臺灣博碩士論文加值系統

在這篇博士論文中，我們應用擬弧長延續法與擬普方法處理多參數的非線性特徵值問題，包含 Rydberg-dressed BEC 問題以及二維的 spin-1 BEC 問題。
首先，我們討論具有三次非線性項的半線性橢圓特徵值問題（SEEP）。接著，我們將阱位能加入方程中，則SEEP問題會變成 GPE 方程。我們利用有效率的預測－修正延續法來計算 SEEP 問題以及 GPE 方程的前幾個解曲線。從數值結果可以看出上述方程的 corank-2 分叉點具有兩個長方形解以及兩個三角形解，這些分叉現象以及節線的結構完全被決定。
其次，我們研究 spin-1 BEC 的線性穩定分析，我們證明這些物理系統的有界解都是中性穩定的，尤其是所有穩定狀態的解以及相關聯的離散穩定狀態的解也都是中性穩定的。接下來我們考慮無磁場外力影響的 spin-1 BEC 系統，針對磁性與反磁性這兩種物理現象，我們分別發展出有效率的多層次擬弧長延續法來處理這些問題。當磁場外力被加入到 spin-1 BEC 系統後，另一種多層次的延續法也被發展出來處理這種具有磁性的例子。我們廣泛的計算出具有磁場作用力以及雷射晶格作用力的 spin-1 BEC 的各種數值結果。
最後，我們利用擬弧長延續法處理 Rydberg-dressed BEC 與二元的 Rydberg-dressed BEC 問題。我們利用分治法的技巧，針對非局部非線性項部份，調整延續法步長的大小。此方法可保證我們能夠精準地逼近想求的解。此外，當此方程加入旋轉項後，基態解的結構也會由一個圓錐變成許多漩渦。在求解的過程中我們亦適時的加入不同的延續參數。我們只需追蹤一個基態解曲線，就能算出GPE方程具有不同係數的非局部非線性項的數值解，因此所提的方法是有效率的。我們所求的數值結果與已發表論文的數值結果是一致的。

In this dissertation, we apply pseudo-arclength continuation methods combined with a spectral collocation method for multi-parameter nonlinear eigenvalue problems. These include Rydberg-dressed Bose-Einstein condensates (BEC), and spin-1 BEC in two-dimension which is governed by a system of three coupled Gross-Pitaevskii equations (GPEs).
We begin with a semilinear elliptic eigenvalue problem (SEEP) with cubic nonlinearity. Then a trapping potential is imposed on the SEEP so that the equation becomes the GPE. An efficient predictor-corrector continuation algorithm is exploited to follow the first few solution curves of the SEEP and the GPE. Our numerical results show that at a corank-2 bifurcation of the equations mentioned above, two rectangular solutions and two triangular solutions branch there. The bifurcation scenarios and their nodal set structures of these equations are completely determined.
Next, We study linear stability analysis for spin-1 BEC. We show that all bounded solutions of this physical system are neutrally stable. In particular, all steady state solutions of the physical system, and the associated discrete steady state solutions are neutrally stable. We consider the physical system without the affect of magnetic field. By exploiting the physical properties of both ferromagnetic and antiferromagnetic cases, we develop efficient multi-level pseudo-arclength continuation algorithms for these two cases, respectively. When the magnetic field is imposed on the physical system, an additional multi-level continuation algorithm is described for the ferromagnetic case. Extensive numerical results for spin-1 BEC in a magnetic field, and in optical lattices are reported.
Finally, we describe pseudo-arclength continuation methods for both Rydberg-dressed BEC, and binary Rydberg-dressed BEC. A divide-and-conquer technique is proposed for dealing with the scaling of the range/ranges of nonlocal nonlinear term/terms, which gives enough information for choosing a proper stepsize. This guarantees that the solution curve we wish to trace can be precisely approximated. In addition, the ground state solution would successfully evolve from one peak to vortices when the rotating term is imposed. Moreover, parameter variables with different number of components are exploited in curve-tracing. The proposed methods have the advantage of tracing the ground state solution curve once to compute the contours for various values of the coefficients of the nonlocal nonlinear term/terms. Our numerical results are consistent with those published in the literatures.

Chapter 1: Introduction . . . . . . . . . . . . . . . . . 1
Chapter 2: Bifurcation scenarios of the GPE . . . . . . . 8
2.1 Introduction . . . . . . . . . . . . . . . . . . . . 8
2.2 Spectral collocation method for the GPE . . . . . . .11
2.3 Predictor-corrector continuation method . . . . . . .12
2.4 Numerical results . . . . . . . . . . . . . . . . . .14
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . .16
Chapter 3: Efficient multi-level continuation algorithms for spin-1 BEC in a magnetic field . . . . . . . . . . . 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . 21
3.2 Spectral collocation method for spin-1 BEC . . . . . 22
3.3 Linear stability analysis . . . . . . . . . . . . . .25
3.4 Efficient multi-level continuation algorithms . . . .26
3.5 Numerical results . . . . . . . . . . . . . . . . . .32
3.6 Analysis of numerical results and conclusions . . . .36
Chapter 4: Pseudo-arclength continuation algorithms for binary Rydberg-dressed BEC . . . . . . . . . . . . . . . 56
4.1 Introduction . . . . . . . . . . . . . . . . . . . . 56
4.2 Spectral collocation method for Rydberg-dressed BEC .57
4.2.1 Trapped Rydberg-dressed BEC . . . . . . . . . . . .57
4.2.2 Rotating Rydberg-dressed BEC . . . . . . . . . . . 60
4.2.3 Binary rotating Rydberg-dressed BEC . . . . . . . .62
4.3 A brief review of the continuation algorithm . . . . 64
4.4 Divide-and-conquer with rescaling . . . . . . . . . .66
4.4.1 One-component Rydberg-dressed BEC . . . . . . . . .66
4.4.2 Two-component Rydberg-dressed BEC . . . . . . . . .68
4.5 Numerical results . . . . . . . . . . . . . . . . . .71
4.6 Analysis of numerical results and conclusions . . . .75
Bibliography . . . . . . . . . . . . . . . . . . . . . . 96

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