Bose-Einstein Condensate Harmonic-Gaussian Self-Similar Dark Solitons And Controlled Vortex Solitons In Three-Dimensional Complex Potentials

Deeper studies of the matter-wave solitons have been promoted as the first ex-perimental realization of Bose-Einstein condensates, especially for dark solitons and vortex solitons. The dark and vortex solitons with special structures and properties will provide multiple choices for the storage and processing of the quantum informa-tion. Considering the importance of the external potentials in constructing solitons, we construct the harmonic-Gaussian self-similar dark solitons with the self-similarity and the controlled vortex solitons with the controlled variations by applying the three-dimensional complex external potentials. Then, we analyze their stabilities and dy-namics in detail. This article contains the following two parts:1. We construct a three-dimensional Bose-Einstein condensate harmonic-Gaussian self-similar dark soliton with the shape of a spherical shell by using an external poten-tial consisting of a three-dimensional harmonic trap and a three-dimensional Gaussian potential in the center. An analytical approximate self-similar dark-soliton solution is found through the standard variational method. The definite expressions of the ampli-tudes, the pulse widths and the spatiotemporal quadratic phases are given considering the energy and boundary conditions. Then, we identify the stability area in the spe-cific parameter space by means of the energy functional method and direct numerical simulations. We pick a specific harmonic-Gaussian self-similar dark soliton from the stable region, investigating its stability, dynamics and the self-similarity in the evolu-tion process. It shows that the amplitude and the corresponding pulse width enlarge si-multaneously. For comparison, the unstable evolution of a specific harmonic-Gaussian self-similar dark soliton in the instability region is displayed. Moreover, we present the comparisons between the analytical self-similar results and numerical simulations, which show great agreement.2. By employing the combined external potential composed of a two-dimensional harmonic trap and a one-dimensional optical lattice, we build a three-dimensional Bose-Einstein condensate controlled vortex soliton with arbitrary atomic number ra-tios under the gain modulations. The cylindrical symmetrical construction of the ex-ternal potential and the oblate shape of every lattice make it to be a multi-vortices soliton with a layer-chain structure. By adopting the energy functional method and direct numerical simulations, we find the stability region of the controlled vortex soli-tons with the atomic number ratio 1:1:1:1 and the linear gain modulations in the specific parameter space. Then, a specific controlled vortex soliton in the stable area is selected. Its stability and dynamics are examined, and the simultaneous manipulations of the controlled variations are realized. We find that the amplitude, the pulse width and its central position can be manipulated synchronously. Additionally, we success-fully realize the simultaneous manipulations of the controlled variations of the vortex solitons with the nonlinear gain modulations.