Nonlinear Waves in Bose-Einstein Condensates: Physical Relevance and Mathematical Techniques
arXiv:0805.0761 · doi:10.1088/0951-7715/21/7/R01
Abstract
The aim of the present review is to introduce the reader to some of the physical notions and of the mathematical methods that are relevant to the study of nonlinear waves in Bose-Einstein Condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyze some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons, as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g., the linear or the nonlinear limit, or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.
69 pages, 10 figures, to appear in Nonlinearity, 2008. V2: new references added, fixed typos
References in corpus (55)
- Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond
- Formation of bright matter-wave solitons during the collapse of Bose-Einstein condensates
- Extension of Bogoliubov theory to quasi-condensates
- On Dispersive and Classical Shock Waves in Bose-Einstein Condensates and Gas Dynamics
- Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities
- Anderson Localization of Expanding Bose-Einstein Condensates in Random Potentials
- Experimental observation of oscillating and interacting matter wave dark solitons
- Deformation of a Trapped Fermi Gas with Unequal Spin Populations
- Observation of Vortex Pinning in Bose-Einstein Condensates
- Stationary and non-stationary fluid flow of a Bose-Einstein condensate through a penetrable barrier
- Vortex Formation by Interference of Multiple Trapped Bose-Einstein Condensates
- Non-Equilibrium Dynamics and Superfluid Ring Excitations in Binary Bose-Einstein Condensates
- Solitary Wave Dynamics in an External Potential
- Bright-Dark Soliton Complexes in Spinor Bose-Einstein Condensates
- Ultracold Bose Gases in 1D Disorder: From Lifshits Glass to Bose-Einstein Condensate
- Deviation from one-dimensionality in stationary properties and collisional dynamics of matter-wave solitons
- Asymmetric vortex solitons in nonlinear periodic lattices
- Effective mean-field equations for cigar-shaped and disk-shaped Bose-Einstein condensates
- Dark solitons in F=1 spinor Bose--Einstein condensate
- Quasi-1D Bose-Einstein condensates in the dimensional crossover regime
- Fully three dimensional breather solitons can be created using Feshbach resonance
- Faraday waves in Bose-Einstein condensates
- Symmetry Breaking in Symmetric and Asymmetric Double-Well Potentials
- Anderson Localization of Bogolyubov Quasiparticles in Interacting Bose-Einstein Condensates
- Localization of Matter Waves in 2D-Disordered Optical Potentials
- Anderson localization of a Bose-Einstein condensate in a 3D random potential
- Gap solitons in Bose-Einstein condensates in linear and nonlinear optical lattices
- Multi-component gap solitons in spinor Bose-Einstein condensates
- Superfluidity versus Anderson localization in a dilute Bose gas
- Pinning of vortices in a Bose-Einstein condensate by an optical lattice
- Effect of interactions on the localization of a Bose-Einstein condensate in a quasi-periodic lattice
- Expansion of a Bose-Einstein Condensate in the Presence of Disorder
- Friction and diffusion of matter-wave bright solitons
- Dark soliton dynamics in Bose-Einstein condensates at finite temperature
- Smoothing effect and delocalization of interacting Bose-Einstein condensates in random potentials
- Nonlinear modes for the Gross-Pitaevskii equation -- demonstrative computation approach
- Dynamics of Vortex Formation in Merging Bose-Einstein Condensate Fragments
- Ground-state properties of trapped Bose-Einstein condensates: Extension of the Thomas-Fermi approximation
- Dark solitons as quasiparticles in trapped condensates
- Solitons in a trapped spin-1 atomic condensate
- Dynamics of Bose-Einstein condensates in cigar-shaped traps
- Solitary Waves Under the Competition of Linear and Nonlinear Periodic Potentials
- Soliton oscillations in collisionally inhomogeneous attractive Bose-Einstein condensates
- Numerical Study of the stability of Skyrmions in Bose-Einstein Condensates
- Temporal dynamics of tunneling. Hydrodynamic approach
- The Mathematics of the Bose Gas and its Condensation
- Stabilization of three-dimensional matter-waves solitons in an optical lattice
- Polarized States and Domain Walls in Spinor Bose-Einstein Condensates
- Cerenkov-like radiation in a binary Schr{ö}dinger flow past an obstacle
- A density-functional approach to fermionization in the 1D Bose gas
- Three-dimensional matter-wave vortices in optical lattices
