几类非线性波动方程行波解分支的研究
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摘要
本文从动力系统分支理论的角度来研究几类非线性波动方程的行波解分支,并充分利用可积行波系统的首次积分和相图来研究非线性波动方程的显式精确行波解,且在难以获得显式精确解的情况下,利用微分方程定性理论对光滑和非光滑行波解的存在性进行分析.全文共分六章.
第一章是绪论,简要阐述了非线性波动方程的发展历史,研究现状和研究意义.
第二章是预备知识,主要介绍了李继彬教授提出的研究奇非线性行波方程的动力系统方法—“三步法”.
第三章,应用“三步法”研究了非线性色散Drinfel’d-Sokolov (D(m,n))系统的行波解分支及其动力学行为.通过“时间尺度”变换,把奇异行波系统D(m,n)系统转化为一个正则系统,利用奇异系统和正则系统的区别和联系,获得了在不同参数条件下各种光滑行波解和非光滑周期波解存在的充分条件,解释了非光滑周期尖波产生的原因.
第四章,应用“三步法”研究了广义Camassa-Holm-KP方程的动力学行为.证明了该方程存在孤立波解,扭结波解和反扭结波解,紧解,无穷多光滑和非光滑的周期波解.并在参数空间的不同区域内,给出了孤立波解,扭结波解和反扭结波解,紧解,无穷多光滑和非光滑的周期波解存在的充分条件,并求出了上述一些精确的参数表达式.
第五章,应用动力系统理论研究了浅水长波近似方程的精确行波解.在参数空间的不同区域内,给出了光滑孤立波解,扭结波和反扭结波解及无穷多光滑周期波解存在的充分条件,并计算出上述一些显式的精确行波解.
第六章是总结与展望,对本文的工作进行了总结,提出了有待于进一步解决的问题.
In this thesis, from the viewpoint of bifurcation theory of dynamical systems, weinvestigate the bifurcations of several classes of nonlinear wave equations. By makingfull use of the first integrals and phase portraits of integrable wave systems, we study theexplicit and exact traveling wave solutions of the nonlinear wave equations. Meanwhile,by using the qualitative theory of di?erential equations, we make an analysis of theexistence of smooth and non-smooth traveling wave solutions which are di?cult toobtain. This thesis consists of six chapters.
In Chapter l, we summarize the historical background, research developments andsignificance of nonlinear wave equations.
In Chapter 2, we mainly introduce the approach of dynamical systems proposed byProfessor Jibin Li for studying singular nonlinear wave equations. We call the methodas“three-step method”.
In Chapter 3, the bifurcations and dynamical behavior of the nonlinear dispersionDrinfel’d-Sokolov (D(m,n)) system is studied by using the three-step method. Aftermaking a transformation of time scale, the singular traveling wave system of D(m,n)system is reduced to a regular dynamical system. Under di?erent parametric condi-tions, various su?cient conditions to guarantee the existence of smooth and non-smoothtraveling wave solutions are obtained by using the relation between the singular systemand the regular system. And how smooth period traveling wave solutions lose theirsmoothness and become non-smooth period traveling wave solutions is explained.
In Chapter 4, the dynamical behavior of the generalized Camassa-Holm-KP equa-tions is studied by the three-step method. The existence of solitary wave solutions,kink and anti-kink wave solutions, compacton solutions, infinitely many smooth andnon-smooth traveling wave solutions are proved. Under di?erent regions of parametricspaces, various su?cient conditions to guarantee the existence of above solutions aregiven. Some exact explicit parametric representations of the above waves are given.
In Chapter 5, the theory of dynamical systems is used to investigate the exacttraveling wave solutions of the shallow long wave approximate equations. In di?erentregions of the parametric space, su?cient conditions to guarantee the existence ofsmooth solitary wave solutions, kink and anti-kink wave solutions and periodic wavesolutions are given. Some exact explicit expressions of the above waves are obtained.
Finally, the summary of this thesis and the prospect of future study are given.
第一章是绪论,简要阐述了非线性波动方程的发展历史,研究现状和研究意义.
第二章是预备知识,主要介绍了李继彬教授提出的研究奇非线性行波方程的动力系统方法—“三步法”.
