几类非线性发展方程的精确行波解的研究
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摘要
非线性现象是自然界最普遍的现象,是自然界的本质.非线性系统的提出和研究,促使不同学科相互渗透融会,大批新兴学科应运而生,逐步诞生了探讨复杂性现象的非线性科学。非线性科学主要包括研究孤子理论、混沌理论、分形理论和耗散结构理论等等,以及这些理论在其他相关学科领域的广泛应用。孤波理论与应用是非线性科学研究的热门课题之一。孤立子相关性质的研究在揭示波的传播规律、准确解释自然现象和科学应用相关技术等方面均具有极大的科学研究价值.非线性发展方程的研究又离不开孤立子理论.大量的非线性问题的研究和解决最终都归结为求解非线性偏微分方程(组)的问题.非线性偏微分方程(组)的求解要远比线性偏微分方程(组)的求解困难得多,很难用统一的方法对前者加以处理.由此,求非线性偏微分方程(组)精确解的工作,就显示了很重要的理论和应用价值.
本文基于此目的,在归纳和总结了现有各种主要的精确求解非线性发展方程方法的基础上,研究了一类具有实际应用物理背景的非线性波动方程,如mBBM方程、MCH方程,Klein-Gardner方程,组合KdV-mKdV方程、广义的BBM方程等.把一些经典的研究方法加以推广和改进,借助于符号计算和数学机械化的方法,来研究这类非线性波动方程的行波解,不但获得了已有的结果,而且得到了一些新的结果.通过研究波动方程的动力学性质,从定性角度数形结合分析,寻求非线性波动方程的解.
全文共分七章.第一章介绍了非线性波动方程提出的研究背景、进展和现状,提出了本课题的研究意义和研究内容.
第二章介绍了几个重要的求解非线性波动方程精确解的方法,并简要阐述了本文主要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.
第三章借助于首次积分法,对常见的mBBM方程、简化形式的MCH方程,Klein-Gardner方程,组合KdV-mKdV方程,进行了全面的分析,得到了一些精确解,并结合简单的直接积分,以及Jacobi椭圆正弦函数展开法,比较了这个方法的优点。
第四章应用(?)展开法,对以上讨论的简化形式的MCH方程,Klein-Gardner方程,组合KdV-mKdV方程,再次作进一步的研究,得到了这些方程的双曲函数形式、三角函数形式的解,丰富了这些方程的讨论。这类方法的应用日益丰富,甚至对一些重要的离散的孤波方程也一样适用.
第五章借助Wazwaz的独创性的工作,结合著名的Hirota法,用一种简单的方式,有效地寻求了(2+1)维Zakharov-Kuznetsov方程(简记为ZK(m,n,k))、破裂孤子方程、Potential Kadomtsev-Petviashvili (PKP)方程以及一个五阶色散方程的解,并且包含多重孤子解和奇异孤子解。本文还尝试利用同宿测试法(homoclinic test method),讨论Hirota法的一些延伸工作。
第六章受启发于Kuru等人利用二阶微分算子的分解理论,直接将Wazwaz等人讨论过的几类广义BBM方程以及两个修正的Boussinesq方程,经过行波变换后,通过Weierstrass椭圆函数的形式,求得这几类方程的行波解,主要是周期解和双曲函数解,且多数解的形式未曾在文献中被发现。
最后,结合已有的一些结论,对各类方法的后续展开,作了初步的展望,并为以后的尝试提供平台。
Nonlinear phenomenon is the most common phenomenon in nature, and it is the essence of nature. When nonlinear systems were proposed and researched, it promoted fusion of different disciplines. A large number of new disciplines emerged, and gradually gave birth to the nonlinear science to systematically study the complexity of the phenomenon. Nonlinear science includes the research of soliton theory, chaos theory and fractal theory and theory of dissipative structures, etc., and the extensive application of these theories in other related disciplines. Solitary wave theory and the application is a hot topic in nonlinear science.
The related properties of soliton play great importance in revealing the wave propagation, in the accurate and scientific explanation of natural phenomenon, and in the related application of engineering techniques. In order to describe these, the nonlinear evolutionary equation (or equations) was introduced, and therefore there are both theoretical and practical meanings both in quantitative research and qualitative research. The research of amount of nonlinear problem comes to the research of the nonlinear evolutionary equation (or equations).However, it is more sophisticated to solve the nonlinear equation than to solve linear one, and generally there isn't a unified approach to deal with the former. Thus, it shows a very important theoretical and practical value for solving nonlinear partial differential equation (or equations).
Based on this, in the summary of the major existing methods for solving nonlinear evolution equations, the paper applies several methods to some typical equations, such as mBBM equation, MCH equation, Klein-Gardner equation, combined KdV-mKdV equation, generalized BBM equations. From the application, the paper got some old solutions and some new ones.
