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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