Resonance excitation of nonlinear dispersive waves

The action of an external wave perturbation on a dispersive medium is investigated in the case where the perturbation spectrum contains harmonics that satisfy the dispersion relation for linear waves of the medium. A perturbation theory is developed that is based on the multiple time-scale method and describes the resonant growth of the induced wave as well as the effects of nonlinear saturation of this growth for the cases of four well-kno~ 1 nonlinear equations. The analytic results are compared with those of numerical computations for the perturbed Korteweg--de Vries equation.
The processes of propagation of nonlinear waves in dispersive media have been attracting considerable attention of investigators. The use of the method of the inverse problem of the scattering theory (for example, see [I]) has made it possible to obtain analytical solutions of the basic nonlinear equations describing the propagation and evolution of waves in different physical media [2]. One of the directions taken in the investigation of nonlinear waves is the exploitation of various methods of the perturbation theory, which are intended to give an approximate solution of the problem in the case when the nonlinear wave equation contains small" perturbations" terms. In this article we develop a perturbation theory describing the excitation of waves in nonlinear dispersive media under the action of an external wave perturbation assuming that the spectrum of the wave contains harmonics whose frequency and wave vectors satisfy a certain resonance condition depending on the characteristics of the medium.