Soliton solutions of nonlinear Schroedinger equation on simple networks

TL;DR

This paper demonstrates the existence and conditions for reflectionless soliton propagation in nonlinear Schrödinger equations on simple network graphs, extending known solutions from 1D chains to complex network structures.

Contribution

It introduces a framework for soliton solutions on networks with bond-dependent nonlinearities, including a sum rule for nonlinearities at vertices, and extends known 1D solutions to complex graph topologies.

Findings

Soliton solutions on networks are scaled versions of 1D solutions.

A sum rule for nonlinearities at vertices ensures reflectionless propagation.

Numerical simulations confirm theoretical predictions.

Abstract

We show soliton solutions of nonlinear Schroedinger equation on simple networks consisting of vertices and bonds, where the strength of cubic nonlinearity is different from bond to bond. We concentrate on reflectionless propagation of Zakharov-Shabat's solitons through a branched chain, namely, a primary star graph consisting of three semi-infinite bonds connected at a vertex. The conservation of the norm and the global current elucidates: (1) the solution on each bond is a part of the universal soliton solution on a simple 1-dimensional (1-d) chain but multiplied by the inverse of square root of bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each vertex must satisfy a sum rule. Under these conditions, all other conservation rules for a simple 1-d chain have proved to hold for multi-soliton solutions on graphs. The argument is extended to other graphs, i.e., general star graphs, tree graphs, loop graphs and their combinations. Numerical evidence is also given on the reflectionless propagation of a soliton through a branched chain.