The full-text is divided into seven chapters. The first chapter describes the research background, progress and nowadays works of nonlinear wave equation.
The second chapter introduces several important methods for looking for exact solutions of nonlinear wave equation, and briefly describes the basic concepts and principles in this paper for solving nonlinear wave equation associated with this article.
In the third chapter, by means of the first integral method, I analyzed some equations, such as mBBM equation, simplified form of MCH equation, Klein-Gardner equation, and combined KdV-mKdV equation. By a comprehensive analysis, I find some exact solutions, combined with a simple direct integration and the expansion method of Jacobi elliptic sine function.
In the next chapter, I apply (G'/G) expansion method to discuss the simplified form of MCH equation, Klein-Gardner equation, combined KdV-mKdV equation, and get many solutions with the forms of hyperbolic function, trigonometric function, which enrich the discussion of these equations. The increasingly rich application of the method is still suitable for some important discrete solitary wave equation.
In the fifth chapter, inspired by Wazwaz's original work with well-known Hirota method, I apply a simple way to effectively search for the solutions of the (2 +1)-dimensional Zakharov-Kuznetsov equation (abbreviated as ZK (m, n, k)), breaking soliton equation, Potential Kadomtsev-Petviashvili (PKP) equation, and a fifth-order dispersion equation, and includes multiple soliton solutions and singular soliton solutions. Still I attempt to utilize the homoclinic test method to discuss the extension work of the Hirota method.
In the sixth chapter, I am enlightened by the work of Kuru's decomposition theory of second order differential operator, and discuss several classes of generalized BBM equations which were researched by Wazwaz and two modified Boussinesq equations. After traveling wave transformation, with the Weierstrass function form, the paper obtains some types of traveling wave solutions, among which are periodic solutions and hyperbolic function solutions, and most of the solution has not been found in the literature.
Finally, there are some of the conclusions of the various methods. I make some summary and forecast. Based on these, I would make future attempts.
本文基于此目的,在归纳和总结了现有各种主要的精确求解非线性发展方程方法的基础上,研究了一类具有实际应用物理背景的非线性波动方程,如mBBM方程、MCH方程,Klein-Gardner方程,组合KdV-mKdV方程、广义的BBM方程等.把一些经典的研究方法加以推广和改进,借助于符号计算和数学机械化的方法,来研究这类非线性波动方程的行波解,不但获得了已有的结果,而且得到了一些新的结果.通过研究波动方程的动力学性质,从定性角度数形结合分析,寻求非线性波动方程的解.
全文共分七章.第一章介绍了非线性波动方程提出的研究背景、进展和现状,提出了本课题的研究意义和研究内容.
第二章介绍了几个重要的求解非线性波动方程精确解的方法,并简要阐述了本文主要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.
第三章借助于首次积分法,对常见的mBBM方程、简化形式的MCH方程,Klein-Gardner方程,组合KdV-mKdV方程,进行了全面的分析,得到了一些精确解,并结合简单的直接积分,以及Jacobi椭圆正弦函数展开法,比较了这个方法的优点。
第四章应用(?)展开法,对以上讨论的简化形式的MCH方程,Klein-Gardner方程,组合KdV-mKdV方程,再次作进一步的研究,得到了这些方程的双曲函数形式、三角函数形式的解,丰富了这些方程的讨论。这类方法的应用日益丰富,甚至对一些重要的离散的孤波方程也一样适用.
第五章借助Wazwaz的独创性的工作,结合著名的Hirota法,用一种简单的方式,有效地寻求了(2+1)维Zakharov-Kuznetsov方程(简记为ZK(m,n,k))、破裂孤子方程、Potential Kadomtsev-Petviashvili (PKP)方程以及一个五阶色散方程的解,并且包含多重孤子解和奇异孤子解。本文还尝试利用同宿测试法(homoclinic test method),讨论Hirota法的一些延伸工作。
第六章受启发于Kuru等人利用二阶微分算子的分解理论,直接将Wazwaz等人讨论过的几类广义BBM方程以及两个修正的Boussinesq方程,经过行波变换后,通过Weierstrass椭圆函数的形式,求得这几类方程的行波解,主要是周期解和双曲函数解,且多数解的形式未曾在文献中被发现。
最后,结合已有的一些结论,对各类方法的后续展开,作了初步的展望,并为以后的尝试提供平台。
Nonlinear phenomenon is the most common phenomenon in nature, and it is the essence of nature. When nonlinear systems were proposed and researched, it promoted fusion of different disciplines. A large number of new disciplines emerged, and gradually gave birth to the nonlinear science to systematically study the complexity of the phenomenon. Nonlinear science includes the research of soliton theory, chaos theory and fractal theory and theory of dissipative structures, etc., and the extensive application of these theories in other related disciplines. Solitary wave theory and the application is a hot topic in nonlinear science.
