Differential and Integral Equations

Traveling waves for the Whitham equation

Mats Ehrnström and Henrik Kalisch

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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.

Article information

Differential Integral Equations, Volume 22, Number 11/12 (2009), 1193-1210.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35C07: Traveling wave solutions 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30] 76B25: Solitary waves [See also 35C11]


Ehrnström, Mats; Kalisch, Henrik. Traveling waves for the Whitham equation. Differential Integral Equations 22 (2009), no. 11/12, 1193--1210. https://projecteuclid.org/euclid.die/1356019412

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