Bright And Dark Soliton Solutions In Growing Bose-Einstein Condensates

In recent years, Bose-Einstein condensation (BEC) is a hot topic in the field of condensed matter physics. This article is starting from the nonlinear Gross-Pitaevskii equation (GPE). In terms of Hirota method, we obtain the exact bright and dark soliton solutions. With these solutions we investigate their dynamics properties and collision characteristic of two solitons. The main contents are as follows.Firstly, we briefly introduce the formation and properties of ideal BEC.Secondly, we explain some properties of weakly interaction Bose gases, and the nonlinear GPE is obtained. Then we reduce the GPE into a quasi-one-dimensional Schr?dinger equation.The realization of the nonlinear matter waves in experiment, such as bright and dark soliton, in Bose-Einstein Condensates is also described.At last, the Hirota method is developed carefully for applying into the growing model of quasi-one-dimensional BECs with attractive and repulsive interaction, respectively. After a tedious calculation we obtain the exact bright and dark soliton solutions analytically. With these solutions we studied the dynamic of solitons including the soliton width, amplitude and velocity. A detail analysis for the asymptotic behavior of two-soliton solutions shows the collision is elastic.