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1.
In this paper, we prove existence, symmetry and uniqueness of standing waves for a coupled Gross–Pitaevskii equations modeling component Bose–Einstein condensates BEC with an internal atomic Josephson junction. We will then address the orbital stability of these standing waves and characterize their orbit.  相似文献   

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We prove that in the fast rotating regime, the three-dimensional Gross–Pitaevskii energy describing the state of a Bose Einstein condensate can be reduced to a two-dimensional problem and that the vortex lines are almost straight. Additionally, we prove that the minimum of this two-dimensional problem can be sought in a reduced space corresponding to the first eigenspace of an elliptic operator. This space is called the Lowest Landau level and is of infinite dimension  相似文献   

4.
We study the generalized (3 + 1)-dimensional Gross–Pitaevskii equation with time-dependent nonlinearity, harmonic trap and gain or loss, and obtain bright and dark spatiotemporal similaritons, whose form originates from the exact balance between the nonlinearity, harmonic trap and the gain/loss. Based on these analytical solutions, we firstly discuss the solitonic similaritons interaction and control in a exponential amplification system. Secondly, we investigate the nonlinear tunnelling of bright and dark similariton pairs passing through the nonlinear barrier and well. Results indicate that the similariton can be compressed to a desired width and amplitude in a controllable manner by the choice of the barrier (or well) parameters.  相似文献   

5.
We propose a time-splitting spectral method for the coupled Gross–Pitaevskii equations, which describe the dynamics of rotating two-component Bose–Einstein condensates at a very low temperature. The new numerical method is explicit, unconditionally stable, time reversible, time transverse invariant, and of spectral accuracy in space and second-order accuracy in time. Moreover, it conserves the position densities in the discretized level. Numerical applications on studying the generation of topological modes and the vortex lattice dynamics for the rotating two-component Bose–Einstein condensates are presented in detail.  相似文献   

6.
We study asymptotic behaviour at time infinity of solutions close to the non-zero constant equilibrium for the Gross–Pitaevskii equation in two and three spatial dimensions. We construct a class of global solutions with prescribed dispersive asymptotic behavior, which is given in terms of the linearized evolution. Submitted: May 24, 2006. Revised: December 21, 2006. Accepted: February 6, 2007.  相似文献   

7.
The possibility of finite-time, dispersive blow-up for nonlinear equations of Schrödinger type is revisited. This mathematical phenomena is one of the conceivable explanations for oceanic and optical rogue waves. In dimension one, the fact that dispersive blow up does occur for nonlinear Schrödinger equations already appears in [9]. In the present work, the existing results are extended in several ways. In one direction, the theory is broadened to include the Davey–Stewartson and Gross–Pitaevskii equations. In another, dispersive blow up is shown to obtain for nonlinear Schrödinger equations in spatial dimensions larger than one and for more general power-law nonlinearities. As a by-product of our analysis, a sharp global smoothing estimate for the integral term appearing in Duhamel's formula is obtained.  相似文献   

8.
We consider the persistence and stability of dark solitons in the Gross–Pitaevskii (GP) equation with a small decaying potential. We show that families of black solitons with zero speed originate from extremal points of an appropriately defined effective potential and persist for sufficiently small strength of the potential. We prove that families at the maximum points are generally unstable with exactly one real positive eigenvalue, while families at the minimum points are generally unstable with exactly two complex-conjugated eigenvalues with positive real part. This mechanism of destabilization of the black soliton is confirmed in numerical approximations of eigenvalues of the linearized GP equation and full numerical simulations of the nonlinear GP equation. We illustrate the monotonic instability associated with the real eigenvalues and the oscillatory instability associated with the complex eigenvalues and compare the numerical results of evolution of a dark soliton with the predictions of Newton’s particle law for its position.  相似文献   

9.
Considering the time-dependent external potential and thermal cloud effects, this paper investigates via symbolic computation the dark-soliton dynamics in a two-species Bose–Einstein condensate (BEC), which can be described by the quasi-one-dimensional coupled Gross–Pitaevskii equations. Under the balance between the harmonic potential and thermal cloud effects, dark multi-soliton solutions are derived for the two-species BEC through the Hirota bilinear method. Regions of the ss-wave scattering lengths are ascertained for the existence of the dark solitons in two species. Influence of the scattering lengths and external potential on the background density, soliton width and velocity is examined. Graphical analysis demonstrates that the harmonic and linear potentials can change the propagation paths, collision positions and collision time of the dark solitons, and that the thermal cloud effects can affect the number of atoms in the two-species BEC.  相似文献   

