Instability of Standing Waves to the Inhomogeneous Nonlinear Schr\"odinger Equation with Harmonic Potential

TL;DR

This paper investigates the instability of standing waves in an inhomogeneous nonlinear Schrödinger equation with harmonic potential, highlighting the interplay between wave frequency and nonlinearity power.

Contribution

It provides a novel analysis of the conditions leading to instability of standing waves in an inhomogeneous NLS with harmonic potential, emphasizing the balance between frequency and nonlinearity.

Findings

Standing waves are unstable under certain conditions.

Instability depends on the relationship between frequency and nonlinearity power.

The results reveal a critical balance point for stability.

Abstract

We study the instability of standing-wave solutions $e^{i\omega t}\phi_{\omega}(x)$ to the inhomogeneous nonlinear Schr\"{o}dinger equation $$i\phi_t=-\triangle\phi+|x|^2\phi-|x|^b|\phi|^{p-1}\phi, \qquad \in\mathbb{R}^N, $$ where $ b > 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 > 0. $