Studies Of Nonautonomous Coupled Soliton Pairs And Their Applications In Bose-Einstein Condensates

Soliton solution in nonlinear Schrodinger (NLS) equation is the most distinguishing fundamental excitations in a wide range of nonlinear sys-tems. Stable excitation mode is of great significance for the experiment and potential applications. Hence the investigation of exact solutions for the NLS equations is one of the fundamental directions in studies of the nonlinear dynamics in Bose-Einstein condensates (BECs). In recent years, with the advancement of science and technology, means of control for non-linear systems become more flexible. Then the NLS equations used to describe the systems can be extended to generalized equations, which are called as nonautonomous NLS equations. Most recently, the studies of these soliton solutions become a focus of concern for us.In our work, we first carefully investigate integrability conditions of single and coupled nonautonomous NLS equations using Painleve analy-sis and their applications in. Then we find the integrability conditions are as the conditions under which the soliton can exist. For the solitons in BECs, existence of them are due to the balance between the different competition features, including the kinetic energy (dispersion) versus the harmonic potential applied and the dispersion versus the nonlinearity. In the double-component ease, due to the inter-species interaction'influence, the competitions between two components become intricate. They includes all possible different combinations between the dispersion and nonlinearity involving intra- and inter-interactions. Furthermore, we discuss the inte-grability conditions by means of similarity transformation and variational method. we first focus on the equations describing the homogenous Bose-Einstein condensations. According to the integrability conditions, on the one hand, we explore the dynamics of single- and two- solitons from the similarity transformation; on the other hand, we discuss the separation dynamics of two wave packets by variational method.In the second part of thesis we study the dynamics of local waves in heteronuclear BECs with time-dependent interaction, external harmonic potential during the simultaneous evaporative cooling and after that. We construct the soliton pair solutions of nonautonomous coupled NLS equa-tions used to describe the systems by using a combination of the homo-geneous balance principle and the F-expansion technique and obtain some restrictive condition on time-dependent parameters and gains. Under the conditions, the interactions and external potentials are controllable in ex-periments. The external potentials for two components are different in the heteronuclear systems, due to the different mass of two components. In addition, the two solitons in each soliton pair are asymmetric. Then we design two different schemes on the interactions to discuss the restrictive conditions. These conditions are found being completely consistent with Painleve integrability conditions and those from the similarity transforma-tion, respectively. Finally, we investigate the collision dynamics of soliton pairs with or without gains by synchronous modulation of the interactions and external potentials. The interactions are tuned via Feshbach reso-nance. The results show that the collisions of each soliton pair is elastic and the solitons ean keep their identities after the collision. Moreover, we find that the collision time without gain is delayed compared with that with gain in the same initial conditions.These investigations provide a possible way to control the dynamies behaviors of the two-component BEC systems in experiments.