Math. Model. Nat. Phenom.
Volume 8, Number 5, 2013Bifurcations
|Page(s)||13 - 30|
|Published online||17 September 2013|
Global Bifurcation for the Whitham Equation
Department of Mathematical Sciences, Norwegian University of
Science and Technology, 7491
2 Department of Mathematics, University of Bergen Postbox 7800, 5020 Bergen, Norway
⋆ Corresponding author. E-mail: firstname.lastname@example.org
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.
Mathematics Subject Classification: 35Q53 / 35C07 / 45K05 / 65M70 / 76B15
Key words: Whitham equation / global bifurcation / traveling waves / spectral projection / cosine transform
© EDP Sciences, 2013
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