Abstract
The fully nonlinear long wave equations describe the motion over a flat bottom of a two‐dimensional inviscid fluid with a free surface in a gravitational field in the long wave approximation. These equations are shown to possess an infinite number of conservation laws (in two space dimensions) in the form urn:x-wiley:00222526:media:sapm197453145:sapm197453145-math-0001 The conserved densities T and the fluxes −X and −Y are polynomials in the height h and the horizontal and vertical components of velocity, u and v, and also in integrals of powers of u. The method of proof is a modification of the method recently devised by D. J. Benney to prove that these same equations possess an infinite number of conservation laws (in one space dimension) in the form urn:x-wiley:00222526:media:sapm197453145:sapm197453145-math-0002 where T and X are polynomials in the height h and integrals of powers of u. Conservation laws which explicitly contain x and t are also given.
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