一类非线性波动方程行波解的研究
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摘要
非线性现象是自然界中普遍存在的一种重要现象。非线性科学是随着研究非线性现象问题而形成的一门科学,它的研究主体是孤立子、混沌和分形。许多实际的非线性问题最终都可归结为非线性系统来描述。
在非线性系统中,非线性波动方程的孤立子理论研究是其中一个重要和热点内容。孤立子理论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。非线性波动方程的精确求解及其解法研究作为非线性科学中的前沿研究课题和热点问题,极具挑战性。目前虽然已经提出和发展了许多求非线性偏微分方程精确解的方法,但由于求解非线性波动方程没有也不可能有统一而普遍适用的方法,因此继续寻找一些有效可行的方法依然是一项十分重要和极有价值的工作。
本文在对非线性波动方程的现有解法进行了较为系统和深入的研究的基础上,对一类有物理背景的非线性波动方程的行波解,分别从定性和定量的角度,做了较为细致的研究,丰富和发展了非线性波动方程解法研究的内容。本文的工作具有一定的理论意义和应用价值。
全文共分八章。第一、二章首先介绍了非线性波动方程提出的历史背景、研究进展和现状,以及几个重要的非线性波动方程,简要阐述了现有的求解非线性波动方程的方法以及与本文相关的基本概念和基本原理、本文的研究意义和主要内容。第三章研究了非线性BBM型方程的行波解。引入非线性强度概念,把一些经典的方法推广到非线性项更复杂的非线性波动方程-充分非线性BBM方程,获得了丰富的孤立波解,如具有双曲正弦、双曲余弦、双曲正切形式的孤立波模型解,以及光滑孤立波解,kink解,anti-kink解,移动孤立波解和尖峰孤立波解。利用辅助方程法,对于OS-BBM方程,我们构造了一种可以确定孤立波解形式与P(u)之间关系的方法,并且获得了尖峰孤立波解(peakon)以及奇异孤立波解。最后分析了P(u)以及方程系数对解的形式的影响。从动力系统分岔理论的角度,研究了ZK-BBM方程和一个一般BBM方程的行波解。对于ZK-BBM方程,通过行波变换将其等价于一个平面系统,由相平面分析得到了系统的所有可能存在的有界行波解及相应的参数条件,分析参数的变化对系统解的结构的影响,写出了这些解的具体表达式。对于一般BBM方程,由对应行波系统的平衡点性质,讨论了当Hamiltonian值变化时,系统解的变化情况,给出了不同情形下有界解的积分表达式。第四章构造了非线性色散波方程的新型Miura变换。给出了构造连结复杂非线性方程与简单方程的变换的新的代数方法。本方法的特点是可直接从较简单方程的解得到目标方程的行波解。另一方面可给出方程有不同解的条件,以非线性色散KdV方程,K(m+1,2)方程,mKdV方程为例。得到K(m+1,2)方程丰富的行波解,包括周期解,衰减的孤立波解,孤立波解,扭结解。第五章研究了一类b族水波方程的显式孤立波解。通过引进一个参数b,得到一个新的b族方程,它以修正的CH方程和DP方程为其特殊情况。利用扩展的tanh方法、有理双曲函数法和有理指数函数法,将现有的一类水波方程的解做了推广,不但能获得已有的结果,且结论更具一般性。第六章探讨了F-展开法的应用。应用F-展开法及其扩展形式得到了Mizhnik-Novikov-Veselov方程,Klein-Gordon方程,Modified Benjamin-Bona-Mahony方程的孤立波解。第七章是几种形式的孤立波解在实际中的应用。最后一章是对研究内容的总结和展望。
Nonlinearity is universal and important phenomenon in nature. Nonlinear Science, which has soliton, fractal and chaos theories as its main parts, is the subject of studying the nonlinearity. Most nonlinear problems can be described by nonlinear equations.
In the nonlinear systems, the soliton theory of the nonlinear wave equations is the important topic. The key problem in soliton theory is to get solutions of the nonlinear equations. As a leading subject and hot interest in nonlinear science, study on the solution method of the nonlinear wave equations has become more and more challenging. At present, although a number of methods are proposed and developed to look for the exact solutions of the nonlinear wave equations, unfortunately, not all these approaches are universally applicable for solving all kinds of nonlinear wave equations directly.As a consequence, it is still a very significant task to go on searching for various powerful and efficient approaches to solve nonlinear wave equations.
This dissertation is based on systematic research and on the existing technique of solving nonlinear wave equations.we do more meticulous research on traveling wave solutions of a class of nonlinear wave equation with the physical context ,from the qualitative and quantitative point of view.The studies enrich and develop the contents of the nonlinear wave equations ,are of some theoretical significance and application value.
This dissertation consists of eight chapters. In Chapter 1 and Chapter 2, we introduce the historical background, study development of nonlinear wave equation and several important nonlinear wave equations . 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 and the primary contents of this dissertation are reported as well.
In Chapter 3, the travelling wave solutions for BBM-like equations with fully dispersion are studied. By introducing the concept of nonlinear intensity, some classical methods are employed to study solutions of nonlinear wave equation with more complex nonlinear terms (BBM-like equations with fully dispersion) and find abundant solitary wave solutions:solitary pattern solutions expressed in terms of the hyperbolic sine, cosine and tangent functions,smooth solitary wave solution , kink solution, anti-kink solution, floating solitary wave solution, peakon solution .
