Stationary States of NLS on Star Graphs

TL;DR

This paper studies stationary solutions of the nonlinear Schrödinger equation on star graphs, analyzing existence, stability, and the construction of traveling waves under various boundary conditions and nonlinearities.

Contribution

It provides a comprehensive analysis of stationary states on star graphs, including existence, stability, and the construction of traveling waves, extending previous work to more general boundary conditions.

Findings

Existence of stationary states for both attractive and repulsive interactions.

Characterization of ground states as minimizers of a constrained action.

Construction of traveling waves from stationary states in the free case for even N.

Abstract

We consider a generalized nonlinear Schr\"odinger equation (NLS) with a power nonlinearity |\psi|^2\mu\psi, of focusing type, describing propagation on the ramified structure given by N edges connected at a vertex (a star graph). To model the interaction at the junction, it is there imposed a boundary condition analogous to the \delta potential of strength \alpha on the line, including as a special case (\alpha=0) the free propagation. We show that nonlinear stationary states describing solitons sitting at the vertex exist both for attractive (\alpha<0, representing a potential well) and repulsive (\alpha>0, a potential barrier) interaction. In the case of sufficiently strong attractive interaction at the vertex and power nonlinearity \mu<2, including the standard cubic case, we characterize the ground state as minimizer of a constrained action and we discuss its orbital stability. Finally we show that in the free case, for even N only, the stationary states can be used to construct traveling waves on the graph.