Henrik Kalisch
Daulet Moldabayev
Olivier Verdier
2017-03-02
<p>In nonlinear dispersive evolution equations, the competing effects of nonlinearity and dispersion make a number of interesting phenomena possible. We describe a Python package which computes traveling-wave solutions of equations of the form <span class="math-tex">\(u_t + [f(u)]_{x} + \mathcal{L} u_x = 0\)</span>, where <span class="math-tex">\(\mathcal{L}\)</span> is a self-adjoint operator, and <span class="math-tex">\(f\)</span> is a real-valued function with <span class="math-tex">\(f(0) = 0\)</span>. We use a continuation method coupled with a spectral projection. For the Whitham equation, we obtain numerical evidence that the main bifurcation branch features three distinct points of interest: a turning point, a point of stability inversion, and a terminal point corresponding to a cusped wave. We also found that two solitary waves may interact in such a way that the smaller wave is annihilated. We also observed that bifurcation curves of periodic traveling-wave of the Benjamin solutions may cross and connect high up on the branch in the nonlinear regime.</p>
<p><br>
</p>
This research was supported by the Research Council of Norway on grant no. 213474/F20 and by the J C Kempe Memorial Fund on grant no. SMK-1238.
https://doi.org/10.5281/zenodo.398855
oai:zenodo.org:398855
Zenodo
https://arxiv.org/abs/arXiv:1606.01465
https://doi.org/
info:eu-repo/semantics/openAccess
Creative Commons Attribution 4.0 International
https://creativecommons.org/licenses/by/4.0/legalcode
Electronic Journal of Differential Equations, 2017(62), 1–23, (2017-03-02)
Traveling Waves
Nonlinear dispersive equations
Solitary waves
Bifurcation
A Numerical Study of Nonlinear Dispersive Wave Models with SpecTraVVave
info:eu-repo/semantics/article