Vortex Solitons And Stability Analysis In Rotating Bose-Einstein Condensate

Bose-Einstein condensate(BEC), as a new kind of state of matter predicted by Bose and Einstein in 1924, is available in a single wave function description. We can study coherent effects and corresponding atom laser and atom optics of the wave through BEC. Not only in basic research, but also in many fields, such as chip technology, precision measurement and nano technology, the ap-plication of Bose Einstein condensation is of importance. In theory, we mainly study and find the stable soliton solution through the high-dimensional Gross-Pitaevskii(GP) equation under mean-field theory. The solitons studied often include bright soliton, dark soliton and vortex soliton. We mainly studies quan-tized vortex solitons of the rotating BEC in this paper. Compared with the traditional vortex solitons which is a typical example of topological defects and exist in many areas of physics, the quantized vortex solitons we studied allows us to understand and control the quantum turbulence better. This thesis mainly include the following two aspects:1. Vortex solitons in a rotating BEC of single component with spatiotempo-rally modulated interactions of two- and three-body. Firstly, exact solutions of the model are obtained by similarity reduction under the harmonic trap and an-harmonic trap respectively. Secondly, We study the evolution properties of the vortex solitons and explore impaction of the two- and three-body interaction-s, external potential and rotating to the vortex solitons using direct numerical simulation method by adjusting the parameters. Finally, we get stable vortex solitons.2. Vortex solitons in a rotating BEC of multiple component with spatiotem-porally modulated interaction of two-body. To begin with, we build the vortex soliton solutions of (2+1)-dimensional coupled GP equations with cubic nonlin-earities and rotating terms. Next evolution images over time are obtained by using the split-step Fourier method, and we analyze the influence of cubic non-linearities and rotating terms to stability. Then we also give the evolution of the vortices against perturbation with an initial random noise. In the end, stable vortex solitons are depicted.Lastly, we summarize the main works in this article, and put forward the outlook for the future research.