一类非线性色散波方程的孤立子解
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摘要
非线性现象是自然界中既普遍又重要的现象。非线性科学是研究非线性现象共性的一门学问,它研究的主体是孤立子,混沌和分形。许多非线性问题的研究最终可归结为非线性系统的描述。
非线性系统的精确解对研究相关的非线性问题非常重要。孤立子研究的一个主要内容,就是寻求非线性系统的解,特别是孤立波解。在过去大约50年中,非线性科学研究领域颇具特色的新成果之一就是创造了求非线性方程的解特别是孤波解的各种精巧方法。
本文正是应用上述方法来研究非线性的色散波方程的孤立子解。在第一、二章首先介绍了非线性波动方程及孤立子理论的研究背景、研究进展和发展现状和意义,总结并分析了现有的求解非线性波动方程的方法。随后介绍了本文研究非线性波动方程孤立波解所用的方法及涉及的相关的概念、定理。
第三章,利用动力系统的定性分析理论,通过相图分析的方法,借助Mathematical软件,通过同宿轨,异宿轨,周期轨,对应解的情况,研究了广义CH方程的孤立子解,并且给出了光滑周期波解,光滑孤立波解,类似扭波解和类似反扭波解的存在条件以及周期尖波解,孤立尖波解的精确表达式。第四章,利用相图分析的方法,研究了Fornberg-Whitham方程的类似扭波解和类似反扭波解,给定参数值,求出了解的精确表达式,画出其图形。
Nonlinear Science,which has solution theory,fractal and chaos as its main parts,is the subject of studying the common futures of nonlinearity. Nonlinearity is universal and important.Most nonlinear problem can be described by nonlinear equations,which generally includes nonlinear.
How to obtain the exact solutions of nonlinear equations is of vital importance to study of the corresponding problem.The key problem in soliton theory is to get solutions of the nonlinear evolution equations, including exact ones or numerical ones.During the past 50 years or so, the scientists have created various ingenious methods to construct exact solutions,especially soliton solutions of nonlinear equations.
In this paper,we should used above methods to study nonlinear dispersive wave equation.In chapter 1 and chapter 2,we introduce the study background,study development and significance of nonlinear wave equation and soliton theory.The methods known up to today for solving the nonlinear wave equation are summarized and analyzed.Then the concerned concepts and theories which used in this paper are introduced.
In the third chapter,the qualitative analysis methods of dynamical system are used to investigate the soliton solutions of the generalizedγ-CH equation.Using the way of the phase portrait bifurcation analysis and mathematical software,we obtain the soliton solutions by the solutions corresponding to the homoclinic orbit, heteroclinic orbit and the periodic orbit.However we give the conditions of there are exist smooth periodic wave solutions,smooth soliton wave solutions,periodic cusp solutions,kink-like wave solutions,antikink-like solutions.And exact explicit parametric representations of periodic cusp solutions and soliton cusp wave solutions are given and the numerical simulation is made.In the forth chapter,using the way of the phase portrait bifurcation analysis,we study kink-like wave solutions and antikink-like solutions of the F-W equation.The exact explicit parametric representations of those solutions are obtained and the numerical simulation is made.
非线性系统的精确解对研究相关的非线性问题非常重要。孤立子研究的一个主要内容,就是寻求非线性系统的解,特别是孤立波解。在过去大约50年中,非线性科学研究领域颇具特色的新成果之一就是创造了求非线性方程的解特别是孤波解的各种精巧方法。
本文正是应用上述方法来研究非线性的色散波方程的孤立子解。在第一、二章首先介绍了非线性波动方程及孤立子理论的研究背景、研究进展和发展现状和意义,总结并分析了现有的求解非线性波动方程的方法。随后介绍了本文研究非线性波动方程孤立波解所用的方法及涉及的相关的概念、定理。
第三章,利用动力系统的定性分析理论,通过相图分析的方法,借助Mathematical软件,通过同宿轨,异宿轨,周期轨,对应解的情况,研究了广义CH方程的孤立子解,并且给出了光滑周期波解,光滑孤立波解,类似扭波解和类似反扭波解的存在条件以及周期尖波解,孤立尖波解的精确表达式。第四章,利用相图分析的方法,研究了Fornberg-Whitham方程的类似扭波解和类似反扭波解,给定参数值,求出了解的精确表达式,画出其图形。
Nonlinear Science,which has solution theory,fractal and chaos as its main parts,is the subject of studying the common futures of nonlinearity. Nonlinearity is universal and important.Most nonlinear problem can be described by nonlinear equations,which generally includes nonlinear.
