非线性薛定谔方程的保结构算法与应用

项目名称： 非线性薛定谔方程的保结构算法与应用

项目编号： No.11301350

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 花巍

作者单位： 沈阳师范大学

项目金额： 22万元

中文摘要： 本项目主要研究辛算法在高阶非线性薛定谔方程及Gross-Pitaevskii(GP)方程中的应用。 首先将辛算法用于含时立方五次方非线性薛定谔方程，结合分步Crank-Nicolson格式，讨论方程解的动力学演化；优化改进的辛打靶法（ISM），用于描述玻色-爱因斯坦凝聚的定态GP方程（具有非线性薛定谔方程的形式），改进解的归一化，研究凝聚体的静态性质。 其次，将辛格式用于含时GP方程，提高精度，研究系统参数周期性变化时，凝聚体的均方根半径的演化；研究凝聚体的相干现象，提出简谐势阱与高斯能垒相结合，数值模拟凝聚体的干涉；研究同一陷俘势中凝聚体的动力学性质，以期高精度的计算能给出精准的周期性演化结果。 再次，研究耦合GP方程的辛结构，构造该类方程的辛格式，数值地研究双阱中凝聚体的隧穿、自囚禁及自发对称性破缺。 总之，以上有直接应用特色的保结构算法研究具有重要的基础理论意义和应用价值。

中文关键词： 哈密顿系统；保结构计算；非线性薛定谔方程；GP方程；玻色-爱因斯坦凝聚动力学

英文摘要： This project mainly involves the application of the symplectic algorithm to the high-order nonlinear Schr?dinger equation and to the Gross-Pitaevskii (GP) equation. Firstly, the symplectic algorithm is used to solve the time-dependent cubic and quintic nonlinear Schr?dinger equation, the split step Crank-Nicolson scheme is also considered, and we discuss the dynamic evolution of its solution; Optimize the improved symplectic shooting method (ISM), and apply it to the time-independent GP equation (whose form is similar with the nonlinear Schr?dinger equation) which is used to descrip the Bose-Einstein condensation, improve the normaliztion of the solution, and investigate the static property of the condensation. Secondly,apply the symplectic scheme to the time-dependent GP equation, increased the accuracy, study the evolution of the root mean square radius of the condensate when the system parameter is varied periodically; Study the interference phenomenon of the condensates. We suggest to use the harmonic potential and the Gaussian energy barrier together, and simulate the interference of the condensates numerically; Study the dynamic property of the condensates within one potential, and it is expected that we can obtain more accurate periodically evolution by computation with high order accuracy. Thirdly, study

英文关键词： Hamilton system；Structure-preserving algorithm；Nonlinear Schrodinger equation；GP equation；Bose-Einstein condensation dynamics