Radiation by weakly nonlinear shallow-water solitons due to higher-order dispersion

Abstract

Nonlinear asymptotic equations for shallow-water waves, with account of high-order dispersion and surface tension [generalized Boussinesq system (GBS) and generalized Korteweg--de Vries (GKdV) equation] are derived. Regular expansions of these equations in powers of a dispersion parameter lead to different types of already used KdV-type equations, in particular to fifth- and higher-order KdV equations. It is shown that the fifth-order KdV equation describes in a good approximation the shape of a shallow-water soliton, but is insufficient for the consistent description of soliton resonant radiation. The latter is caused by the resonant interaction between the soliton and a plane wave with the phase velocity equal to the soliton velocity. It is shown that the resonant 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 radiation significantly changes the results obtained earlier by means of the fifth-order KdV equation. A soliton damping caused by its resonant radiation is investigated by means of the GKdV equation.