On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves
Abstract
Many studies of weakly nonlinear surface waves are based on socalled reduced integrodifferential equations. One of these is the widely used Zakharov fourwave equation for purely gravity waves. But the reduced equations now in use are not Hamiltonian despite the Hamiltonian structure of exact water wave equations. This is entirely due to shortcomings of their derivation. The classical method of canonical transformations, generalized to the continuous case, leads automatically to reduced equations with Hamiltonian structure. In this paper, attention is primarily paid to the Hamiltonian reduced equation describing the combined effects of four and fivewave weakly nonlinear interactions of purely gravity waves. In this equation, for brevity called fivewave, the nonresonant quadratic, cubic and fourthorder nonlinear terms are eliminated by suitable canonical transformation. The kernels of this equation and the coefficients of the transformation are expressed in explicit form in terms of expansion coefficients of the gravitywave Hamiltonian in integralpower series in normal variables. For capillarygravity waves on a fluid of finite depth, expansion of the Hamiltonian in integralpower series in a normal variable with accuracy up to the fifthorder terms is also given.
 Publication:

Journal of Fluid Mechanics
 Pub Date:
 August 1994
 DOI:
 10.1017/S0022112094004350
 Bibcode:
 1994JFM...272....1K
 Keywords:

 Differential Equations;
 Gravity Waves;
 Hamiltonian Functions;
 Mathematical Models;
 Power Series;
 Surface Waves;
 Integral Equations;
 Numerical Analysis;
 Perturbation;
 Transformations (Mathematics);
 Physics (General)