Solitons In Bose-Einstein Condensation

Bose-Einstein condensation is the hot topical of interna-tional reseach in recent years. Because the Gross-Pitaevskii equa-tion, which describes Bose-Einstein condensation by mean-field theory, is a nonlinear Schrodinger equation. Much attention has been focused on its nonlinearity, and its nonlinear phenomena are widely studied. Among them, the investigation of solitons makes a difference for the experimental realization.Solitonic solution and its properties are a principle aspect concerning the' study of nonlinear equation. The Gross-Pitaevskii equation governs the evolution of macroscopic wave function. Its nonlinear properties owe to the interaction between the Bose-Einstein condensates' atoms. The further research in the nonlin-ear issue of Bose-Einstein condensation such as solitons, plays a important role in understanding the main physical features of the condensation, improving theoretical study and the technique.This thesis consists of four chapters. The chapter one in-cludes mainly following three part contents. First, we concisely recall the formation and development of Bose-Einstein condensa-tion's theory and it's experimentation. Secondly, we briefly intro-duce some properties of weakly interactive Bose gases, and derive the nonlinear Gross-Pitaevskii equation that is the condensate macroscopic wave function's equation. Thirdly, we give a exam-pie, an approximation analytical solution of the Gross-Pitaevskii equation in a spherical symmetric harmonic trap is derived by a perturbation theory approach, this solution is consistent with that numerically obtained. In chapter two, we introduce the mod-ern studies and experiments of dark solitons and bright solitons in Bose-Einstein condensates, respectively. In chapter three, we in-troduce the homogeneous balance method and improve some key steps in it, then by using this method we are able to obtain mul-tiple soliton solutions of some nonlinear partial differential equa-tions. We spend the fourth chapter on summary and prospect about solitons of Bose-Einstein condensation.