How to obtain the exact solutions of nonlinear equations is of vital importance to study of the corresponding problem.The key problem in soliton theory is to get solutions of the nonlinear evolution equations, including exact ones or numerical ones.During the past 50 years or so, the scientists have created various ingenious methods to construct exact solutions,especially soliton solutions of nonlinear equations.
In this paper,we should used above methods to study nonlinear dispersive wave equation.In chapter 1 and chapter 2,we introduce the study background,study development and significance of nonlinear wave equation and soliton theory.The methods known up to today for solving the nonlinear wave equation are summarized and analyzed.Then the concerned concepts and theories which used in this paper are introduced.
In the third chapter,the qualitative analysis methods of dynamical system are used to investigate the soliton solutions of the generalizedγ-CH equation.Using the way of the phase portrait bifurcation analysis and mathematical software,we obtain the soliton solutions by the solutions corresponding to the homoclinic orbit, heteroclinic orbit and the periodic orbit.However we give the conditions of there are exist smooth periodic wave solutions,smooth soliton wave solutions,periodic cusp solutions,kink-like wave solutions,antikink-like solutions.And exact explicit parametric representations of periodic cusp solutions and soliton cusp wave solutions are given and the numerical simulation is made.In the forth chapter,using the way of the phase portrait bifurcation analysis,we study kink-like wave solutions and antikink-like solutions of the F-W equation.The exact explicit parametric representations of those solutions are obtained and the numerical simulation is made.
引文
[1]庞小峰。《孤立子理论及其应用》,成都:四川科学技术西出版社,1997
[2]黄景宁,徐济仲,熊吟涛。《孤子概念、原理和应用》,高等教育出版社,2004
[3]陈陆君,梁昌洪。《孤立子理论及其应用》,西安电子科技大学出版社,1997
[4]倪皖荪,魏荣爵。《水槽中的孤波》,上海科技教育出版社,1997
[5]谷超豪等。《孤立子理论与应用》,浙江科学技术出版社,1990
[6]郭柏灵,苏凤秋。《孤立子》,辽宁教育出版社,1998
[7]刘式达,刘式适。《孤波和喘流》,上海科技教育出版社,1994
[8]M.J.Ablowitz,EA.Clarkson.《孤立子、非线性发展方程和逆散射》,世界图书出版公司北京公司,2000
[9]冯培成。《孤立子综述》,零陵师范高等专科学校学报,21(3),27-29
[10]刘正荣。《关于孤立子的研究》,云南大学学报,25(3)(2003)207.211
[11]Guo BL,Liu ZR.Peaked wave solutions of CH-c equation.Sci China Ser A:Math 2003;46(5):696 - 709.
[12]Guo BL,Liu ZR.Two new types of bounded waves of CH-c equation.Sci China Ser A:Math 2005;48(12):1618 - 30.
[13]Tang MY,Zhang WL.Four types of bounded wave solutions of CH-c equation.Sci China Ser A:Math 2007;50(1):132 - 52.
[14]Liu ZR,Qian TF.Peakons and their bifurcation in a generalized Camassa -Holm equation.Int J Bifurcat Chaos 2001;11(3):781 - 92.
[15]Tian LX,Song XY.New peaked solitary wave solutions of the generalized Camassa - Holm equation.Chaos Soliton Fract 2004;19(3):621 - 37.
[16]Khuri SA.New ansatz for obtaining wave solutions of the generalized Camassa - Holm equation.Chaos Soliton Fract 2005;25(3):705 - 10.
[17]Shen JW,Xu W.Bifurcations of smooth and non-smooth traveling wave solutions in the generalized Camassa - Holm equation.Chaos Soliton Fract 2005;26(4):1149 - 62.
[18]Wazwaz AM.Solitary wave solutions for modified forms of Degasperis - Procesi and Camassa - Holm equations.Phys Lett A 2006;352(6):500 - 4.
[19]Wazwaz AM.New solitary wave solutions to the modified forms of Degasperis - Procesi and Camassa - Holm equations.Appl Math Comput 2007;186:130 -41.
[20]Liu ZR,Ouyang ZY.A note on solitary waves for modified forms of Camassa - Holm and Degasperis - Procesi equations.Phys Lett A 2007;366:377 - 81.
[21]G.B.Whitham,Variational methods and applications to water wave,Proc.R.Soc.Lond.Ser.A299(1967) 6 - 25.
[22]B.Fomberg,G.B.Whitham,A numerical and theoretical study of certain nonlinear wave phenomena,Philos.Trans.R.Soc.Lond.Ser.A 289(1978) 373- 404.
[23]R.Ivanov,On the integrability of a class of nonlinear dispersive wave equations,J.Nonlinear Math.Phys.1294(2005) 462-468.
