几类非线性波方程行波解的动力学行为研究
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摘要
非线性波方程是描述自然现象的一类重要数学模型,也是非线性数学物理特别是孤立子理论最前沿的研究课题之一.通过对非线性波方程的求解和定性分析的研究,有助于人们弄清系统在非线性情况下的运动变化规律,合理解释相关的自然现象,更加深刻地描述系统的本质特征,推动相关学科如物理学、力学、应用数学以及工程技术的发展.
随着非线性科学的发展,许多物理、化学和生命科学模型都可以转化为非线性方程,如非线性常微分方程、偏微分方程等.非线性方程的求解已经成为非线性科学领域的一个重要研究课题.
本文主要利用积分因子方法和动力系统分支理论研究了几类非线性波方程的行波解及性质,并进一步研究了一类奇异扰动非线性波方程孤立波解的存在性.全文共有六章组成.
第一章是绪论,对非线性波方程的发展历史、研究现状、研究意义进行了叙述.
第二章是预备知识,主要介绍了与本文相关的一些基础理论和方法.
在第三章,我们用积分因子方法研究了两类非线性波方程,广义Camassa-Holm方程和广义G-P程,求出了它们的孤立尖波,孤子类解和周期解.
在第四章,我们利用动力系统分支理论研究了一类广义双sinh-Gordon方程和一类(N+1)维sine-cosine-Gordon方程,讨论了它们的相图及其分支,给出了明确的参数条件以及参数条件下的相图.并给出了行波解的精确参数表示.
在第五章,我们应用几何奇异扰动定理研究了一类奇异非线性波方程,对奇异扰动mKdV方程孤立波解的存在性进行了证明.
最后,就全文进行了总结,就研究中还没有彻底解决的问题进行了说明,并提出了有待进一步研究的问题.
Nonlinear wave equations are important mathematical models for describing nat-ural phenomena and are one of the forefront topics in the studies of nonlinear math-ematical physics, especially in the studies of soliton theory. The research on findingand analyzing exact solutions of nonlinear wave equations can help us understand themotion laws of the nonlinear systems under the nonlinear interactions, explain thecorresponding natural phenomena reasonably, describe the essential properties of thenonlinear systems more deeply, and greatly promote the development of engineeringtechnology and related subjects such as physics, mechanics and applied mathematics.
With the boom of nonlinear science, many models in physics, chemistry and lifesciences can be converted into nonlinear equations, such as nonlinear ordinary dif-ferential equation and partial di?erential equation. Consequently, solving nonlinearequations has become an important research topic in the field of nonlinear science.
In this thesis, we investigate several types of nonlinear wave equations by usingthe technique of integral method and bifurcation theory of planar dynamical systems,calculate their travelling wave solutions, and further study on the existence of solitarywave solutions of perturbed nonlinear wave equation. This paper is formed by sixchapters.
In Chapter 1, the historical background, research developments and significanceof nonlinear wave equations are summarized.
In Chapter 2, the basic theory and method of nonlinear wave equation are pre-sented.
In Chapter 3, by using the technique of integral factors, the peakons, solitarypatterns and periodic solutions of generalized Camassa-Holm equation and generalizedG-P equation are obtained.
In Chapter 4, we investigate the generalized double sinh-Gordon equation and(N+1)-dimensional sine-cosine-Gordon equation by using the bifurcation theory of pla-nar dynamical systems, discuss and analyze their phase portrait and branches, then,work out the exact solutions of the equation.
In Chapter 5, by taking advantage of singular perturbation theory, the existenceof solitary wave solutions of perturbed mKdV equation is proved.
Lastly, a summarization of the whole paper and the still unsolved problems in theresearch are given. Moreover, the future study is prospected.
随着非线性科学的发展,许多物理、化学和生命科学模型都可以转化为非线性方程,如非线性常微分方程、偏微分方程等.非线性方程的求解已经成为非线性科学领域的一个重要研究课题.
本文主要利用积分因子方法和动力系统分支理论研究了几类非线性波方程的行波解及性质,并进一步研究了一类奇异扰动非线性波方程孤立波解的存在性.全文共有六章组成.
第一章是绪论,对非线性波方程的发展历史、研究现状、研究意义进行了叙述.
第二章是预备知识,主要介绍了与本文相关的一些基础理论和方法.
在第三章,我们用积分因子方法研究了两类非线性波方程,广义Camassa-Holm方程和广义G-P程,求出了它们的孤立尖波,孤子类解和周期解.
