Transverse instability for nonlinear Schr\\"odinger equation with a\n linear potential

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