The related properties of soliton play great importance in revealing the wave propagation, in the accurate and scientific explanation of natural phenomenon, and in the related application of engineering techniques. In order to describe these, the nonlinear evolutionary equation (or equations) was introduced, and therefore there are both theoretical and practical meanings both in quantitative research and qualitative research. The research of amount of nonlinear problem comes to the research of the nonlinear evolutionary equation (or equations).However, it is more sophisticated to solve the nonlinear equation than to solve linear one, and generally there isn't a unified approach to deal with the former. Thus, it shows a very important theoretical and practical value for solving nonlinear partial differential equation (or equations).
Based on this, in the summary of the major existing methods for solving nonlinear evolution equations, the paper applies several methods to some typical equations, such as mBBM equation, MCH equation, Klein-Gardner equation, combined KdV-mKdV equation, generalized BBM equations. From the application, the paper got some old solutions and some new ones.
The full-text is divided into seven chapters. The first chapter describes the research background, progress and nowadays works of nonlinear wave equation.
The second chapter introduces several important methods for looking for exact solutions of nonlinear wave equation, and briefly describes the basic concepts and principles in this paper for solving nonlinear wave equation associated with this article.
In the third chapter, by means of the first integral method, I analyzed some equations, such as mBBM equation, simplified form of MCH equation, Klein-Gardner equation, and combined KdV-mKdV equation. By a comprehensive analysis, I find some exact solutions, combined with a simple direct integration and the expansion method of Jacobi elliptic sine function.
In the next chapter, I apply (G'/G) expansion method to discuss the simplified form of MCH equation, Klein-Gardner equation, combined KdV-mKdV equation, and get many solutions with the forms of hyperbolic function, trigonometric function, which enrich the discussion of these equations. The increasingly rich application of the method is still suitable for some important discrete solitary wave equation.
In the fifth chapter, inspired by Wazwaz's original work with well-known Hirota method, I apply a simple way to effectively search for the solutions of the (2 +1)-dimensional Zakharov-Kuznetsov equation (abbreviated as ZK (m, n, k)), breaking soliton equation, Potential Kadomtsev-Petviashvili (PKP) equation, and a fifth-order dispersion equation, and includes multiple soliton solutions and singular soliton solutions. Still I attempt to utilize the homoclinic test method to discuss the extension work of the Hirota method.
In the sixth chapter, I am enlightened by the work of Kuru's decomposition theory of second order differential operator, and discuss several classes of generalized BBM equations which were researched by Wazwaz and two modified Boussinesq equations. After traveling wave transformation, with the Weierstrass function form, the paper obtains some types of traveling wave solutions, among which are periodic solutions and hyperbolic function solutions, and most of the solution has not been found in the literature.
Finally, there are some of the conclusions of the various methods. I make some summary and forecast. Based on these, I would make future attempts.
引文
[1]郭柏灵,庞小峰.孤立子.科学出版社.1987
[2]李翊神.孤子与可积系统.上海科技教育出版社.1999
[3]刘式达,刘式适.孤波和湍流.上海科技教育出版社.1994
[4]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.上海科学技术出版社.2005
[5]R.Camassa, D.Holm. An integrable shallow equation with peaked solitons. Phy. Rev. Lett.1993 (11):1661-1664.
[6]郭柏灵,田立新,杨灵娥.Camassa-Holm方程.科学出版社.2008
[7]Mingliang Wang, Zhibin Li. Application of homegenous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. Lett. A.1996(67): 216.
[8]Fan E.G, Zhang H.Q. A note on the homogeneous balance method. Phys Lett A.1998(246):403.
[9]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.
[10]A.M. Wazwaz. The tanh method and a variable separated ODE method for solving double sine-Gordon equation. Phys. Lett. A.2006(350):367-370.
[11]Lixin Tian, Xiuying Song. New peaked solitary wave solutions of the generalized Camassa-Holm equation.Chaos, Solitons and Fractals,2004(19):621-639
[12]A. Degasperis and M. Procesi. Asymptotic integrability. In A. Degasperis and G. Gaeta, editors, Symmetry and perturbation theory (Rome,1998), pages 23-37. World Scientific Publishing, River Edge, NJ,1999.
[13]Filiz Tascan, Ahmet Bekir, Murat Koparan, Traveling wave solutions of nonlinear revolution equations by using the first integral method, Commun. Nonlinear. Sci. Numer. Simulat.,2009 (14):1810-1815
[14]Zhaosheng Feng. The first-integral method to study the Burgers-Korteweg-de Vries equation, J. Phys. A:Math. Gen.2002(35):343-349
[15]A.H.AhmedAli, K.R.Raslan. New solutions for some important partial differential equations, International Journal of Nonlinear Science.2007 (4):109.