在第四章,我们利用动力系统分支理论研究了一类广义双sinh-Gordon方程和一类(N+1)维sine-cosine-Gordon方程,讨论了它们的相图及其分支,给出了明确的参数条件以及参数条件下的相图.并给出了行波解的精确参数表示.
在第五章,我们应用几何奇异扰动定理研究了一类奇异非线性波方程,对奇异扰动mKdV方程孤立波解的存在性进行了证明.
最后,就全文进行了总结,就研究中还没有彻底解决的问题进行了说明,并提出了有待进一步研究的问题.
Nonlinear wave equations are important mathematical models for describing nat-ural phenomena and are one of the forefront topics in the studies of nonlinear math-ematical physics, especially in the studies of soliton theory. The research on findingand analyzing exact solutions of nonlinear wave equations can help us understand themotion laws of the nonlinear systems under the nonlinear interactions, explain thecorresponding natural phenomena reasonably, describe the essential properties of thenonlinear systems more deeply, and greatly promote the development of engineeringtechnology and related subjects such as physics, mechanics and applied mathematics.
With the boom of nonlinear science, many models in physics, chemistry and lifesciences can be converted into nonlinear equations, such as nonlinear ordinary dif-ferential equation and partial di?erential equation. Consequently, solving nonlinearequations has become an important research topic in the field of nonlinear science.
In this thesis, we investigate several types of nonlinear wave equations by usingthe technique of integral method and bifurcation theory of planar dynamical systems,calculate their travelling wave solutions, and further study on the existence of solitarywave solutions of perturbed nonlinear wave equation. This paper is formed by sixchapters.
In Chapter 1, the historical background, research developments and significanceof nonlinear wave equations are summarized.
In Chapter 2, the basic theory and method of nonlinear wave equation are pre-sented.
In Chapter 3, by using the technique of integral factors, the peakons, solitarypatterns and periodic solutions of generalized Camassa-Holm equation and generalizedG-P equation are obtained.
In Chapter 4, we investigate the generalized double sinh-Gordon equation and(N+1)-dimensional sine-cosine-Gordon equation by using the bifurcation theory of pla-nar dynamical systems, discuss and analyze their phase portrait and branches, then,work out the exact solutions of the equation.
In Chapter 5, by taking advantage of singular perturbation theory, the existenceof solitary wave solutions of perturbed mKdV equation is proved.
Lastly, a summarization of the whole paper and the still unsolved problems in theresearch are given. Moreover, the future study is prospected.
引文
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[9] P. Rosenau. On nonanlytic solitary waves formed by a nonliearly dispersion[J]. PhysLett A, 1997, 230: 305-318.
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[2] J. S. Russell . Report on waves[C]. Rep. Meet. Birt. Assoc. Adv. Sci. 14th York, London,John Murray. 1844, 311.
[3] D. J. Korteweg, G de Vries. On the change of form long waves advancing in a rectangularcanal, and on a new of type of long stationary waves[J]. Phil Mag, 1895, 39:422-443.
[4] E. Fermi, J. R. Pasta, S. M. Ulam, Studies on nonlinear Problems[C]. Los AlamosScience Laboratory Report ,1955.
[5] N. J. Zabusky, M. D. Kruskal. Interaction of solitons in acollisions plasma and therecurrence of initial states[J]. Phys Rev Lett, 1965, 15: 240-243.
[6] A. Seeger, H. Donth, A. Kochendorfer. Theorie der Versetzungen in eindimensionalenAtomreihen[J],Zeitsehrift fur Physik,1953,134(2):173-193.
[7] R. Camassa, D. D. Holm. An integrable shallow water equation with peaked solitons[J].Phys Rev Lett, 1993, 71(11): 1661-1664.
[8] P. Rosenau, J. M. Hyman. Compactons: Solitons with finite wavelength[J]. Phys RevLett, 1993, 70(5): 564-567.
[9] P. Rosenau. On nonanlytic solitary waves formed by a nonliearly dispersion[J]. PhysLett A, 1997, 230: 305-318.
[10] S. N. Chow, J. K. Hale. Method of Bifurcation Theory[M]. New York: Springer-Verlag,1981.
[11] P. Lawrence. Di?erential Equations and Dynamical Systems[M]. New York: Springer-Verlag, 1991.
[12] J. Li, Z. Liu. Smooth and non-smooth travelling waves in a non-linearly dispersiveequation[J]. Appl Math Model, 2000, 25: 41-56.
