Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity

Liu, Yue; Wang, Xiao-Ping; Wang, Ke

doi:10.1090/S0002-9947-05-03763-3

Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity
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by Yue Liu, Xiao-Ping Wang and Ke Wang PDF

Trans. Amer. Math. Soc. 358 (2006), 2105-2122 Request permission

Abstract:

This paper is concerned with the inhomogeneous nonlinear Shrödinger equation (INLS-equation) \begin{equation*}i u_t + \Delta u + V(\epsilon x) |u|^p u = 0, \; x \in {\mathbf R}^N. \end{equation*} In the critical and supercritical cases $p \ge 4/N,$ with $N \ge 2,$ it is shown here that standing-wave solutions of (INLS-equation) on $H^1({\mathbf R}^N)$ perturbation are nonlinearly unstable or unstable by blow-up under certain conditions on the potential term V with a small $\epsilon > 0.$

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