Three-dimensional Ring Vortex Solitons In A Rotating Bose-einstein Condensate

Bose-Einstein condensates (BECs) is a relatively discovered form of matter that wasobserved in dilute atomic gases for the first time in 1995, then is now a subject ofintensive theoretical and experimental study. Beacuse there are many interesting propertiesin Bose-Einstein condensates.the investigation of BECs has unanticipated impact forpeople to understand and analysis the important and fundamental issue in quantummechanics. In recent ten years, we have get a remarkable achievement in the research ofBECs both theoretically and experimentally. In my work, we focus our investigation thestationary and stability analysis about three-dimensional vortex solitons in a rotatingBose-Einstein condensate.First of all, we present the study of quantum theory model about vortex solitons. weinterpretation the origin about Gross-Pitaevskii equation.and its theoretical basis aboutMathematical physics. It is the most important point in my work. In addition, we researchvortex solitons by Gross-Pitaevskii equation. There are two methods in theory. On the onehand, the vortex solitons can be get though the analytical method. On the other hand, italso can be get by numerical method. In this part, the theoretical background of vortexsolitons and the developments on the vortex soliton are presented.Secondly, we give the initial conditions starting with the linear-limit solutions foriteration. We find a series of 3D ring vortex solitons with different vorticity S and differentradial quantum number. and we give different solution in isotropic or anisotropic harmonicpotentials. then we can conclude the density and phase distribution. At the same time, weknow the energy with different interractions and the total particle number.At last, we discuss the dynamic instability evolution. the numerical method is putforward. And it is time-splitting spectral method. On the low-lying excited states ofmultiple quantum number giant vortex stability are discussed. The highly excited vortexsoliton stability is analysed and computed. We design the numerical method about theBogoliubov-de Gennes equations. this article mainly analysis the three-dimensional vortexsoliton stability through nonlinear waveform evolution. At the same time, it is very important to do the linear-stability analysis. So we also made a simple comparativeanalysis using the linear-stability analysis.