A numerical study of nonlinear dispersive wave models with SpecTraVVave
Date
2017-03-02
Authors
Kalisch, Henrik
Moldabayev, Daulet
Verdier, Olivier
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
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 in a very general form.
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 two solitary waves may 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.
Description
Keywords
Traveling Waves, Nonlinear dispersive equations, Bifurcation, Solitary waves
Citation
Kalisch, H. (2017). A numerical study of nonlinear dispersive wave models with SpecTraVVave. <i>Electronic Journal of Differential Equations, 2017</i>(62), pp. 1-23.
Rights
Attribution 4.0 International