Further decay results on the system of NLS equations in lower order Sobolev spaces

Abstract

The initial value problem of a system of nonlinear Schrödinger equations with quadratic nonlinearities in two space dimensions is studied. We show there exists a unique global solution for this initial value problem which decays like $t^{-1}$ as $t\to +\infty$ in $\mathbf{L}^\infty (\mathbb{R}^2)$ for small initial data in lower order Sobolev spaces.