Research On Soliton Solution And Its Stability In Ultracold Atoms

In recent years, with the development of the atom trapping and laser cooling technology, solitons and their stability in ultracold atomic gas of soliton have be-come one of the hot research topic. Under the mean-field approximation, ultracold condensed bosonic atoms can be described by the nonlinear schrodinger equation. The cubic nonlinear term in the nonlinear schrodinger equation comes from the two-body collision between atoms. In the experiment, one can use the external fields to change the strength of the interatomic interactions by means of Feshbach resonance. Because of long coherence time and controllable system parameters, systems of ultracold condensed bosonic atoms provide a controllable ideal tool to study solitons.It is well-known that in absence of any external potentials and with uniform nonlinearity, self-defocusing nonlinearity (corresponding to repulsive interaction between atoms) and self-focusing (corresponding to attractive interaction between atoms) support dark and bright solitons, respectively. Recently, it has been shown that if the cubic nonlinear term is dependent on the space coordinate, the nonlin-ear schrodinger equation with self-defocusing nonlinearity can support the stable bright solitons. This is a very interesting result. In the past two years, the exis-tence and stability of the bright solitons in the nonlinear systems with spatially inhomogeneous nonlinearity, such as Bose Einstein condensation systems and non-linear optical systems, has attracted extensive interest of research.In this thesis, we mainly studies a special kind of spatially inhomogeneous self-defocusing nonlinearity nonlinear where the nonlinear coefficient is a polynomial function of the space coordinate. We found that for some specific values of the nonlinear parameters, we can get certain exact algebraic bright soliton solutions and vortex soliton solutions. Compared with the well-known hyperbolic secant-type solitons, algebraic bright solitons have a weaker localization. This soliton only appears in the dissipative systems. By applying the linear stability analysis, the stability regions of these algebraic solitons are given numerically. At the same time, we also discuss their dynamical behavior in the unstable regions. In addition, we study another situation where a homogeneous self-defocusing nonlinearity is superposed by a localized self-focusing nonlinearity. It is found that in certain specific conditions, the nonlinearity also supports exact algebraic bright solitons.