- Matter wave solitons at finite temperatures
- Partially incoherent gap solitons in Bose-Einstein condensates
- Density Correlations in Cold Atomic Gases: Atomic Speckles in the Presence of Disorder
- From Feshbach-Resonance Managed Bose-Einstein Condensates to Anisotropic Universes: Some Applications of the Ermakov-Pinney equation with Time-Dependent Nonlinearity
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- Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized PT-symmetric Scarff-II potentials
- Higher-order vector discrete rogue-wave states in the coupled Ablowitz-Ladik equations: exact solutions and stability
- Optical rogue waves in the generalized inhomogeneous higher-order nonlinear Schrodinger equation with modulating coefficients
- Time-Dependent Density Functional Theory and the Real-Time Dynamics of Fermi Superfluids
- Rogue Waves In Nonlinear Schrödinger Models with Variable Coefficients: Application to Bose-Einstein Condensates
- Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials
- Stable multi-ring and rotating solitons in two-dimensional spin-orbit coupled Bose-Einstein condensates with a radially-periodic potential
- Soliton Magnetization Dynamics in Spin-Orbit Coupled Bose-Einstein Condensates
- Matter-wave dark solitons: stochastic vs. analytical results
- Faraday waves in binary non-miscible Bose-Einstein condensates
- Excited states in the Thomas-Fermi limit: a variational approach
- Dark-bright solitons in coupled nonlinear Schrödinger equations with unequal dispersion coefficients
- Motion of solitons in one-dimensional spin-orbit-coupled Bose-Einstein condensates
- Dark spherical shell solitons in three-dimensional Bose-Einstein condensates: Existence, stability and dynamics
- Manipulating matter-rogue waves and breathers in Bose-Einstein condensates
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- Coupled backward- and forward-propagating solitons in a composite right/left-handed transmission line
- Phase Separation and Dynamics of Two-component Bose-Einstein Condensates
- Dynamics and modulation of ring dark soliton in 2D Bose-Einstein condensates with tunable interaction
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- Energetically stable singular vortex cores in an atomic spin-1 Bose-Einstein condensate
- Distribution of eigenfrequencies for oscillations of the ground state in the Thomas--Fermi limit
- Wave patterns generated by a supersonic moving body in a binary Bose-Einstein condensate
- Kuramoto dynamics in Hamiltonian systems
- Coherent atomic soliton molecules for matter-wave switching
- On Symmetries and Exact Solutions of a Class of Non-local Non-linear Schrodinger Equations with Self-induced PT-symmetric Potential
- Quantum Shock Waves and Domain Walls in the Real-Time Dynamics of a Superfluid Unitary Fermi Gas
- Hydrodynamics of cold atomic gases in the limit of weak nonlinearity, dispersion and dissipation
- Matter-wave solutions in the Bose-Einstein condensates with the harmonic and Gaussian potentials
- Solitons and their stability in the nonlocal nonlinear Schroedinger equation with PT-symmetric potentials
- Coherent Cavitation in the Liquid of Light
- Bose-Einstein condensation: Twenty years after
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- Stable multidimensional soliton stripes in two-component Bose-Einstein condensates
- Scattering of atomic dark-bright solitons from narrow impurities
- Dark-bright solitons in Bose-Einstein condensates at finite temperatures
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- Statics and dynamics of atomic dark-bright solitons in the presence of delta-like impurities
- Solitary waves in coupled nonlinear Schrodinger equations with spatially inhomogeneous nonlinearities
- Delayed collapses of BECs in relation to AdS gravity
- Acoustic solitons in waveguides with Helmholtz resonators: transmission line approach
- Nonlinear quantum piston for the controlled generation of vortex rings and soliton trains
- Beating dark-dark solitons and Zitterbewegung in spin-orbit coupled Bose-Einstein condensates
- Dynamics of Vector Solitons in Bose-Einstein Condensates
- Single and Multiple Vortex Rings in Three-Dimensional Bose-Einstein Condensates: Existence, Stability and Dynamics
- Dynamical localization of gap-solitons by time periodic forces
- Azimuthal Modulational Instability of Vortices in the Nonlinear Schrödinger Equation
- Scattering and leapfrogging of vortex rings in a superfluid
- One-dimension cubic-quintic Gross-Pitaevskii equation in Bose-Einstein condensates in a trap potential
