Stability and instability of standing waves for a generalized Choquard equation with potential

TL;DR

This paper investigates the stability and instability of standing waves in a generalized Choquard equation with potential, analyzing how potential and parameters influence wave behavior in a three-dimensional setting.

Contribution

It provides new results on the stability criteria of standing waves for a generalized Choquard equation with potential, considering various conditions on the potential and parameters.

Findings

Conditions for stability of standing waves identified

Conditions for instability of standing waves established

Impact of potential and parameters on wave behavior analyzed

Abstract

We are going to study the standing waves for a generalized Choquard equation with potential: $$ -i\partial_t u-\Delta u+V(x)u=(|x|^{-\mu}\ast|u|^p)|u|^{p-2}u, \ \ \hbox{in}\ \ \mathbb{R}\times\mathbb{R}^3, $$ where $V(x)$ is a real function, $0<\mu<3$, $2-\mu/3<p<6-\mu$ and $\ast$ stands for convolution. Under suitable assumptions on the potential and appropriate frequency $\omega$ , the stability and instability of the standing waves $u=e^{i \omega t}\varphi(x)$ are investigated .