On the Cauchy problem for Schrödinger–improved Boussinesq equations

Abstract

The Cauchy problem for a coupled system of Schrödinger and improved Boussinesq equations is studied. Local well-posedness is proved in $L^2 (\mathbf{R}^n)$ for $n \le 3$. Global well-posedness is proved in the energy space for $n \le 2$. Under smallness assumption on the Cauchy data, the local result in $L^2$ is proved for $n = 4$.