Soliton Dynamics In Low-Dimensional Bose-Einstein Condensates: Darboux Transformation

Since the bright and dark solitons in the Bose-Einstein Condensates (BECs) have been observed experimentally, the field of nonlinear matter waves has received extensive attention. At the mean-field level, the Gross-Pitaevskii (GP) equation may describe some characteristics of BEC after the condensates appeared. The observed maroscopic quantities in GP equation are s-wave scattering length (it may describe interatomic interactions) and the external trap. Meanwhile the interatomic interactions play an important role on the investigations of the physical properties of BEC, and the external traps greatly affect the nonlinear dynamics properties of BEC. So in this thesis, by using Darboux transformation, we study analytically nonlinear dynamics properties of BEC in an invariable external potential or with different growing model, and obtain some significant results. The thesis is organized as follows:In chapter one, we introduce the theory of BEC, the relevant experiments and the significance of BEC. BEC is very important for foundation research, and the applications in the foreground are beautiful. Similar to well-known laser, BEC maybe bring another technology revolution.Then the Darboux transformation method is presented in chapter two. Based on the"seed"solution of nonlinear equation and the solution of its Lax equation, we can obtain the new solution of nonlinear equation with some mathematics operation. Compared to the multiple-scale method and Hirota method, the Darboux transformation is a more simple and useful method for nonlinear equation.In chapter three, we study analytically nonlinear dynamics properties of Bose-Einstein condensates trapped in an invariable external potential with invariable atomic scattering length. It is shown that inter-atomic interaction strength of condensate has an important effect on its nonlinear dynamics properties. When inter-atomic interaction is strong enough, there only exhibits bright soliton. While inter-atomic interaction is less than a critical value, the alternate evolvement phenomenon between bright and dark soliton appears in the condensate.In chapter four, by using Darboux transformation, we study the nonlinear dynamics of a one-dimensional growing BEC with exponential function of time varying scattering length. It is shown that the growing model has an important effect on the amplitude of the soliton in the condensates. In the absence of the growing model, there exhibit the stable alternate bright solitons in the condensates. In the presence of the growing model, our results show that the amplitude of the bright soliton decreases (increases) for the BEC loses (gains) atoms.In chapter five, the interaction characteristics of two bright solitons in a one-dimensional growing Bose-Einstein condensate with time-dependent periodic atomic scattering length has been investigated. It is shown that the interaction between two bright solitons occurs fission and fusion in the presence of both time-dependent periodic atomic scattering length and the growing case, and these intriguing phenomena can be observed under the current experimental condition.Finally, we make a summary of our work and some prospects for future work in this field.