Study on the interaction between long and short waves

点击翻译按钮获取中文摘要

(1) The resonant interaction between long and short waves is considered for two typicalscalings, that is, SH type and SK type. By applying the reductive perturbation method to capillary-gravity waves on water of finite depth, the higher-order approximations arederived for each scaling. From themodulational stability characteristics of the uniformwavetrain solutions of the higher-order equations, it has been found that while for SHtype, the higher-order terms affect significantly the stability, for SK type, they hardly do. Further, from the numerical computations for the higher-order equations, it has beenfound that (i) For the higher-order approximation of SH type, a coupled wave of longand short waves analogous to the SH type soliton is formed along with a despersive wave decaying in time from localized initial data.(ii) Under the periodic boundary condition, while for SH type scaling the higher-order terms affect significantly on the long-time evolution of solutions, for SK type scaling they do little.(2) Propagation and generation of waves are considered in a two-layr fluid with a free surface flowig with the speed V over a localized topography. When V is close to the phase speed c for the long wave limit of the internal wave mode and there is a surfacewave packet with a group velocity close to c, it is shown that the governing equationsare SH type long-short wave resonance equations the extra terms expressing the effect of the difference between V and the resonant wave speed and that of topoqraphy. This equations have the coupled soliton solutions far from the localized topography as well as various steady solutions trapped in the topography. By examining the interactions of an incident soliton and a trapped wave numerically and analytically, it has been found that according to the values ot V-c and the parameters of the incident soliton, it is transmitted, reflected and amplified, or fis ioned.