On the Inverse Scattering Method for Integrable PDEs on a Star Graph

TL;DR

This paper develops a unified inverse scattering framework for integrable PDEs on star graphs, extending the Fokas method to matrix IBV problems, and demonstrates its effectiveness with the nonlinear Schrödinger equation.

Contribution

It introduces a novel approach to formulate the inverse scattering method on star graphs, unifying previous cases and handling general Robin boundary conditions.

Findings

Unified framework for integrable PDEs on star graphs.

Extension of Fokas method to matrix IBV problems.

Efficient solution for nonlinear Schrödinger equation with Robin conditions.

Abstract

We present a framework to solve the open problem of formulating the inverse scattering method (ISM) for an integrable PDE on a star-graph. The idea is to map the problem on the graph to a matrix initial-boundary value (IBV) problem and then to extend the unified method of Fokas to such a matrix IBV problem. The nonlinear Schr\"odinger equation is chosen to illustrate the method. The framework unifies all previously known examples which are recovered as particular cases. The case of general Robin conditions at the vertex is discussed: the notion of linearizable initial-boundary conditions is introduced. For such conditions, the method is shown to be as efficient as the ISM on the full-line.