Center stable manifold for ground states of nonlinear Schr\"odinger equations with internal modes

TL;DR

This paper establishes the existence of a center stable manifold around unstable ground states of nonlinear Schrödinger equations, demonstrating asymptotic stability even with internal modes present.

Contribution

It proves the existence of a local center stable manifold and asymptotic stability for solutions near unstable ground states, allowing for internal modes.

Findings

Existence of a local center stable manifold around ground states.

Asymptotic stability of solutions on the manifold.

Inclusion of internal modes in the analysis.

Abstract

We study the dynamics of solutions of nonlinear Schr\"odinger equation near unstable ground states. The existence of the local center stable manifold around ground states and the asymptotic stability for the solutions on the manifold is proved. The novelty of our result is that we allow the existence of internal modes.