On well-posedness results for the cubic–quintic NLS on T3

Abstract

We consider the periodic cubic–quintic nonlinear Schrödinger equation (CQNLS)$(i{\partial }_t+\Delta )u={\mu }_1{\left|u\right|}^{2}u+{\mu }_2{\left|u\right|}^{4}u$on the three-dimensional torus $T^3$ with ${\mu }_1,{\mu }_2\in R\setminus \left\{0\right\}$. As a first result, we establish the small data well-posedness of for arbitrarily given ${\mu }_1$ and ${\mu }_2$. By adapting the crucial perturbation arguments in Zhang (2006) to the periodic setting, we also prove that is always globally well-posed in $H^1\left(T^3\right)$ in the case ${\mu }_2>0$.