Author： Gallay Thierry Schneider Guido
Publisher： Royal Society of Edinburgh
ISSN： 1473-7124
Source： Proceedings Section A: Mathematics - Royal Society of Edinburgh, Vol.131, Iss.4, 2001-08, pp. : 885-898
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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.
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