Dynamics Of Bose-Einstein Condensate Solitons Confined In Three-Dimensional Complex Potential

In the recent years, the realization of atomic Bose-Einstein condensates(BECs) has inspired physicists to explore various aspects of this interesting phenomenon, not only because of very precise experimental control that exists over the rel-evant experimental procedures, but also because of the close connection of the description of BECs with other areas of physics, such as plasma physics, nonlinear optics. nonlinear wave theory and superfluidity.The main content of this article is listed as follows,In Chapter1, We firstly introduce the basic concept of Bose-Einstein con-densates in ideal gases, and discuss the possibility of Bose-Einstein condensates in lower dimensional geometry. Then we introduce the Gross-Pitaevskii equation which governs the dynamics of Bose-Einstein condensates when the interatomic interaction is turned on. We briefly introduce the matter wave solitons in the Bose-Einstein condensates system. Finally, we discuss that how to control the interatomic interaction and external harmonic potential in experiments.In Chapter2, we report on the results of two-dimensional repulsive Bose-Einstein condensate gap solitons with considering both isotropic and anisotropic ones trapped in a harmonic potential and an optical lattice by means of the extended variational approach. Analytical critical strength ratio of a harmonic to an optical lattice is necessary to support the stability of multiple lattice wells for the condensates. The collapse threshold (defined in terms of the gap soliton’s norm) corresponding with the width plane (ax, ay), is discussed too. Our results also clearly show that the anisotropic gap solitons are much more stable than the isotropic ones in the lattice well. Further we provide a possibility that can guide and manipulate the gap solitons to an arbitrary position via the time-dependent potential. In a board parameter region, the predictions based on the variational dynamic system are found to be in good agreement with full numerical solutions of the Gross-Pitaevskii equation.In Chapter3, we systematically present new exact solutions of the three- dimensional nonlinear generalized Gross-Pitaevskii equation, with time-varying gain or loss, in both attractive and expulsive harmonic confinement regimes by means of using F-expansion method. The approach allows us to obtain soli-tons for a large variety of solutions depending on the time-varying potential and the gain or loss profiles. The dynamics of these matter waves, includ-ing quasi-breathing solitons, double-quasi-breathing solitons and three-quasi-breathing solitons, is discussed. The explicit functions which describe the evolu-tion of the amplitude, width and trajectory of the soliton’s wave center are pre-sented exactly. It is demonstrated that an arbitrary additional time-dependent gain function can be added on the model to control the amplitude and width of the soliton, and the nonlinearity without affecting the motion of the solitons’ wave center. Additionally, a number of exact traveling waves including the Faraday pattern formation have been found.In Chapter4, we present the dynamics and manipulation of three-dimensional Bose-Einstein condensate ground state and vortex in the complex potential with three-dimensional harmonic potential and one-dimensional optical lattice. The changes of the similar ground’s amplitude is given. The expected analytical prop-erties have been confirmed by direct numerical simulation of the model. First we look for the ansatz solution by means of Particle Swarm Optimization which can be used to seek the minimum of energy function with optimizing a range of the parameters, such as the radial width k⊥and the z-axis width kz. Then it is implemented using a Crank-Nicholson implicit scheme. The results show that the similar ground state and vortex are both stable. We observed that the gain function G(t) affects the amplitude and width of the similar ground state and vortex. Furthermore, we manipulate the similar ground state by slowly moving the optical lattice.At the end is the conclusion of this paper and vision of future.