超冷原子系统中的非线性薛定谔方程研究及其应用

随着超冷原子理论的发展，出现了许多新类型的非线性薛定谔方程和方程组。获得这些方程(组)解的详尽性质，是深入理解超冷原子系统各种性质的一个重要手段。本项目将研究几类超冷原子的玻色-爱因斯坦凝聚系统中的非线性薛定谔方程(组)，包括带非对称外势的Gross-Pitaevskii方程，以及带自旋轨道耦合的2-分量、3-分量Gross-Pitaevskii方程组。我们将通过解析解，研究1维系统的量子隧穿、量子磁化、量子相变以及其它性质，并在高维情形用变分方法和临界点理论对临界参数下基态解的性质进行定性刻画。本项目关于具有非对称外势的方程(组)的研究，对理解非互易量子隧穿以及低维材料的量子特性有重要意义。到目前为止，对具有非对称势阱的系统的解析研究几乎还是空白，期望本项目在这方面的探索能够起到一定的引领作用。

With the development of the theory of ultracold atoms, there appear a quantity of new types of nonlinear Schrodinger equations. To comprehend deeply the quantum properties of the ultracold-atom systems, an important method is to get the detailed properties of the solutions of these equations. In this project we will consider several kinds of nonlinear Schrodinger equations arising from Bose-Einstein condensates, including nonlinear Schrodinger equations with asymmetric potential, the 2- and 3-component Gross-Pitaevskii system with spin-orbit coupling. By virtue of analytical solutions, we will investigate the quantum tunneling, quantum magnetism, quantum phase transition and other properties of the 1-D ultracold-atom system. We will also apply variational method and critical point theory to analyze these equations to learn the asymptotic behavior of the ground states near the threshold of the parameters. The study on the equations with asymmetric potential is very important for understanding the nonreciprocal quantum tunneling and the quantum properties of low-dimensional materials. Up to now, very few literatures are devoted to the analytical study of the systems with asymmetric potentials, we hope this project can promote the development of this area.