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KP description of unidirectional long waves. The model case

Published online by Cambridge University Press:  12 July 2007

Thierry Gallay
Affiliation:
Mathématiques, Université de Paris-Sud, 91405 Orsay, France (thierry.gallay@math.u-psud.fr)
Guido Schneider
Affiliation:
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany (guido.schneider@uni-bayreuth.de)

Abstract

The Kadomtsev–Petviashvili (KP) equation can be formally derived as an envelope equation for three-dimensional unidirectional water waves in the limit of long waves. As a first step towards a mathematical justification, we consider here a two-dimensional Boussinesq equation, which is a realistic model for three-dimensional water waves. Using rigorous estimates, we show that part of the dynamics of the KP equation can be found approximately in the two-dimensional Boussinesq equation. On the other hand, there exist initial data for the KP equation such that the corresponding solutions of the two-dimensional Boussinesq equation behave in no way according to the KP prediction. We expect that similar results hold for the three-dimensional water wave problem too.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2001

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