- Flipping-shuttle oscillations of bright one- and two-dimensional solitons in spin-orbit-coupled Bose-Einstein condensates with Rabi mixing
- Impact of anisotropy on vortex clusters and their dynamics
- Stability and dynamics of matter-wave vortices in the presence of collisional inhomogeneities and dissipative perturbations
- Non-autonomous bright-dark solitons and Rabi oscillations in multi-component Bose-Einstein condensates
- A Modulus-Squared Dirichlet Boundary Condition for Time-Dependent Complex Partial Differential Equations and its Application to the Nonlinear Schrödinger Equation
- Effective one-dimensional dynamics of elongated Bose-Einstein condensates
- Numerical study of localized impurity in a Bose-Einstein condensate
- Variational analysis of flat-top solitons in Bose-Einstein condensates
- NLSEmagic: Nonlinear Schrödinger Equation Multidimensional Matlab-based GPU-accelerated Integrators using Compact High-order Schemes
- Manipulation of Vortices by Localized Impurities in Bose-Einstein Condensates
- Vortex Ring Dynamics in Trapped Bose-Einstein Condensates
- Variational approximations for travelling solitons in a discrete nonlinear Schrödinger equation
- Three-dimensional gap solitons in Bose-Einstein condensates supported by one-dimensional optical lattices
- Stabilization of ring dark solitons in Bose-Einstein condensates
- Topological interface physics of defects and textures in spinor Bose-Einstein condensates
- Vector rogue waves and dark-bright boomeronic solitons in autonomous and non-autonomous settings
- Matter-wave dark solitons in box-like traps
- Statics and dynamics of quasi one-dimensional Bose-Einstein condensate in harmonic and dimple trap
- Weyl solitons in three-dimensional optical lattices
- Stability and Tunneling Dynamics of a Dark-Bright Soliton Pair in a Harmonic Trap
- Collapse instability of solitons in the nonpolynomial Schrödinger equation with dipole-dipole interactions
- Solitons Riding on Solitons: Hypersolitons and the Quantum Newton's Cradle
- Gap solitons in elongated geometries: the one-dimensional Gross-Pitaevskii equation and beyond
- Spectral and dynamical analysis of a single vortex ring in anisotropic harmonically trapped three-dimensional Bose-Einstein condensates
- Dark solitons in cigar-shaped Bose-Einstein condensates in double-well potentials
- Nonlinear dynamics of Bose-condensed gases by means of a low- to high-density variational approach
- Dark-bright solitons in spinor polariton condensates under nonresonant pumping
- Several Localized Waves Induced by Linear Interference between a Nonlinear Plane Wave and Bright Solitons
- Collisional-inhomogeneity-induced generation of matter-wave dark solitons
- Matter-wave solitons in the counterflow of two immiscible superfluids
- Diverging probability density functions for flat-top solitary waves
- Josephson tunneling of dark solitons in a double-well potential
- Analytical and numerical study of dirty bosons in a quasi-one-dimensional harmonic trap
- A Nonlinear Adiabatic Theorem for Coherent States
- Stabilization of the Peregrine soliton and Kuznetsov-Ma breathers by means of nonlinearity and dispersion management
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- Solitary waves in mixtures of Bose gases confined in annular traps
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- Bose-Einstein condensates in optical lattices: mathematical analysis and analytical approximate formulae
- Scattering of a dark-bright soliton by an impurity
- Hydrodynamics and two-dimensional dark lump solitons for polariton superfluids
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- Macroscopic quantum escape of Bose-Einstein condensates: Analysis of experimentally realizable quasi-one-dimensional traps
- Multiple-scale analysis for resonance reflection by a one-dimensional rectangular barrier in the Gross-Pitaevskii problem
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- Large stable oscillations due to Hopf bifurcations in amplitude dynamics of colliding soliton sequences
- Particle and wave dynamics of nonlocal solitons in external potentials
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- Curved vortex surfaces in four-dimensional superfluids: I. Unequal-frequency double rotations
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- Numerical Stability of Explicit Runge-Kutta Finite-Difference Schemes for the Nonlinear Schrödinger Equation
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