[16]Xijun Deng. Exact peaked wave solution of CH-y equation by the first-integral method.Appl.Math.Comput.2008 (206):806-809
[17]Xijun Deng. Traveling wave solutions for the generalized Burgers-Huxley equation. Appl.Math.Comput.,204(2008)733-737
[18]Filiz Tascan, Ahmet Bekir, Murat Koparan. Traveling wave solutions of nonlinear evolution equations by using the first integral method. Commun. Nonlinear. Sci. Numer. Simulat..14 (2009) 1810-1815
[19]Filiz Tascan, Ahmet Bekir. Traveling wave solutions of the Cahn-Allen equation by using the first integral method. Appl. Math. Comput. (2008),doi:10.1016/ j.amc.2008.10.031
[20]Huaying Li, Yucui Guo. New exact solutions to the Fitzhugh-Nagumo equation. Appl.Math.Comput.180(2006)524-528
[21]Yaning Tang, Wei Xu, Jianwei Shen. Solitary Wave Solutions to Gardner Equation. Chinese Journal of Engineering Mathematics.2007,24(1),119-127
[22]Zuntao Fu, Shida Liu, Shikuo Liu. New kinds of solutions to Gardner equation. Chaos, Solitons and Fractals.20(2004)301-309
[23]E. Yusufoglu, A. Bekir. The tanh and the sine-cosine methods for exact solutions of the MBBM and the Vakhnenko equations, Chaos, Solitons and Fractals,38(4), 2008,1126-1133
[24]E. Yusufoglu, New solitonary solutions for the MBBM equations using Exp-function method. Phys. Lett. A,372(4),2008,442-446
[25]Shikuo Liu, Zuntao Fu, Shida Liu and Qiang Zhao. Jacobi elliptic function expansion methodand periodic wave solutions of nonlinear wave equations. Phys. Lett. A.2001,289(1-2)
[26]刘式达,刘式适.孤波和湍流.上海：上海科技教育出版社.1994
[27]Shikuo Liu, Zuntao Fu, Shida Liu, Qiang Zhao. Expansion Method about the Jacobi Elliptic Function and its Applications to Nonlinear Wave Equations. Acta Phys. Sin.2001 (50):2068-2073
[28]刘式适,付遵涛,刘式达,赵强.变系数非线性方程的Jacobi椭圆函数展开解.物理学报,2002,(09)
[29]刘式适,付遵涛,刘式达,赵强.一类非线性方程的新周期解.物理学 报.2002,(01)
[30]刘式达,傅遵涛,刘式适,赵强.非线性波动方程的Jacobi椭圆函数包络周期解.物理学报.2002,(04).
[31]闫振亚,张鸿庆,范恩贵.一类非线性演化方程新的显式行波解.物理学报.1999,(01)
[32]张善卿,李志斌.Jacobi椭圆函数展开法的新应用.物理学报.2003,(05)
[33]套格图桑,斯仁道尔吉.构造非线性发展方程Jacobi椭圆函数精确解的一种方法.内蒙古师范大学学报(自然科学汉文版),2008,(01).
[34]Zhenya Yan. The extended Jacobian elliptic function expansion method and its application in the generalized Hirota-Satsuma coupled KdV system. Chaos Solitons and Fractals.15 (2003),575-583
[35]Chaoqing Dai, Jiefang Zhang. Jacobian elliptic function method for nonlinear differential-difference equations, Chaos Solitons and Fractals.27(2006):1042-1047
[36]夏铁成,张鸿庆,李佩春.Boussinesq方程精确解析解研究.大连理工大学学报.2003,(04).
[37]王明亮,李向正,韩淑霞.Nizhnik方程组的一个非线性变换和多重孤子解(英文).应用数学.2005(02)
[38]Wang Mingliang. Solitary wave solutions for variant Boussinesq equations.Phys Lett A,1995,199(34):169-172
[39]Liu Shikuo, Fu Zuntao, Liu Shida. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations.Phys Lett A,2001, 289(12):69-74
[40]Wang Mingliang, Zhou Yubin, Li Zhibin. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics.Phys Lett A,1996,216(15):67-75
[41]Fu Z T, Liu S K,Liu S D. Elliptic equation and new solutions to nonlinear wave equations.Comm Theoret Phys,2004(42):343-346.
[42]Fu Z T, Liu S D,Liu S K. Solving Nonlinear Wave Equations by Elliptic Equation.Commun. Theor. Phys,2003(39):531-536.
