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Transverse instability for nonlinear Schrödinger equation with a linear potential (1505.00421v2)
Published 3 May 2015 in math.AP
Abstract: In this paper we consider the transverse instability for a nonlinear Schr\"odinger 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 Schr\"odinger equation with a linear potential. We regard the standing wave of nonlinear Schr\"odinger equation on $\mathbb{R}$ as a line standing wave of nonlinear Schr\"odinger equation on $\mathbb{R} \times \mathbb{T}_L$. We show the stability of line standing waves for all $L>0$.