Nonlinear Schr\"odinger equation on graphs: recent results and open problems

TL;DR

This paper introduces the study of nonlinear Schrödinger equations on graphs, focusing on solution properties like solitary waves, with specific analysis on star graphs, delta interactions, and soliton scattering at junctions.

Contribution

It provides an overview of recent developments in nonlinear dispersive equations on graphs, highlighting existence, behavior of solutions, and open problems in the field.

Findings

Existence of solitary solutions on graphs.

Analysis of soliton scattering at Y-junctions.

Characterization of standing waves with delta interactions.

Abstract

In the present paper an introduction to the new subject of nonlinear dispersive hamiltonian equations on graphs is given. The focus is on recently established properties of solutions in the case of nonlinear Schr\"odinger equation. Special consideration is given to existence and behaviour of solitary solutions. Two subjects are discussed in some detail concerning NLS equation on a star graph: the standing waves of NLS equation on a graph with a $\delta$ interaction at the vertex; the scattering of fast solitons through an Y-junction in the cubic case. The emphasis is on description of concepts and results and on physical context, without reporting detailed proofs; some perspectives and more ambitious open problems are discussed.