[13] J. Li, Z. Liu. Travelling wave solutions for a class of nonlinear dispersive equations[J].Chin Ann Math, 2002, 23B: 397-418.
[14] Z. Liu, Q. Lin, Q. Li. Integral approach to compacton solutions Of Boussinesq-likeB(m,n) equation with fully nonlinear dispersion[J],Chaos,Solitons and Frac-tals,2004,19(5):1071-1081.
[15] J. Li, H. Dai. On the Study of Singular Nonlinear Travelling Wave Equations: Dynam-ical System Approach[M]. Beijing: Science Press, 2007.
[16] J. Shen, J. Li, W. Xu. Bifurcations of travelling wave solutions in a model of thehydrogen-bonded systems[J]. Appl Math Comput, 2005, 171(1): 242-271.
[17] J. Shen, W. Xu, Y. Xu. Travelling wave solutions in the generalized Hirota-Satsumacoupled KdV system[J]. Appl Math Comput, 2005, 161(2): 365-383.
[18] S. Tang, M. Li. Bifurcations of travelling wave solutions in a class of generalized KdVequation[J]. Appl Math Comput, 2006, 177(2): 589-596.
[19] S. Tang, W. Huang. Bifurcations of travelling wave solutions for the generalized doublesinh-Gordon equation[J]. Appl Math Comput, 2007, 189(2): 1774-1781.
[20] S. Tang, W. Huang. Bifurcations of travelling wave solutions for the K(n,-n,2n) equa-tions[J]. Appl Math Comput, 2008, 203(1): 39-49.
[21] A.Chen, W. Huang, J. Li. Qualitative behavior and exact travelling wave solutions ofthe Zhiber-Shabat equation[J]. Journal of Computational and Applied Mathematics,2009, 230: 559-569.
[22]冯大河.非线性波方程的精确解与分支问题研究[D].昆明理工大学, 2007.
[23]张锦炎,冯贝叶.常微分方程几何理论与分支问题(第二版)[M].北京:北京大学出版社, 1980.
[24]叶彦谦,极限环论(修定版)[M].上海科技出版社, 1984.
[25]张芷芬,丁同仁等.微分方程定性理论[M].北京:科学出版社, 1985.
[26]李继彬.非线性微分方程[M].昆明:云南科技出版社, 1987.
[27]李继彬,李存富.非线性微分方程[M].成都科技大学出版社, 1987.
[28]李继彬.浑沌与Melnikov方法[M].重庆:重庆大学出版社, 1989.
[29]陆启韶.常微分方程定性方法和分叉[M].北京:北京航空航天大学出版社, 1989.
[30]丁同仁,李承治.常微分方程教程[M].北京:高等教育出版社, 1992.
[31]谷超豪等.应用偏微分方程[M].北京:高等教育出版社, 1993.
[32]韩茂安,朱德明.微分方程分支理论[M].煤碳工业出版社, 1994.
[33]王高雄,周之铭等.常微分方程[M].北京:高等教育出版社, 1996.
[34]张芷芬,李承治等.向量场分支理论基础[M].北京:高等教育出版社, 1997.
[35]刘式达,刘式适.物理学中的非线性方程[M].北京:北京大学出版社, 1998.
[36]李诩神.孤子与可积系统[M].上海:上海科技教育出版社, 1999.
[37]张锦炎,冯贝叶.常微分方程几何理论与分支问题[M].北京:北京大学出版社,2001.
[38]罗定军等.动力系统的定性与分支理论[M].北京:科学出版社, 2001.
[39]张芷芬,丁同仁,黄文灶等.微分方程定性理论[M].北京:科学出版社, 2003.
[40]范恩贵.可积系统与计算机代数[M].北京:科学出版社, 2004.
[41]黄念宁,陈世荣.完全可积非线性方程的哈密尔顿理论[M].北京:科学出版社,2005.
[42]谷超豪,胡和生,周子翔.孤立子理论中的达布变换及其几何应用(第二版)[M].上海:上海科学技术出版社, 2005.
[43] Z. Yin, S. Lai.Peakons,solitary patterns and periodic solutions for generalized Camassa-holm equations[J].Physics Letters,2008,A372:4141-4143.
[44] A. Chen, W. Huang, S. Tang. Bifurcations of travelling wave solutions for Gilson-pickering equations[J]. Nonlinear Analysis. 2009,(10):2659-2665.
[45] T. Qian, M. Tang. Peakons and periodic cusp waves in a generalized Camassa-Holmequation[J]. Chaos,Solitions,Fractals,2001,12(7):1347-1360.
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