Hydrodynamic Limit of the Gross-Pitaevskii Equation

We study dynamics of vortices in solutions of the Gross-Pitaevskii equation i? _t u = Δu + ?^?2 u(1 ? |u|²) on ?² with nonzero degree at infinity. We prove that vortices move according to the classical Kirchhoff-Onsager ODE for a small but finite coupling parameter ?. By carefully tracking errors we allow for asymptotically large numbers of vortices, and this lets us connect the Gross-Pitaevskii equation on the plane to two dimensional incompressible Euler equations through the work of Schochet [19 Schochet , S. ( 1996 ). The point vortex method for periodic weak solutions of the 2D Euler equations . Comm. Pure Appl. Math. 49 : 911 – 965 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Mass-imbalanced Bose-Einstein condensed mixtures in rotating perturbed trap

We consider the mass-imbalanced sensibility for the emergence of vortex patterns in the Bose-Einstein condensed binary mixture of rubidium-cesium (⁸⁵Rb-¹³³Cs), confined in quasi-two-dimensional harmonic traps, with one species linearly perturbed in one direction. Non-dipolar coupled species are chosen to highlight mass symmetry effects. We first analyze the condensed mixture in the unperturbed non-rotating regime, where radial phase separation is verified in the immiscible regime, which occurs for large ratio between inter- and intra-species repulsive interactions. By going to the linear perturbed regime, the radial phase separation that occurs in the immiscible condition splits up with the two densities having their maxima at distinct positions. In the rotating regime of both unperturbed and perturbed cases, the minimum rotation is determined in terms of the inter-species interaction to observe vortex structures. In the immiscible regime a dramatic spatial interchange between the species is verified by increasing the rotation.     

Multiple bright and dark soliton solutions in three component spinor Bose-Einstein condensates

We investigate a quasi one dimensional spin-1 Bose-Einstein Condensates (BEC) in the absence of an external confinement governed by a system of three coupled Gross-Pitaevskii (GP) equation. Based on the Lax-pair, we construct one soliton solution employing gauge transformation method. In addition, the multiple bright and dark soliton solutions are obtained by properly choosing amplitude dependent parameter in the Lax-pair. The results of the paper emphasizes the richness in the structure of soliton solutions admitted by the spin components, a phenomenon which has never been brought out to the fore. We have also extended the gauge transformation method to generate two soliton solutions.     

Analytical Solitary Wave Solutions to a （3＋1）-Dimensional Gross-Pitaevskii Equation with Variable Coefficients

A （3＋1）-dimensional Gross-Pitaevskii （GP） equation with time variable coefficients is considered, and is transformed into a standard nonlinear Schrodinger （NLS） equation. Exact solutions of the （3＋1）D GP equation are constructed via those of the NLS equation. By applying specific time-modulated nonlinearities, dispersions, and potentials, the dynamics of the solutions can be controlled. Solitary and periodic wave solutions with snaking and breathing behavior are reported.     

Bose–Einstein condensates under a non-Hermitian spin–orbit coupling

We study the properties of Bose–Einstein condensates under a non-Hermitian spin–orbit coupling(SOC), induced by a dissipative two-photon Raman process. We focus on the dynamics of the condensate at short times, when the impact of decoherence induced by quantum jumps is negligible and the dynamics is coherently driven by a non-Hermitian Hamiltonian. Given the significantly modified single-particle physics by dissipative SOC, the interplay of non-Hermiticity and interaction leads to a quasi-steady-state phase diagram different from its Hermitian counterpart. In particular, we find that dissipation can induce a phase transition from the stripe phase to the plane-wave phase. We further map out the phase diagram with respect to the dissipation and interaction strengths, and finally investigate the stability of quasi-steady states through the time-dependent dissipative Gross–Pitaevskii equation. Our results are readily accessible based on standard experiments with synthetic spin–orbit couplings.     

具阻尼的Gross-Pitaevskii(GP)方程在二维空间中的稳定性

本文讨论出现在吸引玻色-爱因斯坦凝聚中的一类带调和势的阻尼非线性Schr dinger方程.对照玻色-爱因斯坦凝聚的物理性质,证明了阻尼参数存在一个门槛值,即当阻尼参数大于该门槛值时,初值问题的解整体存在;当阻尼参数小于该门槛值时,其初值问题的解将在有限时间内坍塌.     

非线性作用随空间变化的玻色-爱因斯坦凝聚原子的规则与混沌分布

本文研究了非线性作用随空间变化的玻色-爱因斯坦凝聚原子的规则与混沌空间分布.空间变化的凝聚体相位导致系统中存在稳定的原子流.考虑化学势为正、原子间呈排斥作用的系统,构造了系统的一级微扰通解,该通解的有界性条件包含了著名的Mel’nikov混沌判据.在系统不满足微扰条件的情况下,数值模拟表明无论凝聚原子呈混沌分布还是规则分布,原子流的增大都可以破坏凝聚原子分布的空间对称性.     

Nonlinear Double-well Schrödinger Equations in the Semiclassical Limit

We consider time-dependent Schrödinger equations with a double well potential and an external nonlinear, both local and non-local, perturbation. In the semiclassical limit, the finite dimensional eigenspace associated to the lowest eigenvalues of the linear operator is almost invariant for times of the order of the beating period and the dominant term of the wavefunction is given by means of the solutions of a finite dimensional dynamical system. In the case of local nonlinear perturbation, we assume the spatial dimension d=1 or d=2.     

New Families of Soliton and Periodic Solutions of Bose-Einstein Condensation in Linear Magnetic Field and Time-Dependent Laser Field

By using a new generally projective Riccati equation method and with the help of symbolic computation,we consider a nonlinear Gross-Pitaevskii equation with weak bias magnetic and time-dependent laser fields. As a result,some new soliton solutions, rational function solution, and periodic solutions are obtained.     

Non-autonomous wave solutions for the Gross-Pitaevskii (GP) equation with a parabola external potential in Bose-Einstein condensates

A Gross-Pitaevskii (GP) equation with a parabola external potential is considered, and is transformed into a standard nonlinear Schrödinger (NLS) equation. By using the homogeneous balance principle and F-expansion method, we study non-autonomous wave solutions of the GP equation with a parabola external potential. In particular, based on the similarity transformation, several families of non-autonomous wave solutions of the GP equation are presented with snaking behaviors and different amplitude surfaces. These obtained bright-dark soliton solutions can give some potential applications in Bose-Einstein condensates.