By the auxiliary equation method,We establish the relationship between a form of solitary wave solutions and P(u), obtain the peakon solution and the singular solitary wave solution.Finally, we give an analysis of P(u) as well as the coefficient on forms of the solutions.
the traveling wave solutions of ZK-BBM equation and general BBM equation are investigated by qualitative analysis method. By studying the bifurcation of this equation and dynamic characteristics, we give the expressions of the solitary wave solution and periodic solitary wave solution according with the bifurcation theory. The limit of periodic solitary cusp wave solution and solitary wave solution both equal to the peakon solution. Then the expressions of the solitary wave solution and periodic solitary wave solution are given at various parameters conditions. Some figures are presented by numerical simulation.
In Chapter 4, new Miura type transformation between nonlinear dis- persive wave equations is established. A new algebraic method is devised to construct a transformation relating the complicated nonlinear wave equations with the simpler ones. A characteristic feature of our method lies in that the travelling wave solutions of an aimed equation can be obtained by the solutions of a simpler equation directly. Another characteristic feature is that conditions under which different solutions appear can be given. We choose the nonlinear dispersive generalized KdV equation (K(m+1,2)) and the mKdV equations to illustrate our method. As a result, abundant travelling wave solutions of the K(m+1,2) equation are obtained, including periodic solutions, smooth solutions with decay, solitary solutions and kink solutions.
In Chapter 5, the b-family of the modified DP-CH equation are discussed. By introducing a parameter b and using the extended tanh method , the rational hyperbolic functions method and the rational exponential functions method,We extend the solutions of a class of wave equation. Not only some are in very good agreement with those obtained in some literatures, but also the conclusion is more general.
In Chapter 6, application of the F-expansion methed are studied. Using the generalized F-expansion methed, solitary wave solutions of Mizhnik-Novikov-Veselov equation, Klein-Gordon equation, Modified Benjamin-Bona-Mahony equation are obtained.
In Chapter 7, some form of solitary wave solutions in the actual application is investigated.
The last chapter, Chapter 8, is devoted to concluding with a short summary and further commentary.
在非线性系统中,非线性波动方程的孤立子理论研究是其中一个重要和热点内容。孤立子理论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。非线性波动方程的精确求解及其解法研究作为非线性科学中的前沿研究课题和热点问题,极具挑战性。目前虽然已经提出和发展了许多求非线性偏微分方程精确解的方法,但由于求解非线性波动方程没有也不可能有统一而普遍适用的方法,因此继续寻找一些有效可行的方法依然是一项十分重要和极有价值的工作。
本文在对非线性波动方程的现有解法进行了较为系统和深入的研究的基础上,对一类有物理背景的非线性波动方程的行波解,分别从定性和定量的角度,做了较为细致的研究,丰富和发展了非线性波动方程解法研究的内容。本文的工作具有一定的理论意义和应用价值。
全文共分八章。第一、二章首先介绍了非线性波动方程提出的历史背景、研究进展和现状,以及几个重要的非线性波动方程,简要阐述了现有的求解非线性波动方程的方法以及与本文相关的基本概念和基本原理、本文的研究意义和主要内容。第三章研究了非线性BBM型方程的行波解。引入非线性强度概念,把一些经典的方法推广到非线性项更复杂的非线性波动方程-充分非线性BBM方程,获得了丰富的孤立波解,如具有双曲正弦、双曲余弦、双曲正切形式的孤立波模型解,以及光滑孤立波解,kink解,anti-kink解,移动孤立波解和尖峰孤立波解。利用辅助方程法,对于OS-BBM方程,我们构造了一种可以确定孤立波解形式与P(u)之间关系的方法,并且获得了尖峰孤立波解(peakon)以及奇异孤立波解。最后分析了P(u)以及方程系数对解的形式的影响。从动力系统分岔理论的角度,研究了ZK-BBM方程和一个一般BBM方程的行波解。对于ZK-BBM方程,通过行波变换将其等价于一个平面系统,由相平面分析得到了系统的所有可能存在的有界行波解及相应的参数条件,分析参数的变化对系统解的结构的影响,写出了这些解的具体表达式。对于一般BBM方程,由对应行波系统的平衡点性质,讨论了当Hamiltonian值变化时,系统解的变化情况,给出了不同情形下有界解的积分表达式。第四章构造了非线性色散波方程的新型Miura变换。给出了构造连结复杂非线性方程与简单方程的变换的新的代数方法。本方法的特点是可直接从较简单方程的解得到目标方程的行波解。另一方面可给出方程有不同解的条件,以非线性色散KdV方程,K(m+1,2)方程,mKdV方程为例。得到K(m+1,2)方程丰富的行波解,包括周期解,衰减的孤立波解,孤立波解,扭结解。第五章研究了一类b族水波方程的显式孤立波解。通过引进一个参数b,得到一个新的b族方程,它以修正的CH方程和DP方程为其特殊情况。利用扩展的tanh方法、有理双曲函数法和有理指数函数法,将现有的一类水波方程的解做了推广,不但能获得已有的结果,且结论更具一般性。第六章探讨了F-展开法的应用。应用F-展开法及其扩展形式得到了Mizhnik-Novikov-Veselov方程,Klein-Gordon方程,Modified Benjamin-Bona-Mahony方程的孤立波解。第七章是几种形式的孤立波解在实际中的应用。最后一章是对研究内容的总结和展望。
Nonlinearity is universal and important phenomenon in nature. Nonlinear Science, which has soliton, fractal and chaos theories as its main parts, is the subject of studying the nonlinearity. Most nonlinear problems can be described by nonlinear equations.
In the nonlinear systems, the soliton theory of the nonlinear wave equations is the important topic. The key problem in soliton theory is to get solutions of the nonlinear equations. As a leading subject and hot interest in nonlinear science, study on the solution method of the nonlinear wave equations has become more and more challenging. At present, although a number of methods are proposed and developed to look for the exact solutions of the nonlinear wave equations, unfortunately, not all these approaches are universally applicable for solving all kinds of nonlinear wave equations directly.As a consequence, it is still a very significant task to go on searching for various powerful and efficient approaches to solve nonlinear wave equations.
