Stationary scattering from a nonlinear network

Abstract

Transmission through a complex network of nonlinear one-dimensional leads is discussed by extending the stationary scattering theory on quantum graphs to the nonlinear regime. We show that the existence of cycles inside the graph leads to a large number of sharp resonances that dominate scattering. The latter resonances are then shown to be extremely sensitive to the nonlinearity and display multi-stability and hysteresis. This work provides a framework for the study of light propagation in complex optical networks.

4 pages, 4 figures

References in corpus (6)

• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Fast soliton scattering by delta impurities
• Soliton solutions of nonlinear Schroedinger equation on simple networks
• An analytical study of resonant transport of Bose-Einstein condensates
• Barrier transmission for the one-dimensional nonlinear Schrödinger equation: resonances and transmission profiles
• Barrier transmission for the Nonlinear Schrödinger Equation: Surprises of nonlinear transport

Cited by in corpus (36)

• Variational properties and orbital stability of standing waves for NLS equation on a star graph
• Nonlinear Schrödinger equation on graphs: recent results and open problems
• On the structure of critical energy levels for the cubic focusing NLS on star graphs
• NLS ground states on metric graphs with localized nonlinearities
• Bound states of the NLS equation on metric graphs with localized nonlinearities
• Topological Resonances in Scattering on Networks (Graphs)
• On the lack of bound states for certain NLS equations on metric graphs
• Soliton transport in tubular networks: transmission at vertices in the shrinking limit
• Complex active optical networks as a new laser concept
• Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory
• Stationary Nonlinear Schrödinger Equation on Simplest Graphs: Boundary conditions and exact solutions
• Standing waves on quantum graphs
• Normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graphs with localized nonlinearities
• Nonlinear Dirac Equation On Graphs With Localized Nonlinearities: Bound States And Nonrelativistic Limit
• Edge-localized states on quantum graphs in the limit of large mass
• On the nonlinear Dirac equation on noncompact metric graphs
• Transport in simple networks described by integrable discrete nonlinear SchrÄodinger equation
• The LANER: optical networks as complex lasers
• Ground states of the $L^2$-critical NLS equation with localized nonlinearity on a tadpole graph
• Stationary waves on nonlinear quantum graphs II: Application of canonical perturbation theory in basic graph structures
• Threshold phenomena and existence results for NLS ground states on graphs
• Exactly solvable Gross-Pitaevskii type equations
• An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity
• Scattering Statistics in Nonlinear Wave Chaotic Systems
• Nonlinearity-induced Scattering Zero Degeneracies for Spectral Management of Coherent Perfect Absorption in Complex Systems
• Nonlinear Wave Chaos: Statistics of Second Harmonic Fields
• Asymptotic behavior for the long-range nonlinear Schrödinger equation on star graph with the Kirchhoff boundary condition
• Prescribed mass ground states for a doubly nonlinear Schrödinger equation in dimension one
• Transmission through a noisy network
• NLS ground states on graphs
• An introduction to the two-dimensional Schrödinger equation with nonlinear point interactions
• The Ablowitz-Ladik system on a graph
• Variational and stability properties of coupled NLS equations on the star graph
• Constrained energy minimization and orbital stability for the NLS equation on a star graph
• Asymptotic Stability of Solitons to Nonlinear Schrodinger Equations on Star Graphs
• Ground states for the NLS on non-compact graphs with an attractive potential