Global well-posedness for the coupled system of Schr\"odinger and Kawahara equations

Abstract

We study the local and global well-posedness for the coupled system of Schr\"odinger and Kawahara equations on the real line. The Sobolev space $L^{2} \times H^{-2}$ is the space where the lowest regularity local solutions are obtained. The energy space is $H^1 \times H^2$. We apply the Colliander-Holmer-Tzirakis method [7] to prove the global well-posedness in $L^2 \times L^2$ where the energy is not finite. Our method generalizes the method of Colliander-Holmer-Tzirakis in the sense that the operator that decouples the system is nonlinear.