[43]Fu Z T,Liu S K,Liu S D. A New Approach to Solve Nonlinear Wave Equations.Commun Theor Phys.2003(39):27-30.
[44]套格图桑,斯仁道尔吉.双曲函数型辅助方程构造具5次强非线性项的波方程的新精确孤波解.物理学报.2006(01).
[45]套格图桑,斯仁道尔吉.BBM方程和修正的BBM方程新的精确孤立波解.物理学报.2004(12).
[46]Mingliang Wang, XiangzhengLi, JinliangZhang. The (G'/G)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A 2008 (372):417-423
[47]Mingliang Wang, Jinliang Zhang, Xiangzheng Li. Application of the (G'/G)-expansion to travelling wave solutions of the Broer-Kaup and the approximate long water wave equations, Appl. Math. Comput..2008(206): 321-326
[48]Xiangzheng Li, Mingliang Wang. The (G'/G)-expansion method and traveling wave solutions for a higher-order nonlinear schrodinger equation, Appl. Math. Comput. (2009), doi:10.1016/j.amc.2008.12.005
[49]Bekir A,Cevikel AC. New exact travelling wave solutions of nonlinear physical models, Chaos, Solitons & Fractals (2008),doi:10.1016/j.chaos.2008.07.017
[50]Sheng Zhang, Ling Dong, Jin-Mei Ba, Ying-Na Sun. The (G'/G)-expansion method for nonlinear differential-difference equations, Phys. Lett. A 2009 (373): 905-910
[51]R. Hirota. Exact solutions of the Sine-Gordon equation for multiple collisions of solitons, J.Phys.Soc.Jpn.33(5) (1972) 1459-1463
[52]Biswas A. Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion, Commun Nonlinear Sci Numer Simulat, 2009 (14):3503-3506
[53]Wazwaz AM. The tanh-coth and the sine-cosine methods for kinks, solitons, and periodic solutions for the Pochhammer-Chree equations, Appl. Math. Comput 195 (2008) 24-33.
[54]Wazwaz A.M. Solitary wave solutions of the generalized shallow water wave (GSWW) equation by Hirota's method, tanh-coth method and Exp-function method. Appl. Math. Comput.,2008.202(1):275-286.
[55]Wazwaz A.M. Multiple-soliton solutions for the generalized (1+1)-dimensional and the generalized (2+1)-dimensional Ito equations. Appl. Math. Comput.,2008. 202(2):840-849.
[56]Wazwaz A.M. Multiple-soliton solutions for the Lax-Kadomtsev-Petviashvili (Lax-KP) equation. Appl. Math. Comput.,2008.201(1-2):168-174.
[57]Wazwaz A.M. The Hirota's direct method for multiple-soliton solutions for three model equations of shallow water waves. Appl. Math. Comput.,2008.201(1-2): 489-503.
[58]Wazwaz A.M., Multiple-soliton solutions of two extended model equations for shallow water waves. Appl. Math. Comput.,2008.201(1-2):790-799.
[59]Wazwaz A.M. The Hirota's bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equation. Appl. Math. Comput.,2008.200(1):160-166.
[60]Wazwaz A.M. Multiple-front solutions for the Burgers-Kadomtsev-Petviashvili equation. Appl. Math. Comput.,2008.200(1):437-443.
[61]Wazwaz A.M. The Hirota's direct method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Ito seventh-order equation. Appl. Math. Comput.,2008.199(1):133-138.
[62]Wazwaz A.M. Multiple-soliton solutions for the fifth order Caudrey-Dodd-Gibbon (CDG) equation. Appl. Math. Comput.,2008.197(2): 719-724.
[63]Wazwaz A.M. Multiple-soliton solutions for the Lax seventh-order equation. Appl. Math. Comput.,2008.198(2):877-881.
[64]Wazwaz A.M. Four (2+1)-dimensional integrable extensions of the KdV equation: Multiple-soliton and multiple singular soliton solutions. Appl. Math. Comput., 2009.215(4):1463-1476.
[65]Wazwaz A.M. A (3+1)-dimensional nonlinear evolution equation with multiple soliton solutions and multiple singular soliton solutions. Appl. Math. Comput., 2009.215(4):1548-1552.
[66]Wazwaz A.M. Multiple soliton solutions and multiple singular soliton solutions for (2+1)-dimensional shallow water wave equations. Phys. Lett. A,2009. 373(33):2927-2930.
[67]Wazwaz A.M. Multiple kink solutions and multiple singular kink solutions for (2+1)-dimensional nonlinear models generated by the Jaulent-Miodek hierarchy. Phys. Lett. A,2009.373(21):1844-1846.
[68]Wazwaz A.M. Multiple-soliton solutions and multiple-singular soliton solutions for two higher-dimensional shallow water wave equations. Appl. Math. Comput., 2009.211(2):495-501.
