Study on the interaction between long and short waves

(1)考虑了SH型和SK型两种典型尺度的长短波共振相互作用。将约化摄动法应用于有限水深的毛细重力波，得到了每种尺度下的高阶近似解。从高阶方程一致波列解的调制稳定性特征可以看出，对于SH型，高阶项对稳定性影响很大，而对于SK型，高阶项对稳定性影响不大。此外，通过对高阶方程的数值计算，发现：(I)对于SH型的高阶近似，一个类似于SH型孤子的长波和短波的耦合波随着局域初值随时间衰减的绝望波一起形成。(Ii)在周期边界条件下，对于SH型标度，高阶项对解的长时间演化影响很大，而对于SK型标度，高阶项对解的长时间演化影响很小。(2)波的传播和产生被考虑在具有速度V的局部地形上具有自由表面的两层流体中。当V接近内波模式长波极限的相速度c，且存在群速度接近c的表面波包时，控制方程是SH型长短波共振方程，附加项表示V与共振波速之差和地形的影响。该方程有远离局域地形的耦合孤子解和陷于地形中的各种定常解。通过数值和解析研究入射孤子和陷波的相互作用，发现根据V-c的大小和入射孤子的参数，它被传输、反射、放大或分离。

(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.