On asymptotic stability of ground states of some systems of nonlinear Schr\"odinger equations

TL;DR

This paper extends the theory of asymptotic stability of ground states from scalar nonlinear Schrödinger equations to certain systems, addressing the challenges posed by non-commutative symmetry groups.

Contribution

It introduces a novel approach using an appropriate system of modulation coordinates for systems with non-commutative symmetry groups.

Findings

Established asymptotic stability for specific NLS systems

Developed a new modulation coordinate framework

Addressed non-commutative symmetry challenges

Abstract

We extend to a specific class of systems of nonlinear Schr\"odinger equations (NLS) the theory of asymptotic stability of ground states already proved for the scalar NLS. Here the key point is the choice of an adequate system of modulation coordinates and the novelty, compared to the scalar NLS, is the fact that the group of symmetries of the system is non-commutative.