THE SHALLOW WATER EQUATIONS : CONSERVATION LAWS AND SYMPLECTIC GEOMETRY

We consider the system of nonlinear differential equations governing shallow 
water waves over a uniform or sloping bottom. By using the hodograph method we construct solutions, conservation laws, and Bocklund transformations for these equations. We show that these constructions are canonical relative to a symplectic form introduced by Manin.


I. INTRODUCTION.
In recent years there has been a revival of interest in the subject of shallow water wave equations.Numerous articles have been appearing on the subject.This can be explained by all the recent work done on completely integrable systems, their soli- ton solutions and conservation laws.In this article we are incorporating our work on shallow water wave equations which was published in [I], [2] and was announced in [33.
We consider the following system of equations governing two-dimensional shallow water waves over a uniform (homogenous case, B O) or sloping (non-homogenous case, B z O) bottom: u t + UUx + hx gB, (1.1a) where 0 X is the horizontal coordinate, t is the time, u(t, x) is the hori- zontal component of the velocity at the point x at time t and h(t, x) is the depth of the free surface below the point x at time t In both cases the nonlinear equations can be reduced to linear ones in the hodograph plane, upon which the deriva- tion of conservation laws and solutions of these equations is based.It was interesting for us to see that from a single equation one could construct explicit solutions of the homogenous and no,homogenous systems as well as conservation laws and BNcklund trans- formations associated with them [I].
Manin [4] cast the homogenous system into a Hamiltonian formalism and the >ymplecti K eet of the homogenous system (B 0) has been studied by Kupershmidt and Manin [5], and Lebedev and Manin [6].We extend Manin's construction to the sloping bottom case (nonhomogenous system, 8 0) and show that the solution space of the above mentioned wave equation (from which we obtain explicit solutions, conservation laws, and Bcklund transformations) is isotropic relative to the accompanying symplectic form; thus, we simultaneously prove that the conserved quantities are in involution and that the BNcklund transformations are sYmPlectic, i.e., canonical.
Whether the isotropic space of the solutions of the above mentioned wave equation is _La.grangianis an important question.This is, in fact, the problem of 9pmplete inte- r[ability of an infinite dimensional Hamiltonian system.See pp. 7and 9 of Cavalcante and McKean [7] on this important issue.
The system (I.I) with B 0 is a pair of quasi-linear partial differential equa- tions with no explicit (t, x) dependence.Hence, for any region where the Jacobian u x h t u t hx is non-zero, (I.i) can be transformed into an equivalent linear system by interchanging the roles of dependent and independent variables, (u,h) and (t, x), respectively.This is a so-called hodograph transformation.Since the system (I.I) is homogenous for B 0 from u jt h u t -ix h h x -jr u h t jx u we see that the highly nonlinear factor cancels through in (I.I), and we arrive at the following linear differential equations x h -tu + Uth, (2. Ib) where V is the gradient operator (/u, /h) on the (u, h)-plane.By eliminating x in (2.1) we obtain the linear equation whose solutions can easily be found in standard tables.
Since the application of the hodograph transformation depends on the assumption that j 0, solutions for which j 0 cannot be obtained by this method.Such solutions are called simple waves, and they are important tools for the solution of flow problems; for instance, wave-breaking occurs when 0 due to the multivalued- ness, i.e., shocks.As an example, the solution u 2x/3t, h (x/3t) 2, (2.3)   which is found in Nutku [8] by a scale-invariance argument, represents a simple wave.
So, we could not possibly obtain it by the hodograph method.
The system of equations (2.1) can also be written in the following equivalent form These, in return, suggest the existence of potentials (u, h) and (u, h) satisfying u -ht, h x-ut, (2.5a) u x ut, h -t.

NONHOMOGENOUS CASE (SLOPING BOTTOM).
In 1958, Carrier and Greenspan [9] applied Riemann's characteristic forms together with a hodograph transformation to the system (1.1) with 8 0 and obtained solutions for this system in terms of the solutions of the wave equation (2.7) above.Again, we miss the so-called simple-wave solutions which correspond to the case in which v and h are functionally dependent.
It is interesting to note that by letting B 0 in (3.1) we obtain h2 (u2h + with the corresponding solution (2.9) in Section 2 of this article.That is, we do not get all the solutions of the homogenous system by setting the nonhomogenous term to zero in the solutions of the nonhomogenous problem.This is quite contrary to what happens in the linear case.
We note that the equations (2.7) and (3.2) have the same form.Hence, after the necessary relabelling of the variables, a solution of any of them can be used to gen- erate a solution to either of the problems: homogenous and nonhomogenous.In this way, we find a correspondence between the non-simple wave solutions of the two systems.
This correspondence can be thought of as a Bcklund transformation between the homo- genous and nonhomogenous problems.By using the linear nature of the same wave equa- tion, we can also consruct auto-Bcklund transformations for each system.These are, of course, nothing but uPerposiion principles for the nonlinear systems under con- sderation: Given two solutions of (i.i), add the corresponding solutions of (3.2), and then construct the solution of (I.I) corresponding to this sum.
In Akyildiz [I], we were also able to construct polynomial conserved quantities for the nonhomogenous system (I.I) in the form f dx, where is a solution of the same wave equation (3.2). (The transformation equation between this article and [13 is h c2).In this way, via the solutions of (3.2), we establish correspondence between conservation laws, nonsimple wave solutions and Bcklund transformations of both the homogenous and nonhomogenous systems.(Perhaps we ought to call (3.2) the moduli equation for the system (I.i)since the solution space of (3.2) parameterizes the non-simple solutions as well as the conservation laws of the system (I.I)). 4. SYMPLECTIC GEOMETRY.
Finally, we shall cast the system (I.I) into Hamiltonian form and show that the solution space of the wave equation (3.2) is isotropic relative to the accompanying symplectic form (Poisson bracket); hence, proving that all the constructions carried out above are canonical.After absorbing the nonhomogenous term gBt in (l.la) as (u gBt) t + UUx + hx O,  VAJVB dx (4.4) for two functions A and B of the variables v and h It is easy to verify that (4.4) satisfies the Jacobi identity.We assume that the boundary conditions u 0, h 0 are satisfied at x 0 and oo.Since v u gBt, in order to have meaning- ful space integrals in (4.4), we must not have v's appearing on their own in our expressions (because t may become arbitrarily large), v's must always come multi- plied with u's or c's.This was an essential point in constructing conserved quantities for the system (1.1) in our work [I], cf.p. 1727.Now, we shall show that the Poisson bracket (4.4) vanishes on the solution space of the wave equation (3.2), which parameterizes both the non-simple wave solutions and conservation laws of the system (1.1):Let A and B be two solutions of (3.2).
The integrand in (4.4) is Since (AvDB h + AhDBv)dX AvdBh + AhdBv. of the arguments v and h This finishes the proof that the Poisson bracket (4.4) vanishes on the solution space of the wave equation (3.2), from which we obtain solutions, Bcklund transformations and conservation laws for the system (1.1).Thus, we have simultaneously proved that the Bcklund transformations above are symplectic and that conserved quantities found in Akyildiz [I] for the non- homogenous system (1.1) are in involution relative to the symplectlc structure intro- the Hamiltonian operator, H the Hamiltonian, D differentiation with respect to x and V the gradient operator in (v, h)-space with v u gBt.