玻色-爱因斯坦凝聚体的非线性动力学研究

项目名称： 玻色-爱因斯坦凝聚体的非线性动力学研究

项目编号： No.11301106

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： 刘期怀

作者单位： 桂林电子科技大学

项目金额： 23万元

中文摘要： 玻色-爱因斯坦凝聚是一个非常奇异的量子现象，玻色-爱因斯坦凝聚体非线性动力行为的研究对于如何有效控制凝聚体的运动、讨论物质玻色的塌缩与失稳以及合理有效利用凝聚体都具有重要的意义,该问题一直是研究的热点。 本项目以Gross-Pitaevskii方程为模型，用几何的观点来理解玻色-爱因斯坦凝聚体的动力学行为，把问题转化为保面积映射与辛同胚的研究，从理论上运用拓扑、非线性振动、KAM方法和定性理论分析等综合手段来研究周期和拟周期外势下的准一维和多组分玻色-爱因斯坦凝聚系统的动力学行为，对周期和拟周期调制振幅波的存在性、多解性和稳定性进行研究,以及对系统固有形式解的相关性质进行讨论，包括不变环面、有界性等。在理论的基础上，结合平均法和哈密顿扰动方法给出解的高阶近似，并进行数值分析。 本项目的研究有助于理解奇异非线性系统的大范围行为的动力学机制，发展奇异非线性系统的定性方法。

中文关键词： 玻色-爱因斯坦凝聚；周期和拟周期解；调制振幅波；哈密顿系统；弱KAM理论

英文摘要： Bose-Einstein condensation is a very strange quantum phenomenon. Researches on nonlinear dynamics of Bose-Einstein Condensates are of great significance, which help us to control the movement of the condensates, discuss the collapse and instability of Bose matter and use the condensates effectively. Such have always been the hot topics. This project takes Gross-Pitaevskii equation as a model, explicates the qualitative dynamics via the geometric point of view, and transform the problems into studying area-preserving mappings and symplectic homeomorphisms. Based on topological methods, nonlinear vibration, KAM method and qualitative analysis, this project is to study the dynamic behavior of quasi one-dimensional and multi-component Bose-Einstein condensates with periodic or quasi periodic external potential, including the existence, multiplicity and stablility of periodic and quasi-periodic modulation amplitude waves. Related properties of the solutions in the inherent form such as the invariant tori and boundedness will be investigated. Combining with the average method and the Hamiltonian perturbation theory, we will give the higher order approximation for solutions, and numerically simulations will support the theoretical results. By the researches of the selected problems, we will understand the dynamic me

英文关键词： Bose-Einstein Condensates；Periodic and quasi-periodic solutions；Modulation amplitude waves；Hamiltonian system；Weak KAM theory