This dissertation is based on systematic research and on the existing technique of solving nonlinear wave equations.we do more meticulous research on traveling wave solutions of a class of nonlinear wave equation with the physical context ,from the qualitative and quantitative point of view.The studies enrich and develop the contents of the nonlinear wave equations ,are of some theoretical significance and application value.
This dissertation consists of eight chapters. In Chapter 1 and Chapter 2, we introduce the historical background, study development of nonlinear wave equation and several important nonlinear wave equations . 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 and the primary contents of this dissertation are reported as well.
In Chapter 3, the travelling wave solutions for BBM-like equations with fully dispersion are studied. By introducing the concept of nonlinear intensity, some classical methods are employed to study solutions of nonlinear wave equation with more complex nonlinear terms (BBM-like equations with fully dispersion) and find abundant solitary wave solutions:solitary pattern solutions expressed in terms of the hyperbolic sine, cosine and tangent functions,smooth solitary wave solution , kink solution, anti-kink solution, floating solitary wave solution, peakon solution .
By the auxiliary equation method,We establish the relationship between a form of solitary wave solutions and P(u), obtain the peakon solution and the singular solitary wave solution.Finally, we give an analysis of P(u) as well as the coefficient on forms of the solutions.
the traveling wave solutions of ZK-BBM equation and general BBM equation are investigated by qualitative analysis method. By studying the bifurcation of this equation and dynamic characteristics, we give the expressions of the solitary wave solution and periodic solitary wave solution according with the bifurcation theory. The limit of periodic solitary cusp wave solution and solitary wave solution both equal to the peakon solution. Then the expressions of the solitary wave solution and periodic solitary wave solution are given at various parameters conditions. Some figures are presented by numerical simulation.
In Chapter 4, new Miura type transformation between nonlinear dis- persive wave equations is established. A new algebraic method is devised to construct a transformation relating the complicated nonlinear wave equations with the simpler ones. A characteristic feature of our method lies in that the travelling wave solutions of an aimed equation can be obtained by the solutions of a simpler equation directly. Another characteristic feature is that conditions under which different solutions appear can be given. We choose the nonlinear dispersive generalized KdV equation (K(m+1,2)) and the mKdV equations to illustrate our method. As a result, abundant travelling wave solutions of the K(m+1,2) equation are obtained, including periodic solutions, smooth solutions with decay, solitary solutions and kink solutions.
In Chapter 5, the b-family of the modified DP-CH equation are discussed. By introducing a parameter b and using the extended tanh method , the rational hyperbolic functions method and the rational exponential functions method,We extend the solutions of a class of wave equation. Not only some are in very good agreement with those obtained in some literatures, but also the conclusion is more general.
In Chapter 6, application of the F-expansion methed are studied. Using the generalized F-expansion methed, solitary wave solutions of Mizhnik-Novikov-Veselov equation, Klein-Gordon equation, Modified Benjamin-Bona-Mahony equation are obtained.
In Chapter 7, some form of solitary wave solutions in the actual application is investigated.
The last chapter, Chapter 8, is devoted to concluding with a short summary and further commentary.
引文
[1]Russell J.S.Reports on waves.Edinburgh:Proc Royal Soc.1844:311-390
[2]Keteweg DJ,de Vries G.On the change of form long waves advancing in a rectangular canal,and on a new of type of long stationary waves.Phil Mag.1895,39(422-443)
[3]Zabusky N J,Kruskal M D.Interation of solitions in a collisionless plasma and the recurrence of intial states.Phys Rev Lett.1965,15:240-243
[4]Vladimir G M.Soliton Phenomenology.Netherlands:Kluwer Academic Publishers,1990
[5]艾仑伯格 G.孤立子-物理学家用的数学方法.北京:科学出版社,1989
[6]Miura R M.The Korteweg-de Vries equation:A survey of results.SIAM Rev.1976,8:412-459
[7]Kox A J.Korteweg,de Vries,and Dutch science at the turn of the century,cta Applicandae Mathematica.1995,39:91-92
[8]郭柏灵.非线性演化方程.上海:上海科技教育出版社,1995
[9]郭柏灵,庞小峰.孤立子.北京:科学出版社,1987
[10]谷超豪,胡和生and周子翔.孤立子理论中的Darboux变换及其几何应用.上海:上海科技出版社,1999
[11]李诩神.孤子与可积系统.上海:上海科技出版社,1999
