RADIATION BY WEAKLY NONLINEAR SHALLOW-WATER SOLITONS DUE TO HIGHER-ORDER DISPERSION

Nonlinear asymptotic equations for shallow-water waves, with account o f high-order dispersion and surface tension [generalized Boussinesq sy stem (GBS) and generalized Korteweg-de Vries (GKdV) equation] are deri ved. Regular expansions of these equations in powers of a dispersion p arameter lead to different types of already used KdV-type equations, i n particular to fifth- and higher-order KdV equations. It is shown tha t the fifth-order KdV equation describes in a good approximation the s hape of a shallow-water soliton, but is insufficient for the consisten t description of soliton resonant radiation. The latter is caused by t he resonant interaction between the soliton and a plane wave with the phase velocity equal to the soliton velocity. It is shown that the res onant radiation can be correctly described only by equations that take into account dispersive effects to all orders in a region beyond the soliton. The GKdV equation possesses this property and a theory of the soliton resonant radiation, based on the GKdV equation, is developed. It is shown that an account for the full dispersion law for the radia tion significantly changes the results obtained earlier by means of th e fifth-order KdV equation. A soliton damping caused by its resonant r adiation is investigated by means of the GKdV equation. [S1063-651X(98 )03710-6].