Ground States, Vortical Solitons And Modulation Of Breathers In Bose-Einstein Condensates With Three-dimensional Potentials

Soliton theory is one of the important branches of nonlinear science, which has become a very interesting frontier of physics in recent years. Since the pros-perity in experiments in1995, Bose-Einstein condensates have become a very interesting frontier of physics. In this thesis, we consider ground states, vortical solitons of three-dimensional Bose-Einstein condensates in a combined external potential consisted of a bichromatic optical lattice along the z axis and a an-harmonic confinement along the transverse radial direction, and modulation of breathers in Bose-Einstein condensates with three-dimensional paraboloid, hy-perboloid and a combined external potential consisted of paraboloid and hyper-boloid.In Chapter1, we firstly introduce the background of Bose-Einstein conden-sates and mean field theory of Bose-Einstein condensates. Next, we analyze an anharmonic potential and bichromatic optical lattice potential. These work for below research establish the theoretical basis.In Chapter2, we study Bose-Einstein condensates single ground states, dou-ble ground states, single vortical solitons, double vortical solitons and vortical solitons with fundamental vortices of different atomic number under both an anharmonic potential (harmonic-plus-quartic trap) and bichromatic optical lat-tice potential. The energy functional given by the simplified analytic theory is applied to find optimum initial solutions of ground states and vortical solitons. And the stabilities of the ground states, vortical solitons and vortical solitons with fundamental vortices of different atomic number are confirmed by the numerical simulation of the time-dependent Gross-Pitaevskii equation. We also examine the existence and stabilities of the solutions by means of analysing the anharmonic parameters. We further manipulate the ground states and the vortical soliton-s to an arbitrary position, and find a stable region for successful manipulating ground states and vortical solitons without collapse, which are dragged from an initially stationary to a prescribed position by a steadily moving quasiperiodic bichromatic optical lattice. It is noteworthy that the stability and manipulation of the double vortical solitons and vortical solitons with fundamental vortices of different atomic number are the most important findings of this chapter.In Chapter3, we introduce breathers of the three-dimensional nonlinear Gross-Pitaevskii equation with a gain coefficient. We use the similarity trans-formations to reduce the three-dimensional equation with time-dependent and space-dependent coefficients into a one-dimensional equation with constant coef-ficients. It is demonstrated that paraboloid, hyperboloid and a combined external potential consisted of paraboloid and hyperboloid with time-dependent function are emploied to control the amplitude, the period and width of the breather excitations. It is very important work about modulation of breathers, it can provide an effective method for quantum information storing and processing in Bose-Einstein condensates, and many researchers pay more attention to the ap-plication prospect of the high energy laser in nonlinear optics.