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Transverse instability for nonlinear Schrödinger equation with a linear potential. (English) Zbl 1344.35142

In this paper, using the smallness of the line standing wave of the equation \[ i\partial_t u=-\Delta u+V(x)u-|u|^{p-1}u, \quad(t,x,y)\in \mathbb{R}\times\mathbb{R}\times\mathbb{T}_L \] 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 \(L>0\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B32 Bifurcations in context of PDEs
35B35 Stability in context of PDEs
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