Is free-surface hydrodynamics an integrable system?

A strong argument is found in support of the integrability of free-surface hydrodynamics in the one-dimensional case. It is shown that the first term in the perturbation series in powers of nonlinearity is identically equal to zero, the consequences of which are discussed as well. 1. It is well known that the equations describing an ideal fluid with a free surface in a gravity field are completely integrable in several important limiting cases. Integrability occurs for long waves in shallow water (KdV [1 ] and KP [2] equations, the Boussinesq approximation [3], Kaup's approximation [4], the HolmCamassa approximation [5 ] ) and for spectrally narrow wave trains in a fluid of arbitrary depth (nonlinear SchrSdinger equation [6]) . The weakly nonlinear motion of the fluid in the absence of a gravity field is integrable as well [ 7 ]. It is very natural to formulate a conjecture that an arbitrary one-dimensional motion of an ideal fluid in a gravity field is integrable. In this article we give arguments in support of this conjecture. We will consider weakly nonlinear waves on the surface of a fluid o f infinite depth and study their simplest resonant interactions, and we will show that the amplitude of this process is zero. Given the current stage of mathematical physics there are no effective general methods for checking and proving integrability for the nonlinear wave Hamiltonian systems. Proving nonintegrability is a much easier problem. Following Poincarr, one can do that by analysing the perturbation series in powers of the nonlinearity [8]. Terms of this series being limited on their resonant manifolds are identified with the "amplitudes of the nonlinear interactions" in the wave system. Nonintegrability is a quite evident fact. To have nonintegrability, it is enough to prove that at least one of these amplitudes is nonzero. As the complexity o f calculations increases significantly with the order of nonlinearity, much information can be extracted from the consideration of the first (lowest order) nontrivial nonlinear process. For instance, nonintegrability of the nonlinear SchrSdinger equation for d >t 2 is a trivial fact due to the nonzero amplitude for the process 2 --* 2 wave scattering. This scattering is trivial for the integrable case d = 1. One may verify (albeit with much effort) that the amplitude o f the first nontrivial scattering 3 --* 3 is identically zero in this case. 0375-9601/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SSDI0375-9601 (94 )00381-X A.L Dyachenko, V.E. Zakharov / Physics Letters A 190 (1994) 144-148 145 2. A one-dimensional potential flow of an ideal incompressible fluid with a free surface in a gravity field fluid is described by the following set of equations, ~xx + ~z~ = 0 (4~z ~ O,z --, c~) , 1 2 ?lt d'llxOx = ~zlz=q, ~bt "]~(t~x + t~ 2) "{g~ = Olz=r/. (1) Here q(x, t) is the shape of the surface, ~b(x, z , t ) is the stream function and g is the gravitational constant. As was shown in Ref. [9 ], the variables t/(x, t) and ~/(x, t) = ~6 (x, z, t)[z=~ are canonically conjugated, and their Fourier transforms satisfy the equations O~k gH Oqk gH Ot ~rlT,' Ot ~uf," Here H = K + U is the total energy of the fluid with the following kinetic and potential energy terms, ,/ K= f f -00 A Hamiltonian can be expanded in an infinite series in powers of the characteristic wave steepness ktlk << 1 (see Refs. [9,10]), H = Ho + H1+ H2 + . . . . It is convenient to introduce a normal complex variable ak, ~lk = X / ~ / 2 g (ak + aLk), ~k = --iv/-~g/mk (ak -aLk). Here COk = V ~ is the dispersion law for the gravity waves. This variable satisfies the equation Oak . 8 H 0-'-7"~ 1--~--~-~.tJak = O,