Stability of ground-states for a system of $M$ coupled semilinear Schr\"odinger equations

TL;DR

This paper investigates the stability of ground-states in a system of multiple coupled semilinear Schrödinger equations, extending single-equation theory and identifying conditions for stability, instability, and weak instability.

Contribution

It generalizes stability analysis to systems with multiple coupled equations and introduces simplified techniques for these complex cases.

Findings

Stability, instability, and weak instability depend on nonlinearity power.

Results cover three classes of bound-states unique to multi-equation systems.

The approach simplifies existing methods for analyzing coupled Schrödinger systems.

Abstract

We focus on the study of the stability properties of ground-states for the system of $M$ coupled semilinear Schr\"odinger equations with power-type nonlinearities and couplings. Our results are generalizations of the theory for the single equation and the technique for this kind of results is simplified. Depending on the power of the nonlinearity, we may observe stability, instability and weak instability. We also obtain results for three distinct classes of bound-states, which is a special feature of the $M\ge2$ case.