非线性色散波模型的数值研究：SpecTraVVave
Numerical Study of Nonlinear Dispersive Wave Models with SpecTraVVave
AI总结：
本文研究非线性色散方程中行波解的数值近似，通过SpecTraVVave代码分析Whitham方程、修改后的Benjamin-Ono方程和Benjamin方程的行波解特性。
摘要：
在非线性色散演化方程中，非线性和色散的相互作用导致了多种有趣现象。本文聚焦于此类方程行波解的数值近似。我们开发了一种专用的Python代码，能够计算非线性色散方程的行波解。SpectraVVave代码结合了连续方法和谱投影，用于计算该方程的稳态对称解的近似。该代码被用于多种情况以理解行波解。第一个案例是Whitham方程，数值证据表明主要分支有三个关键点：转折点、稳定性反转点和终端点，对应于尖点波。第二个案例是修改后的Benjamin-Ono方程，研究了两个孤立波的相互作用，发现较小的波可能被湮灭。第三个案例是Benjamin方程，其中存在两个竞争的色散算子。在此情况下，周期行波解的分支可能在非线性区域交叉并连接到高处。
英文摘要：
In nonlinear dispersive evolution equations, the competing effects of nonlinearity and dispersion make a number of interesting phenomena possible. In the current work, the focus is on the numerical approximation of traveling-wave solutions of such equations. We describe our efforts to write a dedicated Python code which is able to compute traveling-wave solutions of nonlinear dispersive equations of the general form \begin{equation*} u_t + [f(u)]_{x} + \mathcal{L} u_x = 0, \end{equation*} where $\mathcal{L}$ is a self-adjoint operator, and $f$ is a real-valued function with $f(0) = 0$. The SpectraVVave code uses a continuation method coupled with a spectral projection to compute approximations of steady symmetric solutions of this equation. The code is used in a number of situations to gain an understanding of traveling-wave solutions. The first case is the Whitham equation, where numerical evidence points to the conclusion that the main bifurcation branch features three distinct points of interest, namely a turning point, a point of stability inversion, and a terminal point which corresponds to a cusped wave. The second case is the so-called modified Benjamin-Ono equation where the interaction of two solitary waves is investigated. It is found that is possible for two solitary waves to interact in such a way that the smaller wave is annihilated. The third case concerns the Benjamin equation which features two competing dispersive operators. In this case, it is found that bifurcation curves of periodic traveling-wave solutions may cross and connect high up on the branch in the nonlinear regime.
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