本文探讨了非线性水波在线性剪切流和水平电场这两类物理效应下的行波
解，具体的研究方法包括渐近分析和数值计算，研究的重点被放在这些物理效应
对水波系统的影响，尤其是对色散关系和线性稳定性的影响，以及由此而产生的
各种新型非线性波动现象。对于线性剪切流下的水弹性波，经典的Stokes 展开法
被用来计算周期行波的三阶近似解，在此基础上，通过求解流场中粒子的运动方
程，我们能够给出近似的粒子运动轨迹。对于水平电场下的内孤立波，我们发展
了非线性水波理论中的算子展开法，采用Dirichlet-Neumann (DN) 算子在长波近似下的渐近展开方法得到了内孤立波的弱非线性模型以及强非线性模型，对这
些模型进行理论分析和数值计算可以帮助我们获得对孤立波定性或定量的了解。
在数值计算方面，针对这两个问题我们分别采用了保形变换和边界积分的方法，
将原始的自由边界问题转化为固定边界上的非线性积分方程，然后分别利用快
速Fourier 变换(FFT) 和有限差分格式，采用Newton 迭代法计算了行波解。数值
计算展现出了丰富多彩的结果，我们发现线性剪切流明显地修改了系统的色散
关系，会产生一类具有强色散性的新型水跃结构，还会使水弹性波发展成为多值
函数形状的剖面。此外流场受到剪切流的影响还会形成多族封闭式的“猫眼”型
结构，这些结构的包络线代表了一类特殊的轨迹，位于这些轨迹上的粒子将会产
生竖直方向的净位移。另一方面水平电场表现出了对Rayleigh-Taylor (RT) 不稳
定性的抑制效果，在一些特殊的参数条件下我们计算出了一类新型的孤立波解，
其明显受到了电场对色散关系的影响，此外我们还计算出了和水弹性波类似的
水跃结构。

This paper studies the traveling wave solutions of nonlinear water waves via asymptotic analysis and numerical calculation, under two kinds of physical effects of linear shear current and horizontal electric field. The focus of the research is on the impact of these physical effects on the water wave systems, especially on the dispersion relation and linear stability, as well as the resulting various new types of nonlinear wave
phenomenons. For the hydroelastic waves under a linear shear current, classical Stokes expansion is applied to calculate the third-order approximation of periodic travelling waves. On this basis, we can obtain the approximate particle trajectories by solving the equations of particle motion. For the internal solitary waves under a horizontal electric field, we develop the operator expansion method of nonlinear water wave theory,
and use the asymptotic expansion method of Dirichlet-Neumann (DN) operator under long wave approximation to obtain weak nonlinear models and strong nonlinear models. Theoretical analysis and numerical calculation of these models can help us to obtain qualitative or quantitative understanding of the solitary waves. In the aspect of numerical calculation, we use conformal map and boundary integral method to transform the original free boundary problem into the nonlinear integral equations on a fixed boundary, and apply the fast Fourier transform (FFT) and finite difference scheme to obtain the travelling wave solutions respectively by Newton iterative method. The numerical calculation shows a variety of colorful results. We find that the linear shear current can apparently modify the dispersion relation, and gives rise to a new kind of hydraulic jump with strong dispersion, and makes the hydroelastic wave develop into an overhanging profile. In addition, the linear shear current also leads to ”cat’s eyes” structures, whose envelopes represent a special kind of trajectories, on which the particles have a net displacement in the vertical direction. On the other hand, the horizontal electric field shows a suppression of Rayleigh-Taylor (RT) instability. In some special cases, we can obtain a new kind of solitary waves, which is strongly affected by the modification of dispersion relation due to the electric field. In addition, we also calculate a hydraulic jump similar to those appearing in hydroelastic waves.