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On symmetry and decay of traveling wave solutions to the Whitham equation

Banff International Research Station for Mathematical Innovation and Discovery

The Whitham equation is a nonlocal, nonlinear dispersive wave equation introduced by G. B. Whitham as an alternative wave model equation to the Korteweg-de Vries equation, describing the wave motion at the surface on shallow water. The existence of supercritical solitary wave solutions to the Whitham equation has been shown by Ehrnström, Groves, and Wahlén in 2012. We prove that any such solution decays exponentially, is symmetric and has exactly one crest. Moreover, the structure of the Whitham equation allows to conclude that conversely any classical, symmetric solution constitutes a traveling wave. In fact, the latter result holds true for a large class of partial differential equations sharing a certain structure.

Mathematics; Fluid mechanics; Partial differential equations; Fluid dynamics

video/mp4

Author affiliation: NTNU

2017-05-03

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61461

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/