标题： 孤立波-平均流相互作用在强非线性色散浅水波中 显示英文标题

标题： Solitary wave-mean flow interaction in strongly nonlineardispersive shallow water waves

摘要： 孤立波与Serre-Green-Naghdi（SGN）方程的缓慢变化平均背景或流之间的相互作用使用惠瑟姆调制理论进行了研究。在孤立波极限下，三个SGN-惠瑟姆调制方程的确切形式——其中两个用于解耦的平均水平速度和深度，一个用于孤立波振幅场——被精确获得。尽管这三个方程不可对角化，但完整系统限制于平均方程的简单波时，在黎曼不变量下被对角化。利用黎曼不变量来解析描述孤立波与稀疏波和弥散激波（DSW）的迎面碰撞及超车碰撞，导致了孤立波被捕获或通过平均流的情况。对于超车碰撞的分析结果证明，基于DSW拟合方法的简化近似方法在孤立波振幅的二阶精度上准确，超过了Korteweg-de Vries近似的一阶精度。分析结果还准确预测了SGN DSW的孤立波边缘振幅和速度。分析结果与完整的SGN方程的相应数值解进行了有利的比较。由于SGN方程模型可以模拟强非线性长重力波在平坦底部的双向传播，这里提出的分析描述了浅水波中的大振幅孤立波与平均流的相互作用。 显示英文摘要

摘要： The interaction of a solitary wave and a slowly varying mean background or flow for the Serre-Green-Naghdi (SGN) equations is studied using Whitham modulation theory. The exact form of the three SGN-Whitham modulation equations -- two for the mean horizontal velocity and depth decoupled from one for the solitary wave amplitude field -- are obtained exactly in the solitary wave limit. Although the three equations are not diagonalizable, the restriction of the full system to simple waves for the mean equations is diagonalized in terms of Riemann invariants. The Riemann invariants are used to analytically describe the head-on and overtaking interactions of a solitary wave with a rarefaction wave and dispersive shock wave (DSW), leading to scenarios of solitary wave trapping or transmission by the mean flow. The analytical results for overtaking interactions prove that a simpler, approximate approach based on the DSW fitting method is accurate to the second order in solitary wave amplitude, beyond the first order accurate Korteweg-de Vries approximation. The analytical results also accurately predict the SGN DSW's solitary wave edge amplitude and speed. The analytical results are favourably compared with corresponding numerical solutions of the full SGN equations. Because the SGN equations model the bi-directional propagation of strongly nonlinear, long gravity waves over a flat bottom, the analysis presented here describes large amplitude solitary wave-mean flow interactions in shallow water waves.