Justification of and Long-wave Correction to Davey-stewartson Systems from Quadratic Hyperbolic Systems

We prove that the Davey-Stewartson approximation (which degenerates into a cubic Schrödinger equation in 1D) furnishes a good approximation for the exact solution of a wide class of quadratic hyperbolic systems. This approximation remains valid for large times of logarithmic order. We also consider the general case where the polarized component of the mean field needs not to be well-prepared. This is possible by adding to the Davey-Stewarston approximation a long-wave correction, which consists of a wave freely propagated by the long-wave operator associated to the original system.