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November/December 2009 Traveling waves for the Whitham equation
Mats Ehrnström, Henrik Kalisch
Differential Integral Equations 22(11/12): 1193-1210 (November/December 2009). DOI: 10.57262/die/1356019412


The existence of traveling waves for the original Whitham equation is investigated. This equation combines a generic nonlinear quadratic term with the exact linear dispersion relation of surface water waves of finite depth. It is found that there exist small-amplitude periodic traveling waves with sub-critical speeds. As the period of these traveling waves tends to infinity, their velocities approach the limiting long-wave speed $c_0$. It is also shown that there can be no solitary waves with velocities much greater than $c_0$. Finally, numerical approximations of some periodic traveling waves are presented. It is found that there is a periodic wave of greatest height $\sim 0.642 h_0$. Periodic traveling waves with increasing wavelengths appear to converge to a solitary wave.


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Mats Ehrnström. Henrik Kalisch. "Traveling waves for the Whitham equation." Differential Integral Equations 22 (11/12) 1193 - 1210, November/December 2009.


Published: November/December 2009
First available in Project Euclid: 20 December 2012

zbMATH: 1240.35449
MathSciNet: MR2555644
Digital Object Identifier: 10.57262/die/1356019412

Primary: 35Q53
Secondary: 35C07 , 45K05 , 76B15 , 76B25

Rights: Copyright © 2009 Khayyam Publishing, Inc.


Vol.22 • No. 11/12 • November/December 2009
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