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

Abstract

In this paper we present a general framework for solving the stationary nonlinear Schrödinger 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.

26 pages

References in corpus (10)

• Fast soliton scattering by delta impurities
• Fast solitons on star graphs
• Nonlinear dynamics of a cigar-shaped Bose-Einstein condensate coupled with a single cavity mode
• Dynamics of one-dimensional Bose liquids: Andreev-like reflection at Y-junctions and absence of the Aharonov-Bohm effect
• Coherent backscattering of Bose-Einstein condensates in two-dimensional disorder potentials
• Soliton transport in tubular networks: transmission at vertices in the shrinking limit
• An analytical study of resonant transport of Bose-Einstein condensates
• Barrier transmission for the one-dimensional nonlinear Schrödinger equation: resonances and transmission profiles
• On the Inverse Scattering Method for Integrable PDEs on a Star Graph
• Soliton Propagation in Chains with Simple Nonlocal Defects

Cited by in corpus (17)

• Negative energy ground states for the $L^2$-critical NLSE on metric graphs
• Bifurcations of standing localized waves on periodic graphs
• Standing waves on quantum graphs
• Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs
• Green's function approach for quantum graphs: an overview
• Nonlinear Dirac Equation On Graphs With Localized Nonlinearities: Bound States And Nonrelativistic Limit
• On the nonlinear Dirac equation on noncompact metric graphs
• Sensitivity and spectral control of network lasers
• Transparent nonlinear networks
• Ground states of the $L^2$-critical NLS equation with localized nonlinearity on a tadpole graph
• Nonlinear dynamics on branched structures and networks
• A note on the Dirac operator with Kirchoff-type vertex conditions on noncompact metric graphs
• Stationary waves on nonlinear quantum graphs II: Application of canonical perturbation theory in basic graph structures
• An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity
• NLS Bifurcations on the bowtie combinatorial graph and the dumbbell metric graph
• Spectral properties of nonlinear Schrödinger equation on a ring
• On the nodal structure of nonlinear stationary waves on star graphs