On Local Wellposedness of the Schrödinger-Boussinesq System

Abstract

In this paper we prove that the Schrödinger-Boussinesq system with solution $(u,v,$  $(-\partial_{xx})^{-\frac12} v_t)$ is locally wellposed in $ H^{s}\times H^{s}\times H^{s-1}$, $s\geqslant-{1}/{4}$. The local wellposedness is obtained by the transformation from the problem into a nonlinear Schrödinger type equation system and the contraction mapping theorem in a suitably modified Bourgain type space inspired by the work of Kishimoto, Tsugawa. This result improves the known local wellposedness in $ H^{s}\times H^{s}\times H^{s-1}$, $s>-{1}/{4}$ given by Farah.