Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory

TL;DR

This paper develops a comprehensive framework for solving the stationary nonlinear Schrödinger equation on quantum graphs, utilizing explicit nonlinear transfer functions and perturbation theory to analyze wave behavior on complex networks.

Contribution

It introduces a general method for solving NLSE on metric graphs, including explicit solutions for cubic cases and a perturbative approach for small amplitudes.

Findings

Explicit nonlinear transfer functions using Jacobi elliptic functions

Reduction of differential equations to algebraic equations

Leading nonlinear corrections derived via canonical perturbation theory

Abstract

In this paper we present a general framework for solving the stationary nonlinear Schr\"odinger equation (NLSE) on a network of one-dimensional wires modelled by a metric graph with suitable matching conditions at the vertices. A formal solution is given that expresses the wave function and its derivative at one end of an edge (wire) nonlinearly in terms of the values at the other end. For the cubic NLSE this nonlinear transfer operation can be expressed explicitly in terms of Jacobi elliptic functions. Its application reduces the problem of solving the corresponding set of coupled ordinary nonlinear differential equations to a finite set of nonlinear algebraic equations. For sufficiently small amplitudes we use canonical perturbation theory which makes it possible to extract the leading nonlinear corrections over large distances.