10.
We consider the persistence and stability of dark solitons in the Gross–Pitaevskii (GP) equation with a small decaying potential. We show that families of black solitons with zero speed originate from extremal points of an appropriately defined effective potential and persist for sufficiently small strength of the potential. We prove that families at the maximum points are generally unstable with exactly one real positive eigenvalue, while families at the minimum points are generally unstable with exactly two complex-conjugated eigenvalues with positive real part. This mechanism of destabilization of the black soliton is confirmed in numerical approximations of eigenvalues of the linearized GP equation and full numerical simulations of the nonlinear GP equation. We illustrate the monotonic instability associated with the real eigenvalues and the oscillatory instability associated with the complex eigenvalues and compare the numerical results of evolution of a dark soliton with the predictions of Newton’s particle law for its position.   相似文献   

11.
We consider solutions of the focusing cubic and quintic Gross–Pitaevskii (GP) hierarchies. We identify an observable corresponding to the average energy per particle, and we prove that it is a conserved quantity. We prove that all solutions to the focusing GP hierarchy at the L2L2-critical or L2L2-supercritical level blow up in finite time if the energy per particle in the initial condition is negative. Our results do not assume any factorization of the initial data.  相似文献   

12.
We consider a generalized Fitzhugh–Nagumo equation exhibiting time-varying coefficients and linear dispersion term. By means of specific solitary wave ansatz and the tanh method, a new variety of soliton solutions are derived. The physical parameters in the soliton solutions are obtained as function of the time-dependent model coefficients. The conditions of existence and uniqueness of solitons are presented. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous media that is described by the variable coefficients Fitzhugh–Nagumo equation. Clearly, adaptive methods are straightforward and concise and their applications for the Fitzhugh–Nagumo equation with t-dependent coefficients enable one to construct soliton-like solutions.  相似文献   

13.
The aim of this paper is to carry out a rigorous error analysis for the Strang splitting Laguerre–Hermite/Hermite collocation methods for the time-dependent Gross–Pitaevskii equation (GPE). We derive error estimates for full discretizations of the three-dimensional GPE with cylindrical symmetry by the Strang splitting Laguerre–Hermite collocation method, and for the d-dimensional GPE by the Strang splitting Hermite collocation method.  相似文献   

14.

Bäcklund transformations are applied to study the Gross–Pitaevskii equation. Supported by previous results, a class of Bäcklund transformations admitted by this equation are constructed. Schwarzian derivative as well as its invariance properties turn out to represent a key tool in the present investigation. Examples and explicit solutions of the Gross–Pitaevskii equation are obtained.

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15.
We study the decay of the travelling waves of finite energy in the Gross–Pitaevskii equation in dimension greater than three and prove their uniform convergence to a constant of modulus one at infinity. To cite this article: P. Gravejat, C. R. Acad. Sci. Paris, Ser. I 336 (2003).  相似文献   

16.
In this paper we propose a time–space adaptive method for micromagnetic problems with magnetostriction. The considered model consists of coupled Maxwell's, Landau–Lifshitz–Gilbert (LLG) and elastodynamic equations. The time discretization of Maxwell's equations and the elastodynamic equation is done by backward Euler method, the space discretization is based on Whitney edge elements and linear finite elements, respectively. The fully discrete LLG equation reduces to an ordinary differential equation, which is solved by an explicit method, that conserves the norm of the magnetization.  相似文献   

17.
We present an adaptive sparse grid algorithm for the solution of the Black–Scholes equation for option pricing, using the finite element method. Sparse grids enable us to deal with higher-dimensional problems better than full grids. In contrast to common approaches that are based on the combination technique, which combines different solutions on anisotropic coarse full grids, the direct sparse grid approach allows for local adaptive refinement. When dealing with non-smooth payoff functions, this reduces the computational effort significantly. In this paper, we introduce the spatially adaptive discretization of the Black–Scholes equation with sparse grids and describe the algorithmic structure of the numerical solver. We present several strategies for adaptive refinement, evaluate them for different dimensionalities, and demonstrate their performance showing numerical results.  相似文献   

18.
We analyze the mean-square (MS) stability properties of a newly introduced adaptive time-stepping stochastic Runge–Kutta method which relies on two local error estimators based on drift and diffusion terms of the equation [A. Foroush Bastani, S.M. Hosseini, A new adaptive Runge–Kutta method for stochastic differential equations, J. Comput. Appl. Math. 206 (2007) 631–644]. In the same spirit as [H. Lamba, T. Seaman, Mean-square stability properties of an adaptive time-stepping SDE solver, J. Comput. Appl. Math. 194 (2006) 245–254] and with applying our adaptive scheme to a standard linear multiplicative noise test problem, we show that the MS stability region of the adaptive method strictly contains that of the underlying stochastic differential equation. Some numerical experiments confirms the validity of the theoretical results.  相似文献   

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We prove that the Ginzburg–Landau energy of non-constant travelling waves of the Gross–Pitaevskii equation has a lower positive bound, depending only on the dimension, in any dimension larger or equal to three. In particular, we conclude that there are no non-constant travelling waves with small energy. To cite this article: A. de Laire, C. R. Acad. Sci. Paris, Ser. I 347 (2009).  相似文献   

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