Transverse instability for nonlinear Schrödinger equation with a linear potential

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 [18] 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 $L>0$ by using the argument of the previous paper [26].