請用此 Handle URI 來引用此文件: http://hdl.handle.net/11455/49744
標題: 立波諾夫化簡與延續法處理反應-擴散問題
Liapunov-Schmidt Reduction and Continuation for Reaction-Diffusion Systems
作者: 簡澄陞
施因澤
關鍵字: 數學類
基礎研究
Schrodinger eigenvalue problem
薛丁格特徵值問題
薛丁格–波松特徵值問題
週期邊界條件
雙重網格法
有限元素法
有限差分法
立波諾夫–舒密特化簡
雙重網格法
延續法
非線性的薛丁格方程組
吉義核–曼哈特系統
格雷–史考特系統
羅特加–伏德拉系統
分支
Schrodinger-Poisson eigenvalue problem
periodic boundary conditions
two-grid schemes
finite element methods
finite differences
Liapunov-Schmidt reduction
coupled nonlinear Schrodinger equations,
摘要: 我們提出一個三年的研究計畫,將內容分為兩部分,第一部分處理定義在正方形區域,具週期邊界條件的二階線性橢圓特徵值問題。第二部分處理非線性的薛丁格方程組,並將研究範圍擴展到其他的反應–擴散問題。在第一部分裡,我們先研究與此特徵值問題相對應的變分問題。我們使用泛函分析的技巧證明此問題有無限多個特徵對偶。接著在有限元素子空間的架構下,我們證明相對應的變分特徵值問題存在著有限多個特徵對偶,其特徵向量為此有限維子空間的正交基底。然後我們提出一個雙重網格有限元素的算則,用以計算薛丁格特徵值問題與薛丁格–波松特徵值問題的前幾個最小特徵對偶。在使用六點三角元素逼近時,我們能得到的收斂速率。在此部分我們已完成使用有限差分離散特徵值問題的計算工作。 4()Oh在本計畫的第二部分,我們將使用立波諾夫–舒密特化簡研究非線性的薛丁格方程組正解的性質,主要著重解分支的特性。其次以數值方法諸如雙重網格法與延續法探討其數值解。我們將用同樣的技巧研究其他的反應–擴散問題,諸如吉義核–曼哈特方程組,羅特加–伏德拉系統等。立波諾夫–舒密特化簡的功用在於把一個(或一組)偏微分方程化簡為有限維的問題,再由此有限維的問題來研究PDE解的性質,在PDE的領域裡廣被應用。我們用此方法來探討上述反應–擴散問題解分支的特性。使用此法,我們可以瞭解反應–擴散問題的解在分支點上的局部行為。反應–擴散問題一般皆具有很多的參數,由於解分支的形狀會隨著參數的改變而產生變化,我們將以延續法與雙重網格法與中央差分法為主要架構,來探討反應–擴散問題的數值解,並用以驗證立波諾夫–舒密特化簡所得到的結果。目前我們已利用立波諾夫–舒密特法得到M個非線性薛丁格方程之解分支的一些性質。我們證明此方程組的分支如叉子形狀,而且解分支可向右彎或向左彎,完全由方程組三次項的係數矩陣所決定。其次,我們已用數值延續法驗證理論分析的正確性。我們的數值結果顯示出:當解分支向右彎時,則高峰解出現在正方形區域的四個角落;當解分支向左彎時,則高峰解出現在正方形區域的中央。後者的形狀與物理的實驗結果(見[1, 2])看來是一樣的。
We propose a research project which is composed of two parts. The first part will treat a class of second-order linear elliptic eigenvalue problems defined on a rectangular domain with periodic boundary conditions. In the second part we will study positive solutions of the coupled nonlinear Schrodinger equations, and then extend our study to the other reaction-diffusion systems.In the first part we will show that the eigenvalues of the associated variational problem form an increasing sequence, and the eigenfunctions form an orthonormal basis for the space . Next, we will show that the finite element approximation of the variational problem has a finite sequence of eigenvalues, and the corresponding eigenfunctions form an orthonormal basis for the finite dimensional subspace. Then we will propose a two-gird finite element discretization scheme for computing the first few eigenpairs of the Schrodinger eigenvalue problem and the Schrodinger-Poisson eigenvalue problem. By using the six-node triangular elements to approximate the variational linear eigenvalue problem, we will show the rate of convergence of the proposed algorithm is . 2()LΩ4()OhIn the second part of this project, we will study Liapunov-Schmidt reduction for the qualitative properties of positive solutions of some reaction-diffusion systems. Our model problems include the coupled nonlinear Schrodinger equations, the Gierer-Meinhardt system, Gray-Scott system, the Lotka-Volterra system, and so on.Next, we will exploit numerical methods such as two-grid discretization methods and continuation methods to investigate numerical solutions of the reaction-diffusion systems. The continuation methods are used to trace solution branches of the system, while the two-grid method is used mainly to improve the efficiency of the algorithm we propose. We will discretize the system using centered difference approximations.At this moment we have shown that the bifurcations of the M coupled nonlinear Schrodinger equations are pitchfork. The pitchfork bifurcation can be subcritical or supercritical depending on the coefficients of the cubic terms we choose. The theoretical results have been verified by our numerical experiments using a continuation method. Moreover, our numerical results show that peak solutions exist on the four corners of the square if the bifurcation is supercritical. On the other hand, one single peak solution appears on the center of the square if the bifurcation is subcritical. The shape of the latter resembles the experimental results of Anglin and Kitterle [2], and of Anderson et al. [1].
URI: http://hdl.handle.net/11455/49744
其他識別: NSC95-2115-M231-001-MY3
文章連結: http://grbsearch.stpi.narl.org.tw/GRB/result.jsp?id=1595024&plan_no=NSC95-2115-M231-001-MY3&plan_year=97&projkey=PA9706-0133&target=plan&highStr=*&check=0&pnchDesc=%E7%AB%8B%E6%B3%A2%E8%AB%BE%E5%A4%AB%E5%8C%96%E7%B0%A1%E8%88%87%E5%BB%B6%E7%BA%8C%E6%B3%95%E8%99%95%E7%90%86%E5%8F%8D%E6%87%89%EF%BC%8D%E6%93%B4%E6%95%A3%E5%95%8F%E9%A1%8C
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