最近玻色-愛因斯坦凝結的物理現象在實驗室獲得證實，大家對相關的問題感到興趣。本論文針對有雷射光圍攪磁場的非線性薛丁格方程進行研究，並計算在質量守恆的條件下對應於方程式解的能量泛函。經過時間空間變數分離、實部與虛部分解與有限差分法離散化等步驟，原方程式可轉換成一個大型參數化的多項式方程組。
我們利用連續法求滿足質量守恆條件的解。我們也將探討在曲線上發現的分歧點，與在分歧分支上所找到的其他解。我們的數值計算顯示當圍攪磁場角動量小時，沿著特定曲線延拓可以得到解，且這些曲線均為規則曲線。然而隨著角動量增大，曲線上的分歧點亦隨之增加。

Recently the phenomena of Bose-Einstein condensates have been observed in laboratories, and the related problems are extensively studied. In this paper we consider the nonlinear Schrödinger equation in the laser beam rotating magnetic field and compute its corresponding energy functional under the mass conservative condition. By separating time and space variables, factoring real part and image part, and discretizing via finite difference method, the original equation can be transformed to a large scale parametrized polynomial systems. We use continuation method to find the solutions that satisfy the mass conservative condition. We will also explore bifurcation points on the curves and other solutions lying on bifurcation branches. The numerical results show that when the rotating angular momentum is small, we can find the solutions by continuation method along some particular curves and these curves are regular. As the angular momentum is increasing, there will be more bifurcation points on curves.

1 介紹 . . . . . . . . . . . . . . . . . . . . . . . 1
2 Preliminary . . . . . . . . . . . . . . . . . . 3
2.1 連續法 . . . . . . . . . . . . . . . . . .4
2.2 ›‚ 冪迭代. . . . . . . . . . . . . . . . . . . 4
2.3 Schmidt-Mirsky Theorem . . . .5
3 離散化 . . . . . . . . . . . . . . . .. . . . . . .7
3.1 › 角動量算子. . . . . . . . . . . . . . . . 7
3.2 方程與能量. . . . . . . . . . . . . . . . 8
4 路徑追蹤. . . . .. . . . . . . . . . . . . . . .11
4.1 初始解. .. . . . . . . . . . .. . . . . . . 11
4.2 截取曲線 . . . . . . . . . . . . . . . . . 12
4.3 路徑上的奇異點. . . . . . . . . . . . 13
5 Computing singular vector . . . . . . 15
5.1 最小 kernel vector . . . . . . . . . . . 15
5.2 次小 kernel vector . . . . . . . . . . . 16
6 數值結果”. . . . . . . . .. . . . . . . . .. . . .Œ 17
6.1 minimal energy . . . . . . . . . . . . . 17
6.2 small singular vectors . . . . . . . . 47
7 ”結論 . . . . . . . .. . . . . . . . .. . . .Œ . . . .48
8參考文獻 . . . . . . . .. . . . . . . . .. . . .Œ 49

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