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研究生： 戴天恩
Tai, Tien-En
論文名稱： 黎曼空間及橢圓函數的理論及薛丁格方程的應用
The Theory of Riemann Surfaces and Elliptic Functions with application to Nonlinear Schrodinger Equation
指導教授： 蔡孟傑
Chuah, Meng-Kiat
口試委員: 李榮耀
Lee, Jong-Eao
顏東勇
yan, dung-yung
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2013
畢業學年度： 101
語文別： 英文
論文頁數： 49
中文關鍵詞： 薛丁格方程 、黎曼空間 、橢圓函數
相關次數： 點閱：2  下載：0 
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    • 此論文，我們討論黎曼空間的理論及在黎曼空間上積分的計算，接著討論橢圓函數。
    最後，利用這兩個理論討論非線性薛丁格方程的特殊解。

    In this paper, we study the theory of Riemann Surfaces of genus N and the numerical computation of path integrals on those Riemann Surfaces. We then study the classical theory of the elliptic functions.
    Finally, we apply both theorys to solve some special solutions of Nonlinear Schrodinger Equation.

    Contents Abstract 1 Introduction 2 Riemann Surface 2.1 Construct of Riemann Surfaces of genus N 2.2 The Integrals of f(z) over a,b Cycles with Horizontal Cutstruct 2.3 The Integrals of f(z) over a,b Cycles with Vertical Cutstruct 2.4 The Integrals of f(z) over a,b Cycles with Slant Cutstruct 3 Elliptic Functions 3.1 Elliptic Functions. General theorem and The Weierstrassian Functions 3.2 The Theta Functions 3.3 The Jaobian Elliptic Functions 4 The Jaobian Elliptic Functions and The Nonlinear Schrodinger Equation 5 Conclusion Appendix References

    [1] E.T. Whittaker, and G.N. Waston, A Course of Modern Analysis, Cambridge Uni. Press, 1962.
    [2] Forest, M. G., and Lee, J. E., Geometry and modulatiuon theory for the periodic nonlinear Schrodinger equation, inOscillation Theory, Computattion, and Methods of Compensated Compactness, Dafermos, C., Ercksen, J. L., Kinderlehrer,
    D., and Slemrod, M., eds., The IMA volumes in Math. and Its Appl., 2, pp35-70, Springer-Verlag, NY, 1986.
    [3] Ablowitz, M. J., Kaup, D. J., Newell, A. C., Segur, H.,The inverse scattering transform - fourier analysis for nonlinear problems, Appl., Math. 53, 249-315(1974)
    [4] Zakharov, V. E., and Shabat, A. B.,Exact theory of two dimensinal self-focusing and one-dimensional self-odulations of waves in nonlinear media, So-viet Phsy., JEPT 34, 62-69(1972).
    [5] Forest, M. G. and Lee, J. E.,THe Modulation Theory and the General-ized Benjamine-Feir Instability Analysis of Periodic Focussing and Defocussing Nonlinear Schrodinger Equations, preprint(1988).
    [6] Yun-Ting Wu,Theory of Riemann Surfaces and Its applications to Di erential Equations(2010).
    [7] Lee, J. E.,Two-phase solutions of the Nonlinear Schrodinger Equation, NCTU.

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