Strong instability of standing waves for nonlinear Schrödinger equations with harmonic potential

Abstract

We study strong instability of standing waves $e^{i\omega t}{\varphi }_{\omega }(x)$ for nonlinear Schrödinger equations with $L^2$-supercritical nonlinearity and a harmonic potential, where ${\varphi }_{\omega }$ is a ground state of the corresponding stationary problem. We prove that $e^{i\omega t}{\varphi }_{\omega }(x)$ is strongly unstable if ${\partial }_{\lambda }^{2}E({\varphi }_{\omega }^{\lambda }){|}_{\lambda =1}\le 0$, where $E$ is the energy and $v^{\lambda }(x)={\lambda }^{N/2}v(\lambda x)$ is the $L^2$-invariant scaling.

arXiv comment10 pages