Electron. J. Differential Equations, Vol. 2017 (2017), No. 62, pp. 1-23.

A numerical study of nonlinear dispersive wave models with SpecTraVVave

Henrik Kalisch, Daulet Moldabayev, Olivier Verdier

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.

Submitted November 7, 2016. Published March 2, 2017.
Math Subject Classifications: 35C07, 35Q53, 45J05, 65M70
Key Words: Traveling Waves; nonlinear dispersive equations; bifurcation; solitary waves.

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Henrik Kalisch
Department of Mathematics
University of Bergen, P.O. Box 7800
5020 Bergen, Norway
email: henrik.kalisch@math.uib.no
Daulet Moldabayev
Department of Mathematics
University of Bergen, P.O. Box 7800
5020 Bergen, Norway
email: daulet.moldabayev@math.uib.no
  Olivier Verdier
Department of Mathematics and Statistics
University of Umea, Sweden.
Department of Computing, Mathematics and Physics
Western Norway University of Applied Sciences, Norway
email: olivier.verdier@hvl.no

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