Strong instability of standing waves for nonlinear Schr\"odinger equations with harmonic potential

TL;DR

This paper investigates the conditions under which standing wave solutions to nonlinear Schrödinger equations with harmonic potential become strongly unstable, focusing on the role of energy scaling derivatives.

Contribution

It establishes a criterion for strong instability based on the second derivative of the energy functional with respect to a specific scaling parameter.

Findings

Strong instability occurs when the second derivative of energy at the ground state is non-positive.

The criterion applies to $L^2$-supercritical nonlinearities with harmonic potentials.

Provides a mathematical condition linking energy scaling to wave instability.

Abstract

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