[12]刘式达,刘式适.孤波和湍流.上海:上海科技教育出版社,1994
[13]倪皖荪,魏荣爵.水槽中的孤波.上海:上海科技教育出版社,1997
[14]黄念宁.孤子理论和微扰方法.上海:上海科技教育出版社,1996
[15]Camassa R,Holm D D.An integrable shallow water equation with peaked solitons.Physical Review Letters.1993,71(11):1661-1664
[16] Liu Z R, Wand R Q, Jing Z J. Peaked wave solutions of Comassa-Holm equation.Chaos,Solitons and Fractals. 2004, 19:77-92
[17] Liu Z R, Qian T F. Peakons and their bifurcation in a generalized Camassa - Holm equation. International Journal of Bifurcation and Choas. 2001, 11(3):781-792
[18] Liu Z R, Qian T F. Peakons of the Camassa - Holm equation. Applied Mathematical Modeling. 2002, 26:473-480
[19] Qian T F, Tang M Y. Peakons and periodic cusp waves in a generalized Camassa -Holm equation. Chaos,Solitons and Fractals. 2001, 12:1347-1360
[20] P Rosenau, J. M. Hyman. Compactons:solitons with finite wavelengths. Phys Rev Lett. 1993, 70(5):564-567
[21] P Rosenau. Nonlinear dispersion and compact structures. Phys Rev Lett. 1994,73(13):1737-1741
[22] P Rosenau. On solitons, compactons, and Lagrange maps Physical. Phys Lett A.1996, 211:265-275
[23] Rosenau P. Compact and noncompact dispersive patterns. Phys Lett A. 2000,275:193-203
[24] A M Wazwaz. New solitary wave special solutions with compact support the nonlinear dispersive K(m,n) equations. Chaos Solitons and Fractals. 2002, 13:321-330
[25] A M Wazwaz. Compactons dispersive structures for variants of the variants the K(n,n) and the KP equations. Chaos Solitons and Fractals. 2002, 13:1053-1062
[26] A M Wazwaz. Solutions of compact and noncompact structures for nonlinear Klein-Gordon type. Applied Mathematics and Computation. 2003, 134:487-500
[27] Z Y Yan. Chaos, Solitons and Fractals. 2002, 14:1151-1158
[28] Liu Zhengrong, Chen Can. Compactons in general compressible hyperelastic rod. Chaos,Solitons and Fractals. 2004, 22:627-640
[29] Chen S, Foias C, Holm D D, et al. A connection between the Camassa-Holm equations and turbulent flows in channels and pipes. Phys Fluids. 1999, 11 (8):2343-2353
[30] Fisher M, Schiff J. The Camassa Holm equation: conserved quantities and the initial value problem. Phys Lett A. 1999, 259(5):371-376
[31] Kwek K H, Gao Hongjun, Zhang Weinian, et al. An initial boundary value problem of Camassa-Holm equation. J Math Phys. 2000, 41(12):8279-8285
[32] Kraenkel R A, Senthilvelan M, Zenchuk A I. Lie symmetry analysis and reductions of a two-dimensional integrable generalization of the Camassa-Holm equation. Phys Lett A. 2000, 273(3): 183-193
[33] Constantin A. On the scattering problem for the Camassa-Holm equation. Proc R Soc London A. 2001, 457:953-970
[34] Zenchuk A I. The spectral problem and particular solutions to the (2+2)-dimensional integrable generalization of the Camassa-Holm equation. Advances in nonlinear mathematics and science ,Phys D. 2001, 152/153:178-188
[35] Reyes E G. Geometric integrability of the Camassa-Holm equation. Lett Math Phys.2002, 59(2):117-131
[36] Lenells J. The scattering approach for the Camassa-Holm equation. J Nonliear Math Phys. 2002, 9(4):389-393
[37] Johnson R S. On solutions of the Camassa-Holm equation. Soc Lond Proc Ser A Math Phys Eng Sci. 2003, 459(2035): 1687-1708
[38] Mckean H P. The Liouville correspondence between the Korteweg-de Vries and the Camassa-Holm hierarchies. Comm Pure Appl Math. 2003, 56(7):998-1015
[39] Badanin A, Klein M, Korotyaev E. The Marchenko-Ostrovski mapping and the teace formula for the Camassa-Holm equation. J Funct Anal. 2003, 203(2):449-518
[40] Johnson R S. The Camassa-Holm equation for water waves moving over a shear flow. Fluid Dynam Res. 2003, 33(1-2):97-111
[41] Penskoi A V, Veselov A P. Discrete Lagrangian on the Virasoro group and Camassa- Holm family. Nonlinearity. 2003, 16(2):683-688
[42] Lopes O. Stability of peakons for the generalized Camassa-Holm equation. J Differential Equations. 2003, 5:12
[43] Gorsky J M. The Cauchy problem for a modified Camassa-Holm equation with analytic initial data. Differential Integral Equations. 2004, 17(11-12):1233-1254
[44] Boyd J P. Near-Corner waves of the Camassa-Holm equation. Phys Lett A,. 2005,336(4-5):342-348
[45] Lenells J. Conservation laws of the Camassa-Holm equation. J Phys A. 2005,38(4):869-880
[46] Kalisch H, Lenells J. Numerical study of traveling-wave solutions for the Camassa- Holm equation. Chaos,Solitons and Fractals. 2005, 25:287-298
[47] Khuri S A. New anstz for obtaining wave solutions of the generalized Camassa-Holm equation. Chaos,Solitons and Fractals. 2005, 25(3):705-710
[48] Wahlen E A. Blow-up result for the periodic Camassa-Holm equation. Arch Math(Basel). 2005, 84(4):334-340
[49] Penskoi A V. Canonically conjugate variables for the periodic Camassa-Holm equation. Nonlinearity. 2005, 18(1):415-421
