The problem of two-dimensional capillary-gravity waves on an inviscid fluid of finite depth interacting with a linear shear current is considered. The shear current breaks the symmetry of the irrotational problem and supports simultaneously counter-propagating waves of different types: Korteweg de-Vries (KdV)-type long solitary waves and wave-packet solitary waves whose envelopes are associated with the nonlinear Schrödinger equation. A simple intuition for the broken symmetry is that the current modifies the Bond number differently for left- and right-propagating waves. Weakly nonlinear theories are developed in general and for two particular resonant cases: the case of second harmonic resonance and long-wave/short-wave interaction. Traveling-wave solutions and their dynamics in the full Euler equations are computed numerically using a time-dependent conformal mapping technique, and compared to some weakly nonlinear solutions. Additional attention is paid to branches of elevation generalized solitary waves of KdV type: although true embedded solitary waves are not detected on these branches, it is found that periodic wavetrains on their tails can be arbitrarily small as the vorticity increases. Excitation of waves by moving pressure distributions and modulational instabilities of the periodic waves in the resonant cases described above are also examined by the fully nonlinear computations.

中文翻译：

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有限深度水面上的毛细重力孤立波与线性剪切流相互作用

考虑了二维毛细重力波在有限深度的无粘性流体上与线性剪切流相互作用的问题。剪切电流打破了无旋问题的对称性，同时支持不同类型的反向传播波：Korteweg de-Vries (KdV) 型长孤立波和包络线与非线性薛定谔方程相关的波包孤立波。破坏对称性的一个简单直觉是，电流对向左和向右传播的波以不同的方式修改了债券数。弱非线性理论通常是针对两种特定谐振情况开发的：二次谐波谐振和长波/短波相互作用的情况。全欧拉方程中的行波解及其动力学使用时间相关的共形映射技术进行数值计算，并与一些弱非线性解进行比较。额外注意 KdV 型高程广义孤立波的分支：虽然在这些分支上没有检测到真正的嵌入孤立波，但发现随着涡度的增加，它们尾部的周期波列可以任意小。在上述共振情况下，通过移动压力分布和周期波的调制不稳定性对波的激发也通过完全非线性计算进行了检查。虽然在这些分支上没有检测到真正的嵌入孤立波，但发现随着涡度的增加，它们尾部的周期性波列可以任意小。在上述共振情况下，通过移动压力分布和周期波的调制不稳定性对波的激发也通过完全非线性计算进行了检查。虽然在这些分支上没有检测到真正的嵌入孤立波，但发现随着涡度的增加，它们尾部的周期性波列可以任意小。在上述共振情况下，通过移动压力分布和周期波的调制不稳定性对波的激发也通过完全非线性计算进行了检查。