Differential and Integral Equations

Traveling waves for the Whitham equation

Mats Ehrnström and Henrik Kalisch

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Permanent link to this document

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

Export citation