[50] Liu Yongqin, Wang Weike. Global existence of solution to Camassa-Holm equation.Nonlinear Anal. 2005, 60(5):945-953
[51] Yu Yongjiang, Li Kaitai. Gevrey class regularity for the viscous Camassa-Holm equation. Appl Math Lett. 2005, 18(16):713-719
[52] Bennewitz C. On the spectral problem associated with the Camassa-Holm equation. J Nonlinear Math Phys. 2004, 11(4):422-434
[53] Korotyaev E. nverse spectral problem for the periodic Camassa-Holm equation. J Nonlinear Math Phys. 2004, 11(4):499-507
[54] Molinet L. On well-posedness results for Camassa-Holm equation on the line: a survey. J Nonlinear Math Phys. 2004, 11(4):521-533
[55] Yin Zhaoyang. Well-posedness and blow-up phenomena for periodic generalized Camassa-Holm equation. Commun Pure Appl Anal. 2004, 3(3):501-508
[56] Misiolek G. Classical solutions of the periodic Camassa-Holm equation. Geom Funct Anal. 2002, 12(5):1080-1104
[57]Dullin H R,Gottwald G A,Holm D D.An integrable shallow water equation with linear and nonlinear dispersion.Phys Rev Lett.2002,87(19):4501-4504
[58]范恩贵.可积系统与计算机代数.北京:科学出版社,2004
[59]Kruskal M.D.,Gardner C.S.,Greene J.M.,Miura R.M.Method for solving the Korteweg-de Vries equation.Phys Rev Lett.1967,19:1095
[60]Lax P.D.Integrals of nonlinear equations of evolution and solitary waves.Comm Pure Appl Math.1968,21(467-490)
[61]V.E.Zakharov,Shabat A.B.Exact theory of two dimensional self-focusing and one dimensional self-modulation of waves in nonlinear media.Soviet Phys JETP.1972,34:62
[62]Wadati M.The modified korteweg-de vries equation.J Phys Soc Jp.1972,32:1681
[63]Newell A.C.,Ablowitz M.J.,Kaup D.J.,Segur H.The inverse scattering transform-fourier analysis for nonlinear problems.Studies in Applied Mathematics.1974,53:249
[64]M.J.Ablowitz,P.A.Clarkson.Solitons,Nonlinear Evolution Equations and Inverse Scattering.Cambridge University Press,Cambridge,1991
[65]Mingliang Wang,Zhibin Li.Applicatiuu of homegenous balance method to exactsolutions of nonlinear equations in mathematical physics.Phys Lett A.1996,216:67
[66]Fan E.G,Zhang H.Q.A note on the homogeneous balance method.Phys Lett A.1998,246:403-406
[67]Zhenya Yan,Hongqing Zhang.New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics.Phys Lett A.1999,252(291-296)
[68]Engui Fan.Two new applications of the homogenous halance method.Phys Lett A.2000,(265):353-357
[69]石玉仁.函数展开法及同伦分析法在求解非线性演化方程中的应用.Ph.D.thesis,兰州大学,2007
[70]Ablowitz M.J,Kaup D.J.,et al.Method for solving the sine-Gordon equation.Phys Rev Lett.1973,30:1262-1264
[71]屠规彰.非线性方程的逆散射解法.应用数学与计算数学.1979,24(20):913-917
[72]Kodama Y,Ablowitz M J,Satsuma J.Direct and inverse scattering problems of the nonlinear intermediate long wave equation.J Math Phys.1982,23:564
[73]Fokas A S.On the inverse scattering transform of multidimensional nonlinear equations related to first-order systems in the Dlane.J Math Phvs.1984,25(2494)
[74]谷超豪等.孤立子理论与应用.杭州:浙江科技出版社,1990
[75]Weiss J,Tabor M,Carnvale G.The Painleve property for partial differential equations.J Math Phys.1983,24(3):522-526
[76]Wang M.L.Li Z.B.Travelling solutions to the two-dimensional KdV-Burgers equation.J Phy A.1993,26:6027-6031
[77]Wang M.L,Li Z.B.Proc.of the 1994 Beijing Symposium on nonlinear evolution equations and infinite dimensional dynamics symtems.Zhongshan University Press.1995:181-185
[78]范恩贵,张鸿庆.齐次变换法若干新的应用.数学物理学报.1999,19:286-292
[79]Engui Fan.Auto-Backlund transformation and similarity reductions for general variable coefficient KdV equations.Rev Lett A.2002,294:26-30
[80]曾云波.Toda方程的Backlund变换.数学学报.1992,35:454
[81]Gamze Tanoglou.Hirota method for solving reaction-diffusion equations with generalized nonlinearity.International Journal of Nonlinear Science.2006,1(1):30-36
[82]T B Benjamin,JL Bona,J Mahony.Model equations for long wave in nonlinear dispersive systems.Philos Trans R Soc London A.1972,(272):47-78
[83]Shang Yadong.Explicit and exact special solutions for BBM-like B(m,n equations with fully nonlinear dispersion.Chaos Solitons and Fractals.2005,25:1083-1091
[84]L.A.Ostrovsky.Nonlinear internal waves in a rotating ocean.Okeanologia.1978,18
[85]Yue Liu,Vladimir Varlamov.Stability of solitary waves and weak rotation limit for the Ostrovsky equation.J Differential Equations.2004,203:159-183
[86]Vladimir Varlamov,Yue Liu.Cauchy problem for the Ostrovsky equation.Discrete Dynam Systems.2004,10(3):731-753
[87]殷久利.一类浅水波系统的孤波分析.Ph.D.thesis,江苏大学,2007
[88]Jianwei Shen,Wei Xu,Youming Lei.Smooth and non-smooth travelling waves in a nonlinearly dispersive Boussinesq equation.Chaos,Solitons and Fractals.2004,23(1):117-130
[89]Boling Guo,Z.R.Liu.Periodic cusp wave solutions and is single-solutions for bequation.Chaos,Solitons & Fractals.2005,23:1451-1463
[90]Boling Guo,Z.R.Liu.Peaked wave solutions of CH-r equation.Sci China Ser A.2003,46(5):696-709
