Variational properties and orbital stability of standing waves for NLS equation on a star graph

TL;DR

This paper investigates the existence, classification, and orbital stability of standing waves for a focusing nonlinear Schrödinger equation on a star graph with delta interaction at the vertex, covering various coupling strengths and nonlinear regimes.

Contribution

It establishes the existence of multiple families of standing waves, characterizes ground states as minimizers, and analyzes their orbital stability across different nonlinear and coupling conditions.

Findings

Existence of multiple standing wave families for all coupling signs and frequencies.

Ground states identified as minimizers on the Nehari manifold.

Orbital stability proven for subcritical and critical nonlinearities, and for certain frequency ranges.

Abstract

We study standing waves for a nonlinear Schr\"odinger equation on a star graph {$\mathcal{G}$} i.e. $N$ half-lines joined at a vertex. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\leqslant 0$. The nonlinearity is of focusing power type. The dynamics is given by an equation of the form $ i \frac{d}{dt}\Psi_t = H \Psi_t - | \Psi_t |^{2\mu} \Psi_t $, where $H$ is the Hamiltonian operator which generates the linear Schr\"odinger dynamics. We show the existence of several families of standing waves for every sign of the coupling at the vertex for every $\omega > \frac{\alpha^2}{N^2}$. Furthermore, we determine the ground states, as minimizers of the action on the Nehari manifold, and order the various families. Finally, we show that the ground states are orbitally stable for every allowed $\omega$ if the nonlinearity is subcritical or critical, and for $\omega<\omega^\ast$ otherwise.