A long wave approximation for capillary-gravity waves and the Kawahara equation
by Tatsuo Iguchi  

Vol. 2 No. 2 (2007) P.179~P.220

ABSTRACT

The Kawahara equation is a higher-order Korteweg-de Vries equation with an additional fifth order derivative term. It was derived by Hasimoto as a model of capillary-gravity waves in an infinitely long canal over a at bottom in a long wave regime when the Bond number is nearly one third. In this paper, we give a mathematically rigorous justication of this modeling and show that the solution of the Kawahara equation approximates that of the full problem of capillary-gravity waves in an appropriate sense for a long time interval. We also consider the case where the bottom is not at and derive coupled Kawahara type equations whose solution approximates that of the full problem in that case.

KEYWORDS
Capillary-gravity waves, water waves, Kawahara equation, long wave approximation

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 35Q35, 35Q53, 76B15, 76B45

MILESTONES

Received: 2004-11-15
Revised : 2005-06-02
Accepted:

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