy , (1.22) 0

so that we obtain the commutativity of the diagram (1.6) in the case of the Benney system.

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2. The hodograph transformation for scalar systems

Let xl, ..., XM be the Euclidean coordinates on ~M. Denote by d the ring of smooth functions in x, the moments A, (defined by (1.3) ), and their x-derivatives. The scalar hydrodynamical systems have the form [6]

Y t *

ut =Pmliumug +Rm urn- uy j dy [Pmli(um),i + Qmu m ] , (2. la) 0

ht=PmliAm,i + QmAm , (2.1b)

where ( ),r'=O( )/Oxi, and the coefficients {Pmli, Qm, Rml meZ+} are arbitrary elements of d . It is easy to verify that the moment map (1.3) sends the hydrodynamical system (2.1) into the following (autonomous) evolution system in the moment space,

An.t = nRmAn+ m- l + QmAn+ m + PmliAn+ m,i • (2.2)

Let us now consider the kinetic equation

ft = - ( RmprnjOp + emli pmf, i + Qmpmf , (2.3)

with the same P's, Q's and R's, except that the moments A, are now to be understood via (1.5). Obviously, t~

the moment map (5) sends the kinetic equation (2.3) into the same dynamical system (2.2) in the moment space.

Now we will construct a map from the kinetic system (2.3) into the hydrodynamical system (2.1), with the additional property that

p"fdp= u" dy, YnsZ+; (2.4) - - ~ 0

which establishes the commutativity of the triangular diagram (1.6). Heuristically, we see that we must identify p with u, and f d p with dy. Let us therefore follow the same route as we used for the Benny case, in the introduction. We again define

z = z ( p , x , t ) : = 3 f ( p ' , x , t ) dp ' , 0~<z~<Ao= f ( p ' , x , t ) dp ' , (2.5)

with the inverse map

p=p( z , x , t ) . (2.6)

Then

Oz f (p ,x , t ) = Op (2.7)

so that

pnf dp= pn Oz ~ p d p = p n d z . (2.8) - - ~ - - ~ 0

Hence, upon identifying u with p, y with z, and h with Ao, the hydrodynamical and kinetic moments (1.3) and (1.5) will be identified also. Therefore the coefficients P, Q and R in (2.1) and (2.3) will be identified as well. With this in mind, we start with the kinetic system (2.3) and get

215


Op Op Oz m

Ot Oz Ot P

Op Oz

Op Oz

(by (1.18))

f f ( p ' , x , t ) dp' (by(2.5))

P

f dp' [ - (Rmp"nf ' )p, +Pmiip"~f~+Qmp'mf ' ] --oo

(by (2.3))

(by (2.7)) OP I m OZ I/ ,m 02Z' ' = - Oz -RmP ~p+ f dp'LPmliP Op~-x i "~ Qmlj"trnOZ ~]~JJ

(by (1.18))

P p

Op r Pmli(P m OZ Op tm OZtx~'] Op . f ;z f '

--oo -- oo P

m m Op Op f dp' (P Op'mOp' ~ ,m) Oz' =Rmp +PmliP ~ - O-z \ mli Off OXi +~dmP J ~

. . . . . Op Op dz' P m l i +QmP 'm (by (1.19)) =/~mP -t-rmliP ~ - O-z ~ 0

and this is precisely (2.1 a). Since h is identified with Ao, ( 2. lb ) follows from the n = 0 case of the system ( 2.2 ). Thus the desired map is

y= f tip', x, t) dp', u=p, h= f(p', x, t) alp', (2.9) --oo --~

with the inverse map

p=u, f= (uy) -~ . (2.10)

3. Hamiltonian aspects

Suppose the system (2.2) in the moment space is Hamiltonian, with the coefficients P, Q, R being given by differential operators (in x) acting on the vector {6H/~)Am ] mEZ+}. Since the moment maps ( 1.3 ) and ( 1.5 ) are injective, the Hamiltonian structure in the d-space is uniquely lifted into corresponding Hamiltonian structures in the { u, h} or fspaces respectively. In such a case the general hodograph maps (2.9) and (2.10) become canonical. We shall now write down, for two families of Hamiltonian structures appearing in the theory of ( 1 + 1 )-dimensional integrable systems, the corresponding Hamiltonian structures in the three related forms.

The first family, which arises out of the Poisson bracket on BR 2 of the form

{F, G} =pr(FxGp-FpGx) , r eZ+ , (3.1)

leads [6] to the Hamiltonian matrix

B~nm=nAn+m_l +,O + OmAn+m- l +r , (3.2)

which for r = 0 is just the matrix (1.8). Writing the corresponding motion equations

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An,t =Brnm(Hm)

in the form

An,t = nAn+m- 1 +rnm,x q-An+m- 1 +rmHm,x +An+m- 1 +r,xmHm ,

we find that in this case

R m = H . . . . . . Q m = ( m + l - r ) H m + ~ . . . . . P m = P m l l = ( m + l - - r ) H m + l - r •

