Rigorous derivation of Korteweg-De Vries-type systems from a general class of nonlinear hyperbolic systems

In this paper: we study the long wave approximation for quasilinear symmetric hyperbolic systems. Using the technics developed by Joly-Metivier-Rauch for nonlinear geometrical optics, we prove that under suitable assumptions the long wave limit is described by KdV-type systems. The error estimate if the system is coupled appears to be better. We apply formally our technics to Euler equations with free surface and Euler-Poisson systems. This leads to new systems of KdV-type.