Soliton dynamics for the generalized Choquard equation

TL;DR

This paper studies the behavior of solitons in a generalized nonlinear Schrödinger equation with non-local nonlinearities, focusing on the dynamics and properties of solutions where ground states are not well-understood.

Contribution

It introduces analysis of soliton dynamics for the generalized Choquard equation, addressing the challenges posed by non-uniqueness and degeneracy of ground states.

Findings

Analysis of soliton stability and dynamics

Insights into the structure of ground states

Implications for non-local nonlinear Schrödinger equations

Abstract

We investigate the soliton dynamics for a class of nonlinear Schr\"odinger equations with a non-local nonlinear term. In particular, we consider what we call {\em generalized Choquard equation} where the nonlinear term is $(|x|^{\theta-N} * |u|^p)|u|^{p-2}u$. This problem is particularly interesting because the ground state solutions are not known to be unique or non-degenerate.