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研究生:張書銘
研究生(外文):Shu-Ming Chang
論文名稱:多種玻色愛因斯坦凝聚現象之數值研究
論文名稱(外文):Numerical Study of Multi-Component Bose-Einstein Condensates
指導教授:林文偉林文偉引用關係
指導教授(外文):Wen-Wei Lin
學位類別:博士
校院名稱:國立清華大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2003
畢業學年度:92
語文別:英文
論文頁數:106
中文關鍵詞:多種玻色愛因斯坦凝聚固定點迭代多重輪生Gross-Pitaevskii方程渦流拓樸同步
外文關鍵詞:multi-component Bose-Einstein condensatesfixed point iterationverticillate multiplingGross-Pitaevskii equationvorticestopological synchronization
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第一部分提出固定點迭代法計算描述多種玻色-愛因斯坦凝聚現象的時間變元無關之向量Gross-Pitaevskii方程式的能量態數值解。證明固定點迭代法是局部且線性收斂若且為若對應的最小能量函數問題有嚴格局部極小值。這個迭代方法搭配能量函數亦可計算出基態與有界態之分歧行為。數值實驗佐證這種迭代法的收斂行為是廣域性的,並且是具有10步到20步間的線性收斂。再者,觀察到一個新的現象,多重輪生。這指得是一種隨著超精細個數增加而會一直輪狀衍生的現象。
第二部分,首先由調和捕捉位能時間變元相關之Gross-Pitaevskii方程式推導出渦流的漸近行為方程式。此漸近行為方程式構成一個常微分系統,此系統可視為是標準Kirchhoff問題的擾動。從數值模擬上,發現到三渦流的有界且非碰撞之行為裡有著混沌、準二週期及準三週期的現象。此外,在當中兩個渦流的行為上觀察到一比一的拓樸同步現象。

In Chapter 1, we propose fixed point methods for computing the
energy state solutions of the time-independent vector Gross-Pitaevskii equation (VGPE) which describes a multi-component Bose-Einstein condensate. We prove that the fixed point iterative methods converge locally and linearly to a solution of the VGPE if and only if the associated minimized energy functional problem has a strictly local minimum. The iterative methods can also be used to compute the bifurcation diagram of ground states and bound states, as well as the energy functional. Numerical experience shows that our iterative methods converge globally and linearly in 10 to 20 steps. In particular, we observe a new phenomenon: verticillate multipling, i.e., the generation of multiple verticillate structures.
In Chapter 2, we derive the asymptotic motion equations of vortices for the time-dependent Gross-Pitaevskii equation with a
harmonic trap potential. The asymptotic motion equations form a
system of ordinary differential equations which can be regarded as a perturbation of the standard Kirchhoff problem. From the
numerical simulation on the asymptotic motion equations, we
observe that the bounded and collisionless trajectories of three
vortices form chaotic, quasi 2- or quasi 3-periodic orbits.
Furthermore, a new phenomenon of $1:1$-topological ynchronization is observed in the chaotic trajectories of two vortices.

Chapter 1 Multi-Component Bose-Einstein Condensates 1
1 Introduction 1
2 VGPEs and Nonlinear Eigenvalue Problems (NEPs) 3
3 Nonlinear Algebraic Eigenvalue Problems (NAEPs) 6
3.1 Two-Dimensional Domain . . . . . . 7
3.2 Three-Dimensional Domain . . . .. . . 11
4 Fixed Point Iteration for NAEPs 15
5 Numerical Algorithms and Results 25
5.1 Two-Dimensional Domain for m = 2, 3 . . . 26
5.2 Verticillate Structures . . . . . . . . . 34
5.3 Three-Dimensional Domain . . . . . . . 50
6 Conclusion 52
Chapter 2 Vortices Dynamics in Bose-Einstein Condensates 53
1 Introduction 53
2 Asymptotic Motion Equations of Vortices 56
3 Numerical Study of Three Vortices 66
3.1 Topological Synchronization . .. . . . . 68
3.2 Case (n1, n2, n3) = (1,.1,.1) . . . . 71
3.3 Case (n1, n2, n3) = (1, 1, 1) . . .. . 73
4 Conclusions 75
References 100

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