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whitham equation

Mathematical Modelling of Natural Phenomena (1)

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Global Bifurcation for the Whitham Equation
M. Ehrnström, H. Kalisch
Journal:

Mathematical Modelling of Natural Phenomena / Volume 8 / Issue 5 / 2013

Published online by Cambridge University Press:

17 September 2013, pp. 13-30

Print publication:

2013
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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 ofsolutions in the global branch contains a sequence which converges uniformly to somesolution of Hölder class Cα, α < 1/2. Bifurcation formulas are given, as well as some properties along theglobal bifurcation branch. In addition, a spectral scheme for computing approximations tothose waves is put forward, and several numerical results along the global bifurcationbranch are presented, including the presence of a turning point and a ‘highest’, cuspedwave. Both analytic and numerical results are compared to traveling-wave solutions of theKdV equation.