Study On Nonlinear Matter Wave Solitons And Their Interaction In Bose-Einstein Condensates

The Bose-Einstein condensation (BEC) has become a very interesting frontier of physics in recent years. Since the accomplishment in experiments in 1995, its dynamics, thermodynamics and stability have been studying extensively and we have obtained a large number of knowledge about them. One of the most attractive characteristics is the property of matter-waves in macroscale. Because of the experimental achievement and potential applications in the future, the solitary waves has became one of the main goals nowadays. For the investigation of this subject, much efforts has been done experimentally and theoretically. In this thesis, we try to investigate the characters of the matter-waves in BEC from the point of view of the nonlinear dynamics, and mainly study the stability of solitary waves and their interaction dynamics.In the first part, the interactions of two bright matter-wave solitons in one-dimensional BEC confined by the external potential are investigated analytically and numerically. By using the effective particle approach, which is used to study the solitons in nonlinear optics, the interactions of the matter solitary waves in harmonic trap, double-well trap, and finite potential trap are studied respectively. The various characters of the interactions are analyzed by the effective potential and dynamic equations derived from the effective particle approach. Meanwhile, various outcomes of interactions in harmonic trap are examined systematically by direct simulation of the Gross-Pitaevskii Equation.In the second part, the ring dark solitons and their head-on collisions in homogenous and inhomogenous BEC are studied respectively. The dynamics of this nonplanar solitary wave is studied by the multi-scale perturbation method, and the results show that the system admits a solution with two concentric ring solitons, one moving inwards and the other moving outwards, which in small-amplitude limit, are described by the two cylindrical KdV equations in the respective reference frames. By using the extended Poincare-Lighthill-Kuo (PLK) perturbation method, the analytical phase shifts following the head-on collisions between two ring dark solitary waves are derived. For the inhomogenous BEC, the results are quite different with the homogeneous case.In addition to the interaction between atoms in one component, there are other interactions between the components for two- or multi-components BEC, which make the characters more richer than one-component BEC, and the difficulty of the investigation increases accordingly. Fortunately, the effective particle approach can be generalized to the high-dimensional case and the multi-component case. So in the last part, the stability of the soliton pairs in two-component BEC are studied firstly, and the criterion of the stability for different combination are derived. By using the effective particle approach, the effective potentials and the governing ordinary differential equation, which describe the characteristics of soliton interactions, are obtained, and the dynamics of the interaction are discussed mainly.