Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques (original) (raw)
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We present a complete calculation of solitary waves propagating in a steady state with constant velocity v along a cigar-shaped Bose-Einstein trap approximated as infinitely-long cylindrical. For sufficiently weak couplings (densities) the main features of the calculated solitons could be captured by effective one-dimensional (1D) models. However, for stronger couplings of practical interest, the relevant solitary waves are found to be hybrids of quasi-1D solitons and 3D vortex rings. An interesting hierarchy of vortex rings occurs as the effective coupling constant is increased through a sequence of critical values. The energy-momentum dispersion of the above structures is shown to exhibit characteristics similar to a mode proposed sometime ago by Lieb within a strictly 1D model, as well as some rotonlike features.
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In this paper we study the propagation of solitons in a Bose-Einstein condensate governed by the time dependent one dimensional Gross-Pitaevskii equation managed by Feshbach resonance in a linear external potential. We give the Lax pair of the Gross-Pitaevskii equation in Bose-Einstein condensates and obtain exact N-soliton solution by employing the simple, straightforward Darboux transformation. As an example, we present exact one and two-soliton solution and discuss their transmission, interaction and dynamic properties. We further calculate the particle number, momentum and energy of the solitons and discuss their conservation laws. Knowledge of soliton dynamics helps us in understanding the physical nature of the condensate and in the calculation of the thermodynamic properties.
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We consider the three-dimensional (3D) mean-field model for the Bose-Einstein condensate (BEC), with a 1D nonlinear lattice (NL), which periodically changes the sign of the nonlinearity along the axial direction, and the harmonicoscillator trapping potential applied in the transverse plane. The lattice can be created as an optical or magnetic one, by means of available experimental techniques. The objective is to identify stable 3D solitons supported by the setting. Two methods are developed for this purpose: The variational approximation, formulated in the framework of the 3D Gross-Pitaevskii equation, and the 1D nonpolynomial Schrödinger equation (NPSE) in the axial direction, which allows one to predict the collapse in the framework of the 1D description. Results are summarized in the form of a stability region for the solitons in the plane of the NL strength and wavenumber. Both methods produce a similar form of the stability region. Unlike their counterparts supported by the NL in the 1D model with the cubic nonlinearity, kicked solitons of the NPSE cannot be set in motion, but the kick may help to stabilize them against the collapse, by causing the solitons to shed excess norm. A dynamical effect specific to the NL is found in the form of freely propagating small-amplitude wave packets emitted by perturbed solitons.
Journal of Nonlinear Science, 2002
The cubic nonlinear Schrödinger equation with a lattice potential is used to model a periodic dilute gas Bose-Einstein condensate. Both two-and three-dimensional condensates are considered, for atomic species with either repulsive or attractive interactions. A family of exact solutions and corresponding potential is presented in terms of elliptic functions. The dynamical stability of these exact solutions is examined using both analytical and numerical methods. For condensates with repulsive atomic interactions, all stable, trivial-phase solutions are off-set from the zero level. For condensates with attractive atomic interactions, no stable solutions are found, in contrast to the one-dimensional case .
Exact soliton solutions and nonlinear modulation instability in spinor Bose-Einstein condensates
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We find one-, two-, and three-component solitons of the polar and ferromagnetic ͑FM͒ types in the general ͑nonintegrable͒ model of a spinor ͑three-component͒ model of the Bose-Einstein condensate, based on a system of three nonlinearly coupled Gross-Pitaevskii equations. The stability of the solitons is studied by means of direct simulations and, in a part, analytically, using linearized equations for small perturbations. Global stability of the solitons is considered by means of an energy comparison. As a result, ground-state and metastable soliton states of the FM and polar types are identified. For the special integrable version of the model, we develop the Darboux transformation ͑DT͒. As an application of the DT, analytical solutions are obtained that display full nonlinear evolution of the modulational instability of a continuous-wave state seeded by a small spatially periodic perturbation. Additionally, by dint of direct simulations, we demonstrate that solitons of both the polar and FM types, found in the integrable system, are structurally stable; i.e., they are robust under random changes of the relevant nonlinear coefficient in time.
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We produce several families of solutions for two-component nonlinear Schrödinger/Gross-Pitaevskii equations. These include domain walls and the first example of an antidark or gray soliton in the one component, bound to a bright or dark soliton in the other. Most of these solutions are linearly stable in their entire domain of existence. Some of them are relevant to nonlinear optics, and all to Bose-Einstein condensates (BECs). In the latter context, we demonstrate robustness of the structures in the presence of parabolic and periodic potentials (corresponding, respectively, to the magnetic trap and optical lattices in BECs).
Bose-Einstein condensates and spectral properties of multicomponent nonlinear Schrödinger equations
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We analyze the properties of the soliton solutions of a class of models describing one-dimensional BEC with spin F . We describe the minimal sets of scattering data which determine uniquely both the corresponding potential of the Lax operator and its scattering matrix. Next we give several reductions of these MNLS, derive their N -soliton solutions and analyze the soliton interactions. Finally we prove an important theorem proving that if the initial conditions satisfy the reduction then one gets a solution of the reduced MNLS. 2000 Mathematics Subject Classification. Primary: 35Q51, 37K40; Secondary: 34K17 . Key words and phrases. Bose-Einstein condensates, Multicomponent nonlinear Schrödinger equations, Soliton solutions, Soliton interactions, Reductions of MNLS.
Nonlinear lattice dynamics of Bose-Einstein condensates
arXiv preprint nlin/ …, 2004
The Fermi-Pasta-Ulam (FPU) model, which was proposed 50 years ago to examine thermalization in non-metallic solids and develop "experimental" techniques for studying nonlinear problems, continues to yield a wealth of results in the theory and applications of nonlinear Hamiltonian systems with many degrees of freedom. Inspired by the studies of this seminal model, solitary-wave dynamics in lattice dynamical systems have proven vitally important in a diverse range of physical problemsincluding energy relaxation in solids, denaturation of the DNA double strand, self-trapping of light in arrays of optical waveguides, and Bose-Einstein condensates (BECs) in optical lattices. BECS, in particular, due to their widely ranging and easily manipulated dynamical apparatuses-with one to three spatial dimensions, positive-to-negative tuning of the nonlinearity, one to multiple components, and numerous experimentally accessible external trapping potentials-provide one of the most fertile grounds for the analysis of solitary waves and their interactions. In this paper, we review recent research on BECs in the presence of deep periodic potentials, which can be reduced to nonlinear chains in appropriate circumstances. These reductions, in turn, exhibit many of the remarkable nonlinear structures (including solitons, intrinsic localized modes, and vortices) that lie at the heart of the nonlinear science research seeded by the FPU paradigm. PACS numbers: 05.45.Yv, 03.75.Lm, 03.75.Nt, 05.30.Jp
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