Hence the systems (2.1) and (2.3) become, respectively

Y / I

ut = ( m + 1 - r ) H m + l _ r t t m U x ' + H . . . . . ttm--uy" 1 dy [ ( m + 1 - r ) H m + l _ r U m ] x ,

0

ht = [ (rn+ 1 - r )Hm+l_~Am]x ,

and

f = - [ (Hm_~p"-~)xpr f]p +{[ (m+ 1 - r )Hm+l_rpm-r]p~ f}x

~. __[ ( ~ ) x ( p r j ~ lp_~I Q_~)p(prf) ]x.~_(~_~)p(!Qrjg)x__(~_~)x(Prf)p,

where we have used the formula

5H - - = H m p m . ~f

For r=0, eqs. (3.6) and (3.9) become (1.7) and (1.9) respectively. The second family of Hamiltonian structures is given by the formula [ 6 ]

B (.~P) = ( oln'-}- fl)An+mO"l-O ( olm"l- fl)An+rn ,

where ot and fl are arbitrary constants. In particular,

B1 _R (l,o) nm --~nm • Again, writing the motion equations in the moment space in the form

A.,t = [ ( o m + m ) A . + m O + O ( a m + fl)A.+,.] (H,,,)

= ?lAn+m OlHm,x TAn+ m (olm .-I- 2fl)Hm, x +A.+m,~(am + f l )Hm,

we see that

R m = a H m _ l , x , Q m = ( O t m + 2 f l ) H . . . . P,,,=Pml~ = ( o t m + f l ) H , . ,

so that the evolution systems (2.1) and (2.3) become, respectively,

Y

ut = [ u ' ( o~m + fl)Ux + U"+ lotO ] (Hm) - u y j dy[ ( u " ) x ( a m + fl) + um( o~m + 2 fl)O ] ( H, . ) , 0

ht= [A,n.x( a m + fl) + Am( otm+ 2fl)O ] ( H, . ) ,

and

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

217

Volume 144, number 4,5 PHYSICS LETTERS A

ftt "~ -- ( OLnm-l,xpmf)p + ( am"l- fl)pmnmfx + ( a m + 2fl)Hm,xpmf

t, 8f}p(Pf)x+'8-g-ff<+°zlt-Sf)i,x(Pf)+2fl(-~-f)x f

For a = 1 and fl=0, formulae (3.14) and (3.16) become, respectively, (3.6) and (3.8) with r= 1.

5 March 1990

(3.15)

(3.16)

4. Multicomponent systems

In this section we explain why, in the multicomponent case, the hydrodynamical and the kinetic systems cannot be related directly, although they do map into the same systems in the space of moments.

The general form of free-surface hydrodynamics with N components us, 1 ~s<~N, is [6 ]

Y

us.t =P<iU°Us,i + R,~su ° - Us,y j dy [P<i(u~),i + Qou °] , 0

ht = P< iAo, i + Q~A,~ , (4.1)

where we use multi-index notation (as (7/+)N)

U%=U~I...U~v u for cr=(al,...,aN)e(77 + ) N , (4.2)

h

A~:= I u°dy" (4.3) 0

Again, the moment map (4.3) sends the hydrodynamical system (4.1) into the system:

Au, t=#,R,~A,~+~,_I +Q,,A, ,+u+P<,A,,+u,i , #e(T/ + ) N . (4.4)

This moment system (4.4) may be derived also from the kinetic equation

0 f - - Op~ (R'~P~J)+Q~P~f+P<iP~f'i (4.5)

if the moments are instead defined by

oo

o=f ,4.6,

Since the integrals (4.3) and (4.6) defining the hydrodynamical and kinetic moments are taken over the Eu- clidean spaces of dimensions 1 and N respectively, we cannot expect a transformation between the variables

{Ul , ... , UN, y} and {Pl,-.., PN,J} (4.7)

to satisfy the condition

h cx~

r u d d y = f . . . f p~ ' f dp l . . . dpu . (4.8)

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The correct approach is, following Zakharov [2] , to identify u with p, as before, but to allow f t o be a dis-

t r ibut ion, rather than a function; it seems likely that the ansatz

h t l

f= j ~(p-u(y)) dy (4.9)

0

takes solut ions o f (4.1) into solut ions o f (4.5) , but we have not proved this.

Acknowledgement

One of us (B.A.K.) was par t ia l ly suppor ted by the Nat ional Science Foundat ion . We are also grateful to the

Science and Engineering Research Council for f inancial assistance.

References

[ 1 ] D.J. Benney, Stud. Appl. Math. 52 (1973) 45. [2] V.E. Zakharov, Funct. Anal. Appl. 14 (1980) 15. [ 3 ] B.A. Kupershmidt and Yu.I. Manin, Funct. Anal. Appl. 11 (1977) 188; 12 ( 1978 ) 20. [4] J. Gibbons, Physica D 3 ( 1981 ) 503. [ 5 ] B.A. Kupershmidt, in: Springer Lecture notes in mathematics, Vol. 775 (Springer, Berlin, 1980) p. 162. [6] B.A. Kupershmidt, Phys. Lett. A 121 (1987) 167.

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connections between free-surface systems of hydrodynamical type and generalized kinetic equations

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