无限深水中具有指数密度变率的周期性永形内波渐进解析解

采用同伦分析方法研究了一系列有限振幅的周期深水驻波问题.水密度在垂直方向的分布可以是变化的，假设为指数连续分布.提出一种新形式的偏微分方程作为辅助方程，获得解的新的表达形式来满足底部的边界条件和无限大的刚性假设.给出了解的表达式中系数的递推关系和周期海洋内波形成的永久驻波的显式表达式.得到垂直方向和水平方向的全局收敛解，揭示了密度变量和内波幅度间的关系.同伦分析方法对求解具有指数密度率周期性的永形波解是一致有效的.

Analytical Solutions of Periodic Stationary Internal Waves in Infinitely Deep Water With Exponential Vertical Density Distribution

A train of periodic deep water stationary waves with finite amplitudes were investigated analytically with the homotopy analysis method. The vertical distribution of water density was considered as variable in a continuous exponential trend. A new form of partial differential equations were proposed as the auxiliary equations and the new-form solution expressions were obtained in order to match the level boundary condition at the bottom and the hypothetical infinitely rigid condition. The detailed recursive relation of the coefficient in the solution expression was given and the explicit expressions of the permanent stationary periodic internal waves were presented. The convergent series solutions were obtained for the global domain both in vertical and horizontal directions. The relation between the density variable and the internal wave amplitude was revealed.