Self-Similar Evolution Of BEC Matter Wave And Modulation Instability Analysis

Observation of Bose-Einstein condensates (BEC) in ultracold gases of weakly interacting alkali metals has opened up a new possibility for studying the quantum phenomena in macroscopic scales. In particular, the experimental realization of solitons and vortices in BEC has stimulated intensive studies of the nonlinear excitations of matter waves. In this thesis, we investigate the evolution dynamics of BEC matter waves in the presence of time-dependent scattering length and time-dependent harmonic potential, including the exact self-similar evolution of BEC matter waves and their potential applications, the propagation properties of one-dimensional BEC bright solitons and the modulation instabilities on plane waves. This thesis is organized as follows:In Chapter 1, we firstly introduce the basic concept of BEC in ideal gases, and discuss the possibility of BEC in lower dimensional geometry. Then we introduce the Gross-Pitaevskii equation which governs the dynamic of BEC when the interatomic interaction is turned on, and discuss how to control the interatomic interaction and external harmonic potential in experiment.In Chapter 2, we discover that the profile of BEC stationary state can evolve exact self-similarly when the time-dependent interatomic interaction and harmonic potential satisfy a certain condition both for one-, two- and three-dimensional geometry. Based on this discovery, the accuracy in measuring the stationary-state profile is improved when the direct measurement is not available. Furthermore, we also propose a feasible experiment to squeeze the matter wave into very high local matter density to check the validity of three-dimensional Gross-Pitaevskii equation. Specifically, we also find the self-similar condition for vortex in higher dimensional geometry. Further analysis reveals that the topological charge of vortex does not change during the self-similar evolution.In Chapter 3, we present the integrable condition for one-dimensional Gross-Pitaevskii equation and obtain the exact analytical solution which describes the modulation instability and the propagation of bright solitary wave on a continuous wave background. Moreover, we also find the approximate bright solitary wave solutions under the near-integrable conditions. Both of these solutions show that, the amplitude of bright solitary wave with zero boundary condition depends on the scattering length while its motion depends on the external potential. Dark soliton solution can also found in the similar way, but for simplicity, we are not going to discuss it.In Chapter 4, we investigate the modulation instability of Bose-Einstein condensate with three-body interatomic interaction and external harmonic trapping potential. Both of our analytical and numerical results show that, the external potential will either cause the excitation of modulation instable modes or restrain the modulation instable modes from growing.Conclusions and outlook are given in Chapter 5.