Premium
Conservation Laws for the Fully Nonlinear Long Wave Equations
Author(s) -
Miura Robert M.
Publication year - 1974
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm197453145
Subject(s) - conservation law , inviscid flow , dimension (graph theory) , mathematics , nonlinear system , space (punctuation) , gravitational field , mathematical analysis , motion (physics) , mathematical physics , field (mathematics) , equations of motion , classical mechanics , physics , pure mathematics , quantum mechanics , philosophy , linguistics
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.