[69]Wazwaz A.M. Multiple kink solutions and multiple singular kink solutions for two systems of coupled Burgers-type equations. Commun. Nonlinear. Sci. Numer. Simulat.,2009.14(7):2962-2970.
[70]Monro.S, Parkes EJ. The derivation of a modified Zakharov-Kuznetsov equation and the stability of it ssolutions, J. Plasma Phys1999;62(3):305-17
[71]Zakharov VE, Kuznetsov EA. On three-dimensional solitons. Sov Phys 1974:39:285-8
[72]Biswas A, Zerrad E.1-soliton solution of the Zakharov-Kuznetsov equation with dual-power law nonlinearity, Commun Nonlinear Sci Numer Simulat,14(2009), 3574-3577.
[73]Wazwaz A.M. Nonlinear dispersive special type of the Zakharov-Kuznetsov equation ZK(n,n) with Compact and noncompact structures, Appl. Math. Comput, 161 (2005) 577-590.
[74]王飞.Potential Kadomstev-Petviashvili方程的精确孤子解.郑州大学,2007
[75]H. C.Ma. The auxiliary equation method for solving the Zakharov-Kuznetsov (ZK) equation, Computers and Mathematics with Applications (2009), doi:10.1016/j.camwa.2009.03.036
[76]A Bekir,A Boz. Application of Exp-function method for (2+1)-dimensional nonlinear evolution equations, Chaos, Solitons and Fractals 40 (2009) 458-465
[77]S Zhang. A further improved extended Fan sub-equation method for (2+1)-dimensional breaking soliton equations, Appl. Math. Comput 199(2008)259-267
[78]Ryogo Hitota. The direct method in soliton theory. Cambridge University Press
[79]陈登远.孤子理论.科学出版社.2006
[80]钱贤民,张叶.双线性算子和非线性方程的双线性形式.绍兴文理学院学报(自然科学版).22(4),2002(12)：10-15
[81]Wenxiu Ma, Yuncheng You, Solving the Korteweg-de Vries Equation by Its Bilinear Form:Wronskian Solutions, Trans. Amer. Math. Soc.,357 (2005) 5: 1753-1778
[82]Wenxiu Ma. Wronskians, Generalized Wronskians and Solutions to the Korteweg-de Vries Equation. arXiv:nlin/0303068
[83]Wenxiu Ma. Complexiton solutions to integrable equations, arXiv:nlin/0502035
[84]Wenxiu Ma, Ruguang Zhou, Liang Gao. Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in 2+1 dimensions. arXiv:0812.4316
[85]张大军,邓淑芳.孤子解的Wronskian表示.上海大学学报(自然科学版).2002(3),232-242
[86]张大军,邓淑芳,陈登远.mKdV-SineGordon方程的多孤子解.数学物理学报.2004,24A(3)：257-264
[87]林麦麦,段文山,吕克璞.Hirota方法求解KP方程的多孤子解.西北师范大学学报(自然科学版).2004(03)
[88]陈登远,张大军,毕金钵.AKNS方程的新双Wronski解.中国科学(A辑：数学).2007(11)
[89]赵俊霄.Hirota双线性方法和Pfaff式在孤立子理论中的应用.中国科学院数学与系统科学研究院.博士论文.2005年
[90]S.Kuru. Compactons and kink-like solutions of BBM-like equations by means of factorization. Chaos, Solitons & Fractals.2009,42(1):626-633
[91]P.G. Estevez, S. Kuru, J. Negro, L.M. Nieto.Travelling wave solutions of the generalized Benjamin-Bona-Mahony equation.Chaos, Solitons & Fractals,2009, 40(4):2031-2040
[92]S. Kuru.Traveling wave solutions of the BBM-like equations.J. Phys. A:Math. Theor.42 (2009) 375203 (12pp)
[93]Dengshan Wang, Hongbo Li. Single and multi-solitary wave solutions to a class of nonlinear evolution equations.J. Math. Anal. Appl.343 (2008) 273-298
[94]H.C. Rosu, O. Cornejo-Perez. Supersymmetric pairing of kinks for polynomial nonlinearities. Phys. Rev. E 71 (2005) 046607
[95]P.G. Estevez, S. Kuru, J. Negro, L.M. Negro.Travelling wave solutions of two-dimensional Korteweg-de Vries-Burgers and Kadomtsev-Petviashvili equations, J. Phys. A:Math. Gen.39 (2006) 11441-11452.
