【作者】 杨琴；

【导师】 张解放；

【作者基本信息】 浙江师范大学 ， 理论物理， 2005， 硕士

【摘要】 玻色-爱因斯坦凝聚是近几年来物理学家的一个研究热点。本文主要从描述玻色-爱因斯坦凝聚的Gross-Pitaevskii（GP）方程出发，运用F-展开法以及Hirota方法，得到解析的多孤子解，从而来研究玻色-爱因斯坦凝聚的孤子动力学问题，侧重讨论了Feshbach共振附近孤子的演化过程。主要内容如下： 论文的第一部分从外囚禁势的角度出发。外部条件对玻色-爱因斯坦凝聚物质的宏观行为的影响非常大，主要表现在外囚禁势。外势有许多形式：简谐外势、光晶格外势、椭圆函数外势、双阱外势以及含时线性外势等等。本文着重讨论了含时线性外势下的BEC模型。基于F函数展开法，得到了该模型的一系列Jacobian椭圆函数解。同时，我们在得到解析解的基础上，特别讨论了含时线性势为V_ext=ZF（T）=Z[mg+H cos（w₁T）]的特殊情况。当F（T）为常数时，该模型描述的是在线性非均匀等离子中的朗缪尔波或电磁波。 论文的第二部分从散射长度的角度出发。原子间相互作用极大地影响了BEC的性质，包括静态性质，例如凝聚体的尺寸、形状等等，还包括动力学性质，譬如集体激发谱、孤立子以及涡旋等等。实验上和理论上都证明了利用Feshbach共振，可以改变散射长度，包括大小以及符号的改变，即 a_s（t）=a_∞[1+△／（B₀-B（t））]，（0.1）其中a_∞表示远离共振点的散射长度，△表征共振宽度，B₀是磁场的共振值。基于该研究背景，在本文的第二部分，我们来讨论散射长度随时间变化的BEC的孤子动力学问题。首先将三维的玻色-爱因斯坦凝更多还原

【Abstract】 Bose-Einstein condensates （BEC） is the hot topic of physics research in recent years. Based on F-expansion method a.nd Hirota method, we investigate the Gross-Pitaevskii equation, which describes the Bose-Einstein condensates, and we obtain the exact multi-soliton solutions. Then we discuss the dynamics of solitons in Bose-Einstein condensates. Especially we discuss the evolution of solitons near Feshbach resonance. The paper is organized as follows:Part I starts from external trapping potential, The macroscopic behavior of BEC matter is highly sensitive to external conditions, and primarily to the external trapping potential. The external potential has many forms, such as harmonic oscillator potential, optical lattice potential, elliptic function potential, double well potential, and so on. In the part I of this paper, Bose - Einstein soliton solutions of the nonlinear Schrodinger equation with time-dependent linear potential are considered. Based on the F-expansion method, we present a number of Jacobian elliptic function solutions. Particular cases of these solutions, where the elliptic function modulus equals 1 and 0, are various localized solutions and trigonometric functions, respectively. Specially, for V_ext = ZF（T） = Z[mg + H cos（ω₁T）], we discussed the Bose-Einstein condensates trapped in the coupling external field with considering the effect of gravity; for F（T） = constant, it describes the wave （Langmuir or electromagnetic） in a linearly inhomoge-neous plasma with cubic nonlinearly.Part II begins with scattering length. The interatomic interactions greatly affect a wide variety of BEC properties, including static properties, such as the size, shape and stability of the condensate, and dynamic properties, like the collective excitation spectrum and soliton and vortex behavior. Theoretical and experimental studies have demonstrated that variation of the s-wave scattering length a_s for Bose and Fermi gases, including a possibility to change its sign, can be achieved by using the Feshbach reso-更多还原