The Dynamic Properties Of Lower-dimensional Bose-Einstein Condensates

In this thesis, the dynamic properties of quasi-one-dimensional Bose-Einstein condensates are considered. We study the dynamic properites of soliton and breather in Bose-Einstein condensation. The soliton and breather respectively are discussed by the exact solutions of the nonlinear Schrodinger equations. It is found that the interatomic interaction has an important effect on dynamic properties of Bose-Einstein condensates. So that, we give some suggestions for manipulating experimentally solitons or breathers of Bose-Einstein condensates. It can be summarized as follows:1. The phenomenon of Bose-Einstein condensation is discussed in a quantum statistical perspective. It is given that the quantum statistical properties are the essential aspect for Bose-Einstein condensation. The mean-field theory for the dilute ultra-cold Bose-Einstein condensates are described in detail. The model is established in quasi-one-dimesional Bose-Einstein condesates. It is obtained that the nonlinear Schrodinger equations with time-dependent harmonic trap.2. The exact N-soliton solutions of the nonlinear Schrodinger equation with time-dependent linear potential are presented by employing the Darboux transformation method. As the special cases, soliton-solutions are obtained in Bose-Einstein condensates, and their properties and interaction between solitons are discussed as well. It is found that the bright solitons can keep stable energy state after collision. The phenomenon indicates that the bright solitons in Bose-Einstein condesates have properties of the macroscopic quantum. Moreover, we can see the gravity will change the displaced value of soliton.3. The exact breaher solutions of the nonlinear Schrodinger equation with time-dependent parabolic trap with a complex potential by employing the Darboux transformation method. Specialiy, it is found that the breather can be reflected by the parabolic potential or split into many humps and valleys with the time evolution. The nonlinear tunneling behavior of breather colliding on the parabolic potential is observed. It is shown by studying the breather nonlinear tunneling that the breather obey uncertainty principle in Bose-Einstein condensates, but we also find that the breather with "classic" kinetic energies are reflected elastically and soltions appear like ordinary Newtonian particles.However, we get the exact solutions of the quasi-one-dimesional nonlinear Schrodinger equation by employing the Darboux transformation method on AKNS system. By means of discussing the solutions as solitons and breathers,we find some new dynamic properties of Bose-Einstein condesates. Example for, the effect of the gravity,the collision of the solitons, and the quantum tunneling of the breathers,etc.The results provide many possibilities to manipulate Bose atoms experimentally in the Bose-Einstein condensate system.