Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy

TL;DR

This paper proves that certain symmetric standing waves on star graphs with nonlinear Schrödinger equations are orbitally stable because they are local energy minimizers at fixed mass, even when not global ground states.

Contribution

It introduces a new technique to establish orbital stability of standing waves as local energy minimizers without relying on traditional variational or linearization methods.

Findings

Orbital stability holds for all masses, regardless of ground state status.

The new method reduces the problem to a finite-dimensional analysis using soliton pieces.

Symmetric standing waves are shown to be local minimizers of the energy at fixed mass.

Abstract

On a star graph made of $N \geq 3$ halflines (edges) we consider a Schr\"odinger equation with a subcritical power-type nonlinearity and an attractive delta interaction located at the vertex. From previous works it is known that there exists a family of standing waves, symmetric with respect to the exchange of edges, that can be parametrized by the mass (or $L^2$-norm) of its elements. Furthermore, if the mass is small enough, then the corresponding symmetric standing wave is a ground state and, consequently, it is orbitally stable. On the other hand, if the mass is above a threshold value, then the system has no ground state. Here we prove that orbital stability holds for every value of the mass, even if the corresponding symmetric standing wave is not a ground state, since it is anyway a {\em local} minimizer of the energy among functions with the same mass. The proof is based on a new technique that allows to restrict the analysis to functions made of pieces of soliton, reducing the problem to a finite-dimensional one. In such a way, we do not need to use direct methods of Calculus of Variations, nor linearization procedures.