[96]刘式适,刘式达.特殊函数.气象出版社.1988
[97]Taogetusang, Sirendaoreji. New exact solitary wave solutions to the BBM and mBBM equations. Acta Physica Sinica,2004(53),4052-4060
[98]J.P.Boyd. Peakons and cashoidal waves:traveling wave solutions of the Camassa-Holm Equation. Appl.Math.Comput.81(2-3)(1997)173-187
[99]Wazwaz A.M. New compact and noncompact solutions for two variants of a modified Camassa-Holm equation. Appl.Math.Comput.163 (3) (2005) 1165-1179.
[100]Teng Xiaoyan, Zhang Weiguo, Feng Liping. Qualitative Analysis and Exact Solutions of Klein-Gordon Equation. Mathematica Applicata.21(1),2008,44-48
[101]Wazwaz A.M. The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation.Appl. Math. Comput.,167(2),2005,1179-1195
[102]M.F. El-Sabbagh, A.T. Ali. New generalized Jacobi elliptic function expansion method, Commun. Nonlinear. Sci. Numer. Simulat..13(9),2008,1758-1766
[103]Calogero F, Degasperis A. Nuovo Cimento B.1977:39-54.
[104]M. Senthilvelan. On the extended applications of Homogenous Balance Method. Appl. Math. Comput.123(2001)381-388
[105]Ibrahim E. Inan, Dogan Kaya. Some exact solutions to the potential Kadomtsev-Petviashvili equation and to a system of shallow water wave equations. Phys. Lett. A.355 (2006) 314-318
[106]Dogan Kaya, Salah M. El-Sayed. Numerical soliton-like solutions of the potential Kadomtsev-Petviashvili equation by the decomposition method, Phys. Lett. A.320 (2003) 192-199
[107]D. Xian, H.Chen, Symmetry reduced and new exact non-traveling wave solutions of potential Kadomtsev-Petviashvili equation with p-power, Appl. Math. Comput In Press
[108]Wazwaz A.M.Exact solutions with compact and noncompactstructures for the one-dimensional generalized Benjamin-Bona-Mahony equation, Commun. Nonlinear. Sci. Numer. Simulat.2005 (10) 855-867
[109]Abdul-Majid Wazwaz.New travelling wave solutions of different physical structures to generalized BBM equation, Phys. Lett. A 2006 (355):358-362
[110]Yaning Tang, Wei Xu, Liang Gao, Jianwei Shen. An algebraic method with computerized symbolic computation for the one-dimensional generalized BBM equation of any order. Chaos, Solitons and Fractals 2007 (32):1846-1852
[111]H.A. Abdusalam, E.S. Fahmy. Exact solution for the generalized Telegraph Fisher's equation, Chaos, Solitons and Fractals.2009 (41):1550-1556
[112]E.S.Fahmy. Travelling wave solutions for some time-delayed equations through factorizations. Chaos, Solitons and Fractals.2008 (38):1209-1216
[113]Whittaker, E.T., G.N.Watson. A course of modern analysis.北京：世界图书出版公司.2008
[114]Zhenya Yan, George Bluman. New compacton soliton solutions and solitary patterns solutions of nonlinearly dispersive Boussinesq equations, Computer Physics Communications,2002,149(3):11-18
[115]Mustafa Inc. New solitary wave solutions with compact support and Jacobi elliptic function solutions for the nonlinearly dispersive Boussinesq equations. Chaos, Solitons & Fractals.2008,37(3):792-798
[116]Yonggui Zhu. Exact special solutions with solitary patterns for Boussinesq-like B(m,n) equations with fully nonlinear dispersion. Chaos, Solitons & Fractals. 2004,22(1):213-220
[117]Lijun Zhang, Liqun Chen, Xuwen Huo. New exact compacton, peakon and solitary solutions of the generalized Boussinesq-like B(m,n) equations with nonlinear dispersion. Nonlinear Analysis:Theory, Methods & Applications,2007, 67(12):3276-3282
[2]李翊神.孤子与可积系统.上海科技教育出版社.1999
[3]刘式达,刘式适.孤波和湍流.上海科技教育出版社.1994
[4]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.上海科学技术出版社.2005
[5]R.Camassa, D.Holm. An integrable shallow equation with peaked solitons. Phy. Rev. Lett.1993 (11):1661-1664.
[6]郭柏灵,田立新,杨灵娥.Camassa-Holm方程.科学出版社.2008
[7]Mingliang Wang, Zhibin Li. Application of homegenous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. Lett. A.1996(67): 216.
[8]Fan E.G, Zhang H.Q. A note on the homogeneous balance method. Phys Lett A.1998(246):403.
[9]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.
[10]A.M. Wazwaz. The tanh method and a variable separated ODE method for solving double sine-Gordon equation. Phys. Lett. A.2006(350):367-370.
