Transverse instability for nonlinear Schrödinger equation with a linear potential (Q286096)

In this paper, using the smallness of the line standing wave of the equation NEWLINE\[NEWLINEi\partial_t u=-\Delta u+V(x)u-|u|^{p-1}u, \quad(t,x,y)\in \mathbb{R}\times\mathbb{R}\times\mathbb{T}_LNEWLINE\]NEWLINE and the expansion of the standing wave with respect to the parameter \(\omega\), the author weakens the nonlinear structure of the Lyapunov functional around the line standing wave of the equation. Therefore, they can evaluate a value of the integral and make a close investigation into the stability for all exponents \(p\geq 2\), i.e., for all \(L>0\).