Comparaison entre modèles d'ondes de surface en dimension 2

Abstract

On the basis of the principle of conservation of mass and fundamental principle of dynamics, we find the Euler equation enabling us to describe the asymptotic models of waves propagation in shallow water in dimension 1. To describe the waves propagation in dimension 2, a linear perturbation of the KdV equation is used by Kadomtsev and Petviashvili. But that does not specify if the equations thus obtained derive from the Euler equation, that is shown by Ablowitz and Segur. We will insist, in same manner, on the fact that the equations of KP-BBM can be also obtained starting from the Euler equation, and up to what point they describe the physical model. In a second time, we take again the method introduced in the article of Bona, Pritchard and Scott in which the solutions of long water waves in dimension 1, namely the solutions of KdV and BBM, are compared, to show here that the solutions of KP-II and KP-BBM-II are close for a time scale inversely proportional to the waves amplitude. From the point of view of modelling, it will be clear according to the first part, that only the model described by KP-BBM-II is well posed, and since from the physical point of view, KP-II and KP-BBM-II describe the small amplitude long waves when the surface tension is neglected, it is interesting to compare them. Moreover, we will see that the method used here remains valid for the periodic problems.