Mathematical Modelling of Natural Phenomena

Research Article

Global Bifurcation for the Whitham Equation

M. Ehrnströma1 and H. Kalischa2 c1

a1 Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway

a2 Department of Mathematics, University of Bergen Postbox 7800, 5020 Bergen, Norway

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.

(Online publication September 17 2013)

Key Words:

  • Whitham equation;
  • global bifurcation;
  • traveling waves;
  • spectral projection;
  • cosine transform

Mathematics Subject Classification:

  • 35Q53;
  • 35C07;
  • 45K05;
  • 65M70;
  • 76B15

Correspondence

c1 Corresponding author. E-mail: henrik.kalisch@math.uib.no

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