On well-posedness results for the cubic-quintic NLS on $\mathbb{T}^3$

TL;DR

This paper proves small data well-posedness for the cubic-quintic nonlinear Schrödinger equation on the three-dimensional torus, and establishes global well-posedness in H^1 for positive quintic coefficient, adapting perturbation methods to the periodic setting.

Contribution

It extends well-posedness results for the cubic-quintic NLS on ^3, including small data and global well-posedness in H^1 for ^3 with positive quintic term, using adapted perturbation techniques.

Findings

Small data well-posedness for arbitrary ^3 coefficients.

Global well-posedness in H^1 for ^3 when _2 > 0.

Perturbation arguments adapted to the periodic setting.

Abstract

We consider the periodic cubic-quintic nonlinear Schr\"odinger equation \begin{align}\label{cqnls_abstract} (i\partial_t +\Delta )u=\mu_1 |u|^2 u+\mu_2 |u|^4 u\tag{CQNLS} \end{align} on the three-dimensional torus $\mathbb{T}^3$ with $\mu_1,\mu_2\in \mathbb{R} \setminus\{0\}$. As a first result, we establish the small data well-posedness of \eqref{cqnls_abstract} for arbitrarily given $\mu_1$ and $\mu_2$. By adapting the crucial perturbation arguments in \cite{zhang2006cauchy} to the periodic setting, we also prove that \eqref{cqnls_abstract} is always globally well-posed in $H^1(\mathbb{T}^3)$ in the case $\mu_2>0$.