Conservation Laws for the Fully Nonlinear Long Wave Equations

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 formThe 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 formwhere 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.