Variational and stability properties of coupled NLS equations on the star graph

TL;DR

This paper investigates the existence and stability of coupled nonlinear Schrödinger equations on star graphs with delta coupling, extending known results to graph structures and analyzing various standing wave solutions.

Contribution

It provides the first variational and stability analysis of coupled NLS equations on graphs, including existence of ground states and stability of multiple standing wave configurations.

Findings

Existence of ground states as energy minimizers for cubic case.

Stability analysis of one-component and two-component standing waves.

Extension of results from line to star graph structures.

Abstract

We consider variational and stability properties of a system of two coupled nonlinear Schr\"{o}dinger equations on the star graph $\Gamma$ with the $\delta$ coupling at the vertex of $\Gamma$. The first part is devoted to the proof of an existence of the ground state as the minimizer of the constrained energy in the cubic case. This result extends the one obtained recently for the coupled NLS equations on the line. In the second part, we study stability properties of several families of standing waves in the case of a general power nonlinearity. In particular, we study one-component standing waves $e^{i\omega t}(\Phi_1(x), 0)$ and $e^{i\omega t}(0, \Phi_2(x))$. Moreover, we study two-component standing waves $e^{i\omega t}(\Phi(x), \Phi(x))$ for the case of power nonlinearity depending on a unique power parameter $p$. To our knowledge, these are the first results on variational and stability properties of coupled NLS equations on graphs.