Modulational Instability And Dynamics Of Solitons In The Ultracold Atomic Gases

In recent years, ultracold atomic gases are a hot research topic, offers a lot of interest-ing physical phenomena. After realizing Bose-Einstein condensation (BEC) phenomenon in the experimental, a number of nonlinear dynamic behavior were found in condensates, such as soliton generation, the formation of domains, from Ultra-flow changes of the in-sulator. In the understanding of these dynamic behavior, modulational instability (MI) have an important influence. MI is a common phenomenon in nonlinear physics, it can be formed spontaneously in uniform and non-uniform media, and has a wide range of applications in plasma physics, nonlinear optics, fluid dynamics, Therefore, MI and dy-namics of solitons in the ultracold atomic gases has aroused great concern. In this paper, depending on a large number of theoretical and experimental research, we study MI and dynamics of solitons in the ultracold atomic gases. The main contents and conclusions are as follows:Firstly, the concept of phenomenon of BEC、the experiments of realizing BEC、the dynamic equation (G-P equation) of BEC、Fermi condensation and the phenomena of MI and solitons are briefly introduced.Secondly, the MI and dynamics of solitons in the ultracold atomic gases are detailedly studied in the following two chapters. In chapter III, we study the MI of BEC with higher-order interaction. We consider the MI of BEC described by a modified Gross-Pitaevskii (G-P) equation with higher-order nonlinearity both analytically and numerically. A new explicit time-dependent criterion for exciting the MI is obtained. It is shown that the higher-order term can either suppress or enhance the MI, which is interesting for control of the system, instability. Importantly, we predict that with the help of the higher-order nonlinearity, the MI can also take place in a BEC with repulsively contact interactions. The analytical results are confirmed by direct numerical simulations.In chapter IV, an analytical study on the dynamics of dark solitons in superfluid Fermi gases. By using the modified lens-type transformation, the dynamics equation of superfluid Fermi gases is re-duced to a modified one-dimensional nonlinear Shordinoger equation (NLSE). Once again, the use of reductive perturbation method, the NLSE is reduced to a standard Korteweg-de Vries (KdV) equations which may be useful to understand the dynamics of dark solitons in superfluid Fermi gases. It is shown that the existence of soliton solutions in the Fermi gases and obtain a set of solutions. Manipulating and controlling the scattering length between Fermi atomics of different components and the external potential can change soliton’s parameters (amplitude, width). This may be helpful to design experiment. Finally, we summarize the main results and give an outlook of the future in this field.