Transverse instability for nonlinear Schrödinger equation with a linear potential

Abstract

In this paper we consider the transverse instability for a nonlinear Schrödinger equation with a linear potential on $\mathbb{R}\times {\mathbb{T}}_L$, where $2\pi L$ is the period of the torus ${\mathbb{T}}_L$. Rose and Weinstein showed the existence of a stable standing wave for a nonlinear Schrödinger equation with a linear potential. We regard the standing wave of nonlinear Schrödinger equation on $\mathbb{R}$ as a line standing wave of nonlinear Schrödinger equation on $\mathbb{R}\times {\mathbb{T}}_L$. We show the stability of line standing waves for all $L>0$.