第三章,应用“三步法”研究了非线性色散Drinfel’d-Sokolov (D(m,n))系统的行波解分支及其动力学行为.通过“时间尺度”变换,把奇异行波系统D(m,n)系统转化为一个正则系统,利用奇异系统和正则系统的区别和联系,获得了在不同参数条件下各种光滑行波解和非光滑周期波解存在的充分条件,解释了非光滑周期尖波产生的原因.
第四章,应用“三步法”研究了广义Camassa-Holm-KP方程的动力学行为.证明了该方程存在孤立波解,扭结波解和反扭结波解,紧解,无穷多光滑和非光滑的周期波解.并在参数空间的不同区域内,给出了孤立波解,扭结波解和反扭结波解,紧解,无穷多光滑和非光滑的周期波解存在的充分条件,并求出了上述一些精确的参数表达式.
第五章,应用动力系统理论研究了浅水长波近似方程的精确行波解.在参数空间的不同区域内,给出了光滑孤立波解,扭结波和反扭结波解及无穷多光滑周期波解存在的充分条件,并计算出上述一些显式的精确行波解.
第六章是总结与展望,对本文的工作进行了总结,提出了有待于进一步解决的问题.
In this thesis, from the viewpoint of bifurcation theory of dynamical systems, weinvestigate the bifurcations of several classes of nonlinear wave equations. By makingfull use of the first integrals and phase portraits of integrable wave systems, we study theexplicit and exact traveling wave solutions of the nonlinear wave equations. Meanwhile,by using the qualitative theory of di?erential equations, we make an analysis of theexistence of smooth and non-smooth traveling wave solutions which are di?cult toobtain. This thesis consists of six chapters.
In Chapter l, we summarize the historical background, research developments andsignificance of nonlinear wave equations.
In Chapter 2, we mainly introduce the approach of dynamical systems proposed byProfessor Jibin Li for studying singular nonlinear wave equations. We call the methodas“three-step method”.
In Chapter 3, the bifurcations and dynamical behavior of the nonlinear dispersionDrinfel’d-Sokolov (D(m,n)) system is studied by using the three-step method. Aftermaking a transformation of time scale, the singular traveling wave system of D(m,n)system is reduced to a regular dynamical system. Under di?erent parametric condi-tions, various su?cient conditions to guarantee the existence of smooth and non-smoothtraveling wave solutions are obtained by using the relation between the singular systemand the regular system. And how smooth period traveling wave solutions lose theirsmoothness and become non-smooth period traveling wave solutions is explained.
In Chapter 4, the dynamical behavior of the generalized Camassa-Holm-KP equa-tions is studied by the three-step method. The existence of solitary wave solutions,kink and anti-kink wave solutions, compacton solutions, infinitely many smooth andnon-smooth traveling wave solutions are proved. Under di?erent regions of parametricspaces, various su?cient conditions to guarantee the existence of above solutions aregiven. Some exact explicit parametric representations of the above waves are given.
In Chapter 5, the theory of dynamical systems is used to investigate the exacttraveling wave solutions of the shallow long wave approximate equations. In di?erentregions of the parametric space, su?cient conditions to guarantee the existence ofsmooth solitary wave solutions, kink and anti-kink wave solutions and periodic wavesolutions are given. Some exact explicit expressions of the above waves are obtained.
Finally, the summary of this thesis and the prospect of future study are given.
引文
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[2] Russell J S. Report on waves[C]. Edinburgh: Proc Royal Soc. 1844,311-390.
[3] Korteweg D J, de Vries G. 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] Fermi E, Pasta J R, Ulam S M. Studies on nonlinear Problems[C]. Los Alamos ScienceLaboratory Report, 1955.
[5] Zabusky N J, Kruskal M D. Interaction of solitons in acollisions plasma and the recur-rence of initial states[J]. Phys Rev Lett, 1965,15:240-243.
[6] Camassa R, Holm D D. An integrable shallow water equation with peaked solitons[J].Phys Rev Lett, 1993,71(11):1661-1664.
[7] Rosenau P, Hyman J M. Compactons: Solitons with finite wavelength[J]. Phys RevLett, 1993,70(5):564-567.
[8] Adomian G. Review of the decomposition method in applied mathematics[J]. J MathAppl, 1988,135:501-544.
[9] Adomian G. Solving frontier problem of physics: The Decomposition Method[M]. MA:Kluwer Academic Publisher, Boston, 1994.
[10] Baker G A, Graves-Morris P. Essentials of Pad′e approximant[M]. Cambridge: Com-bridge University Press, 1996.
