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THE LONG WAVE LIMIT FOR A GENERAL CLASS OF 2D QUASILINEAR HYPERBOLIC PROBLEMS

ABSTRACT This paper is dedicated to the study of long wave approximation for quasilinear symmetric hyperbolic systems. We use ansatz with multiple scales in order to capture both dispersive and nonlinear effects, occuring at the same time scale as a relatively small transverse correction of the unidirectional propagation. Using the techniques developped by Joly, Métivier and Rauch for nonlinear geometric optics, we prove that the exact solutions split up into two wave packets, one moving to the left and one to the right, and whose amplitude satisfy a system of Kadomtsev-Petviashvili equations. Whether the limit system is coupled or not is critical to the error estimate obtained. In the last part, we formally apply these techniques to derive both uncoupled and coupled KP systems from the Euler equations with free surface in the context of water waves.