The Discrete Nonlinear Schrödinger equation - 20 Years on

Abstract

We review work on the Discrete Nonlinear Schrödinger (DNLS) equation over the last two decades.

24 pages, 1 figure, Proceedings of the conference on "Localization and Energy Transfer in Nonlinear Systems", June 17-21, 2002, San Lorenzo de El Escorial, Madrid, Spain; to be published by World Scientific

References in corpus (13)

• Quantum Coherent Atomic Tunneling between Two Trapped Bose-Einstein Condensates
• Discrete Breathers
• Discrete Solitons and Breathers with Dilute Bose-Einstein Condensates
• Nonlinear excitations in arrays of Bose-Einstein condensates
• Statistical mechanics of a nonlinear discrete system
• Energy thresholds for discrete breathers in one-, two- and three-dimensional lattices
• Discreteness-Induced Oscillatory Instabilities of Dark Solitons
• Growth and decay of discrete nonlinear Schroedinger breathers interacting with internal modes or standing-wave phonons
• Solitons in Triangular and Honeycomb Dynamical Lattices with the Cubic Nonlinearity
• Moving discrete breathers?
• Moving lattice kinks and pulses: an inverse method
• Discrete Nonlinear Schr{ö}dinger Breathers in a Phonon Bath
• A Novel Mechanism for Superfluidity Breakdown in Weakly Coupled Bose-Einstein Condensates

Cited by in corpus (30)

• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Dark solitons in atomic Bose-Einstein condensates: from theory to experiments
• Solitons in a chain of PT-invariant dimers
• Breathers and surface modes in oscillator chains with Hertzian interactions
• Nonlinear localized modes in two-dimensional electrical lattices
• Discrete breathers in a nonlinear electric line: Modeling, Computation and Experiment
• Discrete skew selfadjoint canonical systems and the isotropic Heisenberg magnet model
• Unstaggered-staggered solitons in two-component discrete nonlinear Schrödinger lattices
• Multibreathers in Klein-Gordon chains with interactions beyond nearest neighbors
• Photon-atomic solitons in a Bose-Einstein condensate trapped in a soft optical lattice
• Flat bands and PT-symmetry in quasi-one-dimensional lattices
• Arrayed and checkerboard optical waveguides controlled by the electromagnetically-induced transparency
• Superposition of Elliptic Functions as Solutions For a Large Number of Nonlinear Equations
• Intrinsic Localized Modes of a Classical Discrete Anisotropic Heisenberg Ferromagnetic Spin Chain
• Non-Nearest-Neighbor Interactions in Nonlinear Dynamical Lattices
• Breathers for the Discrete Nonlinear Schrödinger equation with nonlinear hopping
• A discrete Schrodinger equation via optimal transport on graphs
• Discrete solitons in an array of quantum dots
• Symmetry breaking and restoring wave transmission in diode-antidiode double chains
• Dark lattice solitons in one-dimensional waveguide arrays with defocusing saturable nonlinearities and alternating couplings
• Solitons and Vortices in Two-dimensional Discrete Nonlinear Schrodinger Systems with Spatially Modulated Nonlinearity
• Heteroclinic standing waves in defocussing DNLS equations -- Variational approach via energy minimization
• A note on N-soliton solutions for the viscid incompressible Navier-Stokes differential equation
• High order explicit symplectic integrators for the Discrete Non Linear Schrödinger equation
• Asymptotic Behavior of Solutions of Complex Discrete Evolution Equations: The Discrete Ginzburg-Landau Equation
• Editorial (a preface to a Special Issue of journal Applied Sciences on "Guided-Wave Optics")
• Nonreciprocal wave propagation through open, discrete nonlinear Schroedinger dimers
• Static configurations of an exciton field in interaction with a deformable two-dimensional hexagonal lattice
• A remark on the existence of breather solutions for the Discrete Nonlinear Schrödinger Equation in infinite lattices: The case of site dependent anharmonic parameter
• Bright Discrete Solitons in Spatially Modulated DNLS Systems