[91]Micu S.On the controllability of the linearized Benjamin-Bona-Mahony equation.SIAM J Control Optim.29(6):1677-1696
[92]Bona J.On solitary waves and their role in the evolution of long waves,Applications of nonlinear analysis,Boston,MA:Pitman,1981
[93]V.E.Zakharov,E.A.Kuznetsov.On three-dimensional solitons.Soviet Phys.1974,39:285-288
[94]A M Wazwaz.Compact and noncompact physical structures for the ZK-BBM equation.Appl Math Comput.2005,169:713-725
[95]A M Wazwaz.The extended tanh method for new compact and noncomapct solutions for the KP-BBM and the ZK-BBM equations.Chaos Solitons and Fractals.2007:DOI:10.1016/j.chaos.2007.01.135
[96]Jianwei Shen,Wei Xu,Wei Li.Bifurcations of traveling wave solutions in a new integrable equation with peakon and compacton.Chaos,Solitons and Fractals.2006,27:413-425
[97] R.M. Miura. J Math Phys. 1968, 9:1202
[98] Dullin H R, Gottwald G A, Holm D D. Phys Rev Lett. 2001, 87:1945
[99] W. Ames. Nonlinear Partial Differential Equations. New York: Academic Press,1967
[100] H. Ono. J Phys Soc Jpn. 1973, 34:1073
[101] V. Ziegler. Chaos Soliton Fract. 2001, 12:1719
[102] T.S. Komatsu, S.I. Sasa. Phys Rev E. 1995, 52:5574
[103] Fan E.G. Chaos Soliton Fract. 2003, 16:819
[104] Z. Yan. Phys A. 2003, 326:344
[105] Abdul Majid Wazwaz. New solitary wave solutions to the modified forms of De-gasperis - Procesi and Camassa - Holm equations. Applied Mathematics and Computation. 2007, 186:130-141
[106] W. Malfliet, W. Hereman. The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Physica Scripta. 1996, 54:563-568
[107] W. Malfliet, W. Hereman. The tanh method: II. Perturbation technique for conservative systems. Physica Scripta. 1996, 54:569-575
[108] M Lakshmanan, P Kaliappan. Lie transformations, nonlinear evolutions and Painleve forms. J Math Phys. 1983, 24(4):795-806
[109] R K Dodd. Solitons and nonlinear wave equations. London: Academic Press, 1982
[110] V G Makhakov. Phys Rep. 1978, 35:1
[2]Keteweg DJ,de Vries G.On the change of form long waves advancing in a rectangular canal,and on a new of type of long stationary waves.Phil Mag.1895,39(422-443)
[3]Zabusky N J,Kruskal M D.Interation of solitions in a collisionless plasma and the recurrence of intial states.Phys Rev Lett.1965,15:240-243
[4]Vladimir G M.Soliton Phenomenology.Netherlands:Kluwer Academic Publishers,1990
[5]艾仑伯格 G.孤立子-物理学家用的数学方法.北京:科学出版社,1989
[6]Miura R M.The Korteweg-de Vries equation:A survey of results.SIAM Rev.1976,8:412-459
[7]Kox A J.Korteweg,de Vries,and Dutch science at the turn of the century,cta Applicandae Mathematica.1995,39:91-92
[8]郭柏灵.非线性演化方程.上海:上海科技教育出版社,1995
[9]郭柏灵,庞小峰.孤立子.北京:科学出版社,1987
[10]谷超豪,胡和生and周子翔.孤立子理论中的Darboux变换及其几何应用.上海:上海科技出版社,1999
[11]李诩神.孤子与可积系统.上海:上海科技出版社,1999
[12]刘式达,刘式适.孤波和湍流.上海:上海科技教育出版社,1994
[13]倪皖荪,魏荣爵.水槽中的孤波.上海:上海科技教育出版社,1997
[14]黄念宁.孤子理论和微扰方法.上海:上海科技教育出版社,1996
[15]Camassa R,Holm D D.An integrable shallow water equation with peaked solitons.Physical Review Letters.1993,71(11):1661-1664
[16] Liu Z R, Wand R Q, Jing Z J. Peaked wave solutions of Comassa-Holm equation.Chaos,Solitons and Fractals. 2004, 19:77-92
[17] Liu Z R, Qian T F. Peakons and their bifurcation in a generalized Camassa - Holm equation. International Journal of Bifurcation and Choas. 2001, 11(3):781-792
[18] Liu Z R, Qian T F. Peakons of the Camassa - Holm equation. Applied Mathematical Modeling. 2002, 26:473-480
[19] Qian T F, Tang M Y. Peakons and periodic cusp waves in a generalized Camassa -Holm equation. Chaos,Solitons and Fractals. 2001, 12:1347-1360
[20] P Rosenau, J. M. Hyman. Compactons:solitons with finite wavelengths. Phys Rev Lett. 1993, 70(5):564-567
[21] P Rosenau. Nonlinear dispersion and compact structures. Phys Rev Lett. 1994,73(13):1737-1741
[22] P Rosenau. On solitons, compactons, and Lagrange maps Physical. Phys Lett A.1996, 211:265-275
[23] Rosenau P. Compact and noncompact dispersive patterns. Phys Lett A. 2000,275:193-203
[24] A M Wazwaz. New solitary wave special solutions with compact support the nonlinear dispersive K(m,n) equations. Chaos Solitons and Fractals. 2002, 13:321-330
[25] A M Wazwaz. Compactons dispersive structures for variants of the variants the K(n,n) and the KP equations. Chaos Solitons and Fractals. 2002, 13:1053-1062
[26] A M Wazwaz. Solutions of compact and noncompact structures for nonlinear Klein-Gordon type. Applied Mathematics and Computation. 2003, 134:487-500
[27] Z Y Yan. Chaos, Solitons and Fractals. 2002, 14:1151-1158
[28] Liu Zhengrong, Chen Can. Compactons in general compressible hyperelastic rod. Chaos,Solitons and Fractals. 2004, 22:627-640
[29] Chen S, Foias C, Holm D D, et al. A connection between the Camassa-Holm equations and turbulent flows in channels and pipes. Phys Fluids. 1999, 11 (8):2343-2353
