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研究生：	張續寶 Jang, Shiu-Bau
論文名稱：	札克洛夫方程組到立方非線性薛丁格方程的收斂 The Convergence of Zakharov Equations to the Cubic Nonlinear Schrödinger Equation
指導教授：	方永富 Fang, Yung-fu
學位類別：	碩士 Master
系所名稱：	理學院 - 數學系應用數學碩博士班 Department of Mathematics
論文出版年：	2017
畢業學年度：	105
語文別：	英文
論文頁數：	49
中文關鍵詞：	札克洛夫方程組、薛丁格方程、離子聲速
外文關鍵詞：	Zakharov equations, Schrödinger equation, ion acoustic speed
相關次數：	點閱：107 下載：1
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在本論文中，我們詳細闡述了Schochet 和Weinstein 在《The Nonlinear Schrödinger Limit of Zakharov Equations Governing Langmuir Turbulence》一文中的工作來研究札克洛夫方程組到立方非線性薛丁格方程的收斂性。我們觀察到：當札克洛夫方程組中的參數（正比於離子聲速）趨近於無限大時，札克洛夫方程組在形式上約化為立方非線性薛丁格方程。因此，我們期待札克洛夫方程組與立方非線性薛丁格方程的解會隨著參數的增長而有更為相似的表現行為。為了驗證這一個猜想，我們證明：在適當的初始條件下，札克洛夫方程組的解存在。並且，隨著參數趨近於無限大，札克洛夫方程組的解會收斂到立方非線性薛丁格方程的解。總的來說，我們補充了Schochet 和Weinstein 工作討論中所省略的細節，並修正其論文中諸如打印等方面的小錯誤。

In this thesis, we elaborate on Schochet and Weinstein’s work “The Nonlinear Schrödinger Limit of Zakharov Equations Governing Langmuir Turbulence” to study the convergence of Zakharov equations to the cubic nonlinear Schrödinger equation. We oberve that the Zakharov equations reduce formally to the cubic nonlinear Schrödinger equation as the parameter, proportional to the ion acoustic speed, approaches to infinite. Therefore we expect that it should be more and more similar that the behavior of the solutions of the Zakharov equations and the cubic nonlinear Schrödinger equation with the increase of the parameter in the Zakharov equations. To present a justification of this expectation, we show that for suitable initial data solutions of Zakharov equations exist and converge to a solution of the cubic nonlinear Schrödinger equation as the parameter approaches to infinity. On the whole, we fill up the details skipped in the discussion of Schochet and Weinstein’s work and correct some minor mistakes such as typos in their paper.

Contents 
Abstract in Chinese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 
Abstract in English. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II 
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III 
Table of Contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV 
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 
2 Preliminaries and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 
2.1 Notation . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 
2.2 Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 
2.3 Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 
2.4 Functional Analysis and Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . 7 
2.5 Physical Systems of Conservation Laws. . . . . . . . . . . . . . . . . . . . . . . . . 9 
2.6 Results about Harmonic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 
3 Results about the Classical Solutions of Zakharov equations. . . . . . . . . . 17 
4 The Alternative Systems for Zakharov equations . . . . . . . . . . . . . . . . . . .  19 
5 Proofs of the Two Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 
5.1 Proof of Theorem 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 
5.2 Proof of Theorem 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
                                    

[1] Aubin, J. P. Applied Functional Analysis. Wiley: New York, 2000.

[2] Evans, L. C. Partial Differential Equations. American Math. Society: Providence, R. I., 2010.

[3] Folland, G. B. Introduction to Partial Differential Equations. Princeton University Press: Princeton, N.J., 1995.

[4] Guo, B.; Gan, Z.; Kong, L.; Zhang, J. The Zakharov System and its Soliton Solutions. Springer Singapore: Singapore, 2016.

[5] Majda, A. Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer-Verlag: New York, Berlin, Heidelberg, Tokyo, 1984.

[6] Majda, A. J.; Bertozzi A. L. Vorticity and Incompressible Flow. Cambridge University Press: Cambridge, 2002.

[7] Royden, H. L.; Fitzpatrick, P. M. Real Analysis. Prentice Hall: Boston, 2010.

[8] Schochet, S. H.; Weinstein, M. I. The Nonlinear Schrödinger Limit of Zakharov Equations Governing Langmuir Turbulence. Commun. Math. Phys., 106, 569-580, 1986.

[9] Toro, E. F. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer-Verlag Berlin Heidelberg: Berlin, Heidelberg, 2009.

[10] Zakharov, V. E. Collapse of Langmuir waves. Sov. Phys. JETP, 35, 908-912, 1972.

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