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Global Bifurcation for the Whitham Equation

Published online by Cambridge University Press:  17 September 2013

M. Ehrnström  and
H. Kalisch
M. Ehrnström
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
H. Kalisch*
Affiliation:
Department of Mathematics, University of Bergen Postbox 7800, 5020 Bergen, Norway
*
⋆ Corresponding author. E-mail: henrik.kalisch@math.uib.no
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Abstract

We prove the existence of a global bifurcation branch of 2π-periodic, smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset of solutions in the global branch contains a sequence which converges uniformly to some solution of Hölder class Cα, α < 1/2. Bifurcation formulas are given, as well as some properties along the global bifurcation branch. In addition, a spectral scheme for computing approximations to those waves is put forward, and several numerical results along the global bifurcation branch are presented, including the presence of a turning point and a ‘highest’, cusped wave. Both analytic and numerical results are compared to traveling-wave solutions of the KdV equation.

Keywords

Whitham equationglobal bifurcationtraveling wavesspectral projectioncosine transform
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 5: Bifurcations , 2013 , pp. 13 - 30
Copyright
© EDP Sciences, 2013

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