The Dynamics Of Localized Waves In Bose-Einstein Condensate

In nonlinear physical systems, soliton and rogue wave are well-known localized waves. They can be described by analytic solutions of nonlinear partial differential equa-tions. Soliton is a stable localized wave packet, which possesses particle-like properties demonstrated by its invariant shape after colliding with another one. Rogue wave usually appear abruptly and disappear without any trace. There is a steep hump which is much higher than the around waves when it emerges. In this thesis, we mainly study on these nonlinear waves in Bose-Einstein condensate systems. We mainly investigate their dy-namics based on exact analytical solutions. The thesis mainly concludes one introduction chapter and six particular research chapters.In chapter1, a brief introduction on Bose-Einstein condensate is given. We intro-duce and describe the main properties of the well-known scalar nonlinear waves, such as bright soliton, dark soliton, and even rogue wave.In chapter2, interference patterns associated with soliton-soliton interaction are in-vestigated based on the well-known two-soliton solution. It is found that the temporal and spatial interference patterns exhibit quite different characteristics. The period of the spatial interference pattern is determined by the relative velocity of the solitons, and the temporal pattern behavior is determined by the peak amplitudes and the relative kinetic energy of the solitons. Analytical expressions for the periods of the interference patterns are obtained. A method for classifying the nonlinearity of many nonlinear systems is proposed. As an example, we discuss possibilities to observe these properties of solitons in a nonlinear planar waveguide.In chapter3, we discuss how to manipulate these nonlinear waves in Bose-Einstein condensate system. The explicit ways are presented theoretically through deriving their properties expressions. Particularly, we obtain one way to "catch" rogue waves through managing nonlinear interaction and atoms exchange between thermal atom clouds and condensates. This would help us to investigate rogue wave conveniently.In chapter4, we study kinds of nonlinear waves in a two-component Bose-Einstein condensate. We obtain bright-dark soliton, vector breather, vector rogue wave solutions. They can be used to describe the interactions between different kind localized waves. Analytic dark rogue wave is given for the first time in the coupled model. Moreover, we find two rogue waves can emerge on the temporal-spatial distribution.In chapter5, we investigate rogue wave solutions in a three-component coupled nonlinear Schrodinger equation. With the certain requirements on the backgrounds of components, we construct a new multi-rogue wave solution that exhibits a structure like a four-petaled flower in temporal-spatial distribution, in contrast to the eye-shaped struc-ture in one-component or two-component systems. The results could be of interest in such diverse fields as Bose-Einstein condensate, nonlinear fibers and super fluid.In chapter6, we investigate pair-tunneling dynamics of vector bright soliton, Akhmediev breather, and rogue wave based on the exact solution derived by Darboux transformation. We find the tunneling behaviors are quite distinctive for them. The tun-neling behavior do not happen for bright soliton. Nonlinear Josephson-like oscillation emerge for Akhmediev breather which is periodic in time. The oscillation just happen once for rogue wave and Akhmediev breather which is periodic in space. Based on the generalized solution, we plot an interesting phase diagram for these well-known nonlin-ear waves.