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

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 Schrödinger 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.

26 pages, final version. v2: Summary of results in introduction, new section 4.3 to improve the discussion on the global relation, complete discussion of the example in Section 5.1.1. v3: titles of references added

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• Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs
• The Korteweg-de Vries equation on a metric star graph
• Linear instability of stationary solutions for the Korteweg-de Vries equation on a star graph
• Nonlocal Nonlinear Schrodinger Equation on Metric Graphs
• Discrete Nonlocal Nonlinear Schroedinger equation on Metric Graphs: Dynamics of PT-Symmetric Solitons in Discrete Networks
• The Ablowitz-Ladik system on a finite set of integers
• The Ablowitz-Ladik system on a graph