Instability of standing waves to the inhomogeneous nonlinear Schrödinger equation with harmonic potential

Abstract

We study the instability of standing-wave solutions $e^{i\omega t}\phi_{\omega}(x)$ to the inhomogeneous nonlinear Schrödinger equation \[ i\varphi_t=-\triangle\varphi+|x|^2\varphi-|x|^b|\varphi |^{p-1}\varphi, \quad x\in\mathbb{R}^N, \] where $b \gt 0$ and $\phi_{\omega}$ is a ground-state solution. The results of the instability of standing-wave solutions reveal a balance between the frequency $\omega$ of wave and the power of nonlinearity $p$ for any fixed $b \gt 0$.