[11] He J H, Wu X H. Construction of solitary solutionand compacton- 1ike solution byvariatonal iteration method[J]. Chaos, Solitons and Fractals, 2006, 29:108-113.
[12] He J H. Variational principles for some nonlinear partial di?erential equations withvariable coe?cients[J]. Chaos, Solitons and Fraetals, 2004,19:847一851.
[13] He J H. Application of homotopy perturbation method to nonlinear Wave equations[J].Chaos, Solitons and Fraetals, 2005,26(3):695.
[14] Wu W, Liao S J. Solving solitary waves with discontinuity by means of the homotopyanalysis method[J]. Chaos, Solitons and Fractals, 2005,26:177-185.
[15] Wu W, Wang C, Liao S J. Solving the one-loop soliton solution of the Vakhnenkoequation by means of the homotopy analysis method[J]. Chaos, Solitons and Fractals,2005,23:1733-1740.
[16] Li J B, Liu Z H. Smooth and non-smooth travelling waves in a non-linearly dispersiveequation[J]. Appl Math Model, 2000,25:41-56.
[17] Li J B, Liu Z H. Travelling wave solutions for a class of nonlinear dispersive equations[J].Chin Ann Math, 2002,23B:397-418.
[18] Li J B, Dai H H. On the Study of Singular Nonlinear Travelling Wave Equations:Dynamical System Approach[M]. Beijing: Science Press, 2007.
[19] Shen J, Li J, Xu W. Bifurcations of travelling wave solutions in a model of the hydrogen-bonded systems[J]. Appl Math Comput, 2005,171(1):242-271.
[20] Shen J, Xu W, Xu Y. Travelling wave solutions in the generalized Hirota-Satsumacoupled KdV system[J]. Appl Math Comput, 2005,161(2):365-383.
[21] Tang S, Li M. Bifurcations of travelling wave solutions in a class of generalized KdVequation[J]. Appl Math Comput, 2006,177(2):589-596.
[22] Tang S, Huang W. Bifurcations of travelling wave solutions for the generalized doublesinh-Gordon equation[J]. Appl Math Comput, 2007,189(2):1774-1781.
[23] Tang S, Huang W. Bifurcations of travelling wave solutions for the K(n,-n,2n) equa-tions[J]. Appl Math Comput, 2008,203(1): 39-49.
[24]申建伟.非线性波方程行波解分岔及其动力学行为的研究[D].西北工业大学, 2006.
[25]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.
[26]王兆娟.广义ZK方程的行波解分支[D].桂林电子科技大学, 2009.
[27]田丽霞.一类非线性波动方程行波解的研究[D].江苏大学, 2008.
[28]李志斌.非线性数学物理方程的行波解[M].北京:科学出版社, 2007.
[29]刘式适,刘式达.特殊函数[M].北京:气象出版社, 2002.
[30]王竹溪,郭敦仁.特殊函数概论[M].北京:北京大学出版社, 2000.
[31] Xie F D, Yan Z Y. New exact solution profiles of the nonlinear dispersion Drinfel’d-Sokolov(D(m,n)) system[J]. Chaos, Solitons and Fractals, 2009,39:866-72.
[32] Deng X J, Cao J L , Li X. Travelling wave solutions for the nonlinear disper-sion Drinfel’d-Sokolov(D(m,n))system[J]. Commun Nonlinear Sci Numer Simulat,2010,15(2),281-290.
[33] Wang J P. A list of 1+1 dimensional integrable equations and their properties[J]. JNonlinear Math Phys, 2002,9:213-33.
[34] Goktas U, Hereman E. Symbolic computation of conserved densities for systems ofnonlinear evolution equations[J]. J Symb Comput, 1997,24(5):591-622.
[35] Wazwaz A M. Exact and explicit travelling wave solutions for the nonlinear Drinfel’d-Sokolov system[J]. Commun Nonlinear Sci Numer Simul, 2006,11:311-25.
[36] Wazwaz A M. The Cole-Hopf transformation and multiple soliton solutions for the inte-grable sixth-order Drinfel’d-Sokolov-Satsuma-Hirota equation[J]. Appl Math Comput,2009,207:248-55.
[37] Guckenheimer J, Holmes P. Nonlinear Oscillations , Dynamical Systems and Bifurca-tions of Vector Fields[M]. New York:Springer-Verlag,1983.
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