Henrik Kalisch, Daulet Moldabayev, Olivier Verdier
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.
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 |
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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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