[30] Fisher M, Schiff J. The Camassa Holm equation: conserved quantities and the initial value problem. Phys Lett A. 1999, 259(5):371-376
[31] Kwek K H, Gao Hongjun, Zhang Weinian, et al. An initial boundary value problem of Camassa-Holm equation. J Math Phys. 2000, 41(12):8279-8285
[32] Kraenkel R A, Senthilvelan M, Zenchuk A I. Lie symmetry analysis and reductions of a two-dimensional integrable generalization of the Camassa-Holm equation. Phys Lett A. 2000, 273(3): 183-193
[33] Constantin A. On the scattering problem for the Camassa-Holm equation. Proc R Soc London A. 2001, 457:953-970
[34] Zenchuk A I. The spectral problem and particular solutions to the (2+2)-dimensional integrable generalization of the Camassa-Holm equation. Advances in nonlinear mathematics and science ,Phys D. 2001, 152/153:178-188
[35] Reyes E G. Geometric integrability of the Camassa-Holm equation. Lett Math Phys.2002, 59(2):117-131
[36] Lenells J. The scattering approach for the Camassa-Holm equation. J Nonliear Math Phys. 2002, 9(4):389-393
[37] Johnson R S. On solutions of the Camassa-Holm equation. Soc Lond Proc Ser A Math Phys Eng Sci. 2003, 459(2035): 1687-1708
[38] Mckean H P. The Liouville correspondence between the Korteweg-de Vries and the Camassa-Holm hierarchies. Comm Pure Appl Math. 2003, 56(7):998-1015
[39] Badanin A, Klein M, Korotyaev E. The Marchenko-Ostrovski mapping and the teace formula for the Camassa-Holm equation. J Funct Anal. 2003, 203(2):449-518
[40] Johnson R S. The Camassa-Holm equation for water waves moving over a shear flow. Fluid Dynam Res. 2003, 33(1-2):97-111
[41] Penskoi A V, Veselov A P. Discrete Lagrangian on the Virasoro group and Camassa- Holm family. Nonlinearity. 2003, 16(2):683-688
[42] Lopes O. Stability of peakons for the generalized Camassa-Holm equation. J Differential Equations. 2003, 5:12
[43] Gorsky J M. The Cauchy problem for a modified Camassa-Holm equation with analytic initial data. Differential Integral Equations. 2004, 17(11-12):1233-1254
[44] Boyd J P. Near-Corner waves of the Camassa-Holm equation. Phys Lett A,. 2005,336(4-5):342-348
[45] Lenells J. Conservation laws of the Camassa-Holm equation. J Phys A. 2005,38(4):869-880
[46] Kalisch H, Lenells J. Numerical study of traveling-wave solutions for the Camassa- Holm equation. Chaos,Solitons and Fractals. 2005, 25:287-298
[47] Khuri S A. New anstz for obtaining wave solutions of the generalized Camassa-Holm equation. Chaos,Solitons and Fractals. 2005, 25(3):705-710
[48] Wahlen E A. Blow-up result for the periodic Camassa-Holm equation. Arch Math(Basel). 2005, 84(4):334-340
[49] Penskoi A V. Canonically conjugate variables for the periodic Camassa-Holm equation. Nonlinearity. 2005, 18(1):415-421
[50] Liu Yongqin, Wang Weike. Global existence of solution to Camassa-Holm equation.Nonlinear Anal. 2005, 60(5):945-953
[51] Yu Yongjiang, Li Kaitai. Gevrey class regularity for the viscous Camassa-Holm equation. Appl Math Lett. 2005, 18(16):713-719
[52] Bennewitz C. On the spectral problem associated with the Camassa-Holm equation. J Nonlinear Math Phys. 2004, 11(4):422-434
[53] Korotyaev E. nverse spectral problem for the periodic Camassa-Holm equation. J Nonlinear Math Phys. 2004, 11(4):499-507
[54] Molinet L. On well-posedness results for Camassa-Holm equation on the line: a survey. J Nonlinear Math Phys. 2004, 11(4):521-533
[55] Yin Zhaoyang. Well-posedness and blow-up phenomena for periodic generalized Camassa-Holm equation. Commun Pure Appl Anal. 2004, 3(3):501-508
[56] Misiolek G. Classical solutions of the periodic Camassa-Holm equation. Geom Funct Anal. 2002, 12(5):1080-1104
[57]Dullin H R,Gottwald G A,Holm D D.An integrable shallow water equation with linear and nonlinear dispersion.Phys Rev Lett.2002,87(19):4501-4504
[58]范恩贵.可积系统与计算机代数.北京:科学出版社,2004
[59]Kruskal M.D.,Gardner C.S.,Greene J.M.,Miura R.M.Method for solving the Korteweg-de Vries equation.Phys Rev Lett.1967,19:1095
[60]Lax P.D.Integrals of nonlinear equations of evolution and solitary waves.Comm Pure Appl Math.1968,21(467-490)
[61]V.E.Zakharov,Shabat A.B.Exact theory of two dimensional self-focusing and one dimensional self-modulation of waves in nonlinear media.Soviet Phys JETP.1972,34:62
[62]Wadati M.The modified korteweg-de vries equation.J Phys Soc Jp.1972,32:1681
[63]Newell A.C.,Ablowitz M.J.,Kaup D.J.,Segur H.The inverse scattering transform-fourier analysis for nonlinear problems.Studies in Applied Mathematics.1974,53:249
[64]M.J.Ablowitz,P.A.Clarkson.Solitons,Nonlinear Evolution Equations and Inverse Scattering.Cambridge University Press,Cambridge,1991
[65]Mingliang Wang,Zhibin Li.Applicatiuu of homegenous balance method to exactsolutions of nonlinear equations in mathematical physics.Phys Lett A.1996,216:67