[24]J.Zhou,L.Tian,A type of bounded traveling wave solutions for the Fornberg - Whitham equation,J.Math.Anal.Appl.306(2008) 10 - 16
[25]范恩贵。《可积系统与计算机代数》,科学出版社,2004
[26]庞小峰。《孤子物理学》,四川科学技术出版社,2003
[27]Fisher M.Sshiff J.The Camassa-Holm equation:conserved quantities and initial value problem.Phys Lett A 259(3)(1999) 371-376
[28]Li Jibin and Liu Zhengrong[2002],"Traveling wave solutions for a class of nonlinear disperisive equations," Chinese Ann.Math.Ser.B23(3),397-418.
[29]Jibin Li and Zhengrong Liu,Smooth and non-smooth traveling waves in a nonlinearly dispersive equation.Applied Math Modelling 2000;25:41-56.
[2]黄景宁,徐济仲,熊吟涛。《孤子概念、原理和应用》,高等教育出版社,2004
[3]陈陆君,梁昌洪。《孤立子理论及其应用》,西安电子科技大学出版社,1997
[4]倪皖荪,魏荣爵。《水槽中的孤波》,上海科技教育出版社,1997
[5]谷超豪等。《孤立子理论与应用》,浙江科学技术出版社,1990
[6]郭柏灵,苏凤秋。《孤立子》,辽宁教育出版社,1998
[7]刘式达,刘式适。《孤波和喘流》,上海科技教育出版社,1994
[8]M.J.Ablowitz,EA.Clarkson.《孤立子、非线性发展方程和逆散射》,世界图书出版公司北京公司,2000
[9]冯培成。《孤立子综述》,零陵师范高等专科学校学报,21(3),27-29
[10]刘正荣。《关于孤立子的研究》,云南大学学报,25(3)(2003)207.211
[11]Guo BL,Liu ZR.Peaked wave solutions of CH-c equation.Sci China Ser A:Math 2003;46(5):696 - 709.
[12]Guo BL,Liu ZR.Two new types of bounded waves of CH-c equation.Sci China Ser A:Math 2005;48(12):1618 - 30.
[13]Tang MY,Zhang WL.Four types of bounded wave solutions of CH-c equation.Sci China Ser A:Math 2007;50(1):132 - 52.
[14]Liu ZR,Qian TF.Peakons and their bifurcation in a generalized Camassa -Holm equation.Int J Bifurcat Chaos 2001;11(3):781 - 92.
[15]Tian LX,Song XY.New peaked solitary wave solutions of the generalized Camassa - Holm equation.Chaos Soliton Fract 2004;19(3):621 - 37.
[16]Khuri SA.New ansatz for obtaining wave solutions of the generalized Camassa - Holm equation.Chaos Soliton Fract 2005;25(3):705 - 10.
[17]Shen JW,Xu W.Bifurcations of smooth and non-smooth traveling wave solutions in the generalized Camassa - Holm equation.Chaos Soliton Fract 2005;26(4):1149 - 62.
[18]Wazwaz AM.Solitary wave solutions for modified forms of Degasperis - Procesi and Camassa - Holm equations.Phys Lett A 2006;352(6):500 - 4.
[19]Wazwaz AM.New solitary wave solutions to the modified forms of Degasperis - Procesi and Camassa - Holm equations.Appl Math Comput 2007;186:130 -41.
[20]Liu ZR,Ouyang ZY.A note on solitary waves for modified forms of Camassa - Holm and Degasperis - Procesi equations.Phys Lett A 2007;366:377 - 81.
[21]G.B.Whitham,Variational methods and applications to water wave,Proc.R.Soc.Lond.Ser.A299(1967) 6 - 25.
[22]B.Fomberg,G.B.Whitham,A numerical and theoretical study of certain nonlinear wave phenomena,Philos.Trans.R.Soc.Lond.Ser.A 289(1978) 373- 404.
[23]R.Ivanov,On the integrability of a class of nonlinear dispersive wave equations,J.Nonlinear Math.Phys.1294(2005) 462-468.
[24]J.Zhou,L.Tian,A type of bounded traveling wave solutions for the Fornberg - Whitham equation,J.Math.Anal.Appl.306(2008) 10 - 16
[25]范恩贵。《可积系统与计算机代数》,科学出版社,2004
[26]庞小峰。《孤子物理学》,四川科学技术出版社,2003
[27]Fisher M.Sshiff J.The Camassa-Holm equation:conserved quantities and initial value problem.Phys Lett A 259(3)(1999) 371-376
[28]Li Jibin and Liu Zhengrong[2002],"Traveling wave solutions for a class of nonlinear disperisive equations," Chinese Ann.Math.Ser.B23(3),397-418.
[29]Jibin Li and Zhengrong Liu,Smooth and non-smooth traveling waves in a nonlinearly dispersive equation.Applied Math Modelling 2000;25:41-56.