[66]Fan E.G,Zhang H.Q.A note on the homogeneous balance method.Phys Lett A.1998,246:403-406
[67]Zhenya Yan,Hongqing Zhang.New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics.Phys Lett A.1999,252(291-296)
[68]Engui Fan.Two new applications of the homogenous halance method.Phys Lett A.2000,(265):353-357
[69]石玉仁.函数展开法及同伦分析法在求解非线性演化方程中的应用.Ph.D.thesis,兰州大学,2007
[70]Ablowitz M.J,Kaup D.J.,et al.Method for solving the sine-Gordon equation.Phys Rev Lett.1973,30:1262-1264
[71]屠规彰.非线性方程的逆散射解法.应用数学与计算数学.1979,24(20):913-917
[72]Kodama Y,Ablowitz M J,Satsuma J.Direct and inverse scattering problems of the nonlinear intermediate long wave equation.J Math Phys.1982,23:564
[73]Fokas A S.On the inverse scattering transform of multidimensional nonlinear equations related to first-order systems in the Dlane.J Math Phvs.1984,25(2494)
[74]谷超豪等.孤立子理论与应用.杭州:浙江科技出版社,1990
[75]Weiss J,Tabor M,Carnvale G.The Painleve property for partial differential equations.J Math Phys.1983,24(3):522-526
[76]Wang M.L.Li Z.B.Travelling solutions to the two-dimensional KdV-Burgers equation.J Phy A.1993,26:6027-6031
[77]Wang M.L,Li Z.B.Proc.of the 1994 Beijing Symposium on nonlinear evolution equations and infinite dimensional dynamics symtems.Zhongshan University Press.1995:181-185
[78]范恩贵,张鸿庆.齐次变换法若干新的应用.数学物理学报.1999,19:286-292
[79]Engui Fan.Auto-Backlund transformation and similarity reductions for general variable coefficient KdV equations.Rev Lett A.2002,294:26-30
[80]曾云波.Toda方程的Backlund变换.数学学报.1992,35:454
[81]Gamze Tanoglou.Hirota method for solving reaction-diffusion equations with generalized nonlinearity.International Journal of Nonlinear Science.2006,1(1):30-36
[82]T B Benjamin,JL Bona,J Mahony.Model equations for long wave in nonlinear dispersive systems.Philos Trans R Soc London A.1972,(272):47-78
[83]Shang Yadong.Explicit and exact special solutions for BBM-like B(m,n equations with fully nonlinear dispersion.Chaos Solitons and Fractals.2005,25:1083-1091
[84]L.A.Ostrovsky.Nonlinear internal waves in a rotating ocean.Okeanologia.1978,18
[85]Yue Liu,Vladimir Varlamov.Stability of solitary waves and weak rotation limit for the Ostrovsky equation.J Differential Equations.2004,203:159-183
[86]Vladimir Varlamov,Yue Liu.Cauchy problem for the Ostrovsky equation.Discrete Dynam Systems.2004,10(3):731-753
[87]殷久利.一类浅水波系统的孤波分析.Ph.D.thesis,江苏大学,2007
[88]Jianwei Shen,Wei Xu,Youming Lei.Smooth and non-smooth travelling waves in a nonlinearly dispersive Boussinesq equation.Chaos,Solitons and Fractals.2004,23(1):117-130
[89]Boling Guo,Z.R.Liu.Periodic cusp wave solutions and is single-solutions for bequation.Chaos,Solitons & Fractals.2005,23:1451-1463
[90]Boling Guo,Z.R.Liu.Peaked wave solutions of CH-r equation.Sci China Ser A.2003,46(5):696-709
[91]Micu S.On the controllability of the linearized Benjamin-Bona-Mahony equation.SIAM J Control Optim.29(6):1677-1696
[92]Bona J.On solitary waves and their role in the evolution of long waves,Applications of nonlinear analysis,Boston,MA:Pitman,1981
[93]V.E.Zakharov,E.A.Kuznetsov.On three-dimensional solitons.Soviet Phys.1974,39:285-288
[94]A M Wazwaz.Compact and noncompact physical structures for the ZK-BBM equation.Appl Math Comput.2005,169:713-725
[95]A M Wazwaz.The extended tanh method for new compact and noncomapct solutions for the KP-BBM and the ZK-BBM equations.Chaos Solitons and Fractals.2007:DOI:10.1016/j.chaos.2007.01.135
[96]Jianwei Shen,Wei Xu,Wei Li.Bifurcations of traveling wave solutions in a new integrable equation with peakon and compacton.Chaos,Solitons and Fractals.2006,27:413-425
[97] R.M. Miura. J Math Phys. 1968, 9:1202
[98] Dullin H R, Gottwald G A, Holm D D. Phys Rev Lett. 2001, 87:1945
[99] W. Ames. Nonlinear Partial Differential Equations. New York: Academic Press,1967
[100] H. Ono. J Phys Soc Jpn. 1973, 34:1073
[101] V. Ziegler. Chaos Soliton Fract. 2001, 12:1719
[102] T.S. Komatsu, S.I. Sasa. Phys Rev E. 1995, 52:5574
[103] Fan E.G. Chaos Soliton Fract. 2003, 16:819
[104] Z. Yan. Phys A. 2003, 326:344
[105] Abdul Majid Wazwaz. New solitary wave solutions to the modified forms of De-gasperis - Procesi and Camassa - Holm equations. Applied Mathematics and Computation. 2007, 186:130-141
[106] W. Malfliet, W. Hereman. The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Physica Scripta. 1996, 54:563-568
[107] W. Malfliet, W. Hereman. The tanh method: II. Perturbation technique for conservative systems. Physica Scripta. 1996, 54:569-575
[108] M Lakshmanan, P Kaliappan. Lie transformations, nonlinear evolutions and Painleve forms. J Math Phys. 1983, 24(4):795-806
[109] R K Dodd. Solitons and nonlinear wave equations. London: Academic Press, 1982
[110] V G Makhakov. Phys Rep. 1978, 35:1
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