[11]Lixin Tian, Xiuying Song. New peaked solitary wave solutions of the generalized Camassa-Holm equation.Chaos, Solitons and Fractals,2004(19):621-639
[12]A. Degasperis and M. Procesi. Asymptotic integrability. In A. Degasperis and G. Gaeta, editors, Symmetry and perturbation theory (Rome,1998), pages 23-37. World Scientific Publishing, River Edge, NJ,1999.
[13]Filiz Tascan, Ahmet Bekir, Murat Koparan, Traveling wave solutions of nonlinear revolution equations by using the first integral method, Commun. Nonlinear. Sci. Numer. Simulat.,2009 (14):1810-1815
[14]Zhaosheng Feng. The first-integral method to study the Burgers-Korteweg-de Vries equation, J. Phys. A:Math. Gen.2002(35):343-349
[15]A.H.AhmedAli, K.R.Raslan. New solutions for some important partial differential equations, International Journal of Nonlinear Science.2007 (4):109.
[16]Xijun Deng. Exact peaked wave solution of CH-y equation by the first-integral method.Appl.Math.Comput.2008 (206):806-809
[17]Xijun Deng. Traveling wave solutions for the generalized Burgers-Huxley equation. Appl.Math.Comput.,204(2008)733-737
[18]Filiz Tascan, Ahmet Bekir, Murat Koparan. Traveling wave solutions of nonlinear evolution equations by using the first integral method. Commun. Nonlinear. Sci. Numer. Simulat..14 (2009) 1810-1815
[19]Filiz Tascan, Ahmet Bekir. Traveling wave solutions of the Cahn-Allen equation by using the first integral method. Appl. Math. Comput. (2008),doi:10.1016/ j.amc.2008.10.031
[20]Huaying Li, Yucui Guo. New exact solutions to the Fitzhugh-Nagumo equation. Appl.Math.Comput.180(2006)524-528
[21]Yaning Tang, Wei Xu, Jianwei Shen. Solitary Wave Solutions to Gardner Equation. Chinese Journal of Engineering Mathematics.2007,24(1),119-127
[22]Zuntao Fu, Shida Liu, Shikuo Liu. New kinds of solutions to Gardner equation. Chaos, Solitons and Fractals.20(2004)301-309
[23]E. Yusufoglu, A. Bekir. The tanh and the sine-cosine methods for exact solutions of the MBBM and the Vakhnenko equations, Chaos, Solitons and Fractals,38(4), 2008,1126-1133
[24]E. Yusufoglu, New solitonary solutions for the MBBM equations using Exp-function method. Phys. Lett. A,372(4),2008,442-446
[25]Shikuo Liu, Zuntao Fu, Shida Liu and Qiang Zhao. Jacobi elliptic function expansion methodand periodic wave solutions of nonlinear wave equations. Phys. Lett. A.2001,289(1-2)
[26]刘式达,刘式适.孤波和湍流.上海：上海科技教育出版社.1994
[27]Shikuo Liu, Zuntao Fu, Shida Liu, Qiang Zhao. Expansion Method about the Jacobi Elliptic Function and its Applications to Nonlinear Wave Equations. Acta Phys. Sin.2001 (50):2068-2073
[28]刘式适,付遵涛,刘式达,赵强.变系数非线性方程的Jacobi椭圆函数展开解.物理学报,2002,(09)
[29]刘式适,付遵涛,刘式达,赵强.一类非线性方程的新周期解.物理学 报.2002,(01)
[30]刘式达,傅遵涛,刘式适,赵强.非线性波动方程的Jacobi椭圆函数包络周期解.物理学报.2002,(04).
[31]闫振亚,张鸿庆,范恩贵.一类非线性演化方程新的显式行波解.物理学报.1999,(01)
[32]张善卿,李志斌.Jacobi椭圆函数展开法的新应用.物理学报.2003,(05)
[33]套格图桑,斯仁道尔吉.构造非线性发展方程Jacobi椭圆函数精确解的一种方法.内蒙古师范大学学报(自然科学汉文版),2008,(01).
[34]Zhenya Yan. The extended Jacobian elliptic function expansion method and its application in the generalized Hirota-Satsuma coupled KdV system. Chaos Solitons and Fractals.15 (2003),575-583
[35]Chaoqing Dai, Jiefang Zhang. Jacobian elliptic function method for nonlinear differential-difference equations, Chaos Solitons and Fractals.27(2006):1042-1047
[36]夏铁成,张鸿庆,李佩春.Boussinesq方程精确解析解研究.大连理工大学学报.2003,(04).
[37]王明亮,李向正,韩淑霞.Nizhnik方程组的一个非线性变换和多重孤子解(英文).应用数学.2005(02)
[38]Wang Mingliang. Solitary wave solutions for variant Boussinesq equations.Phys Lett A,1995,199(34):169-172
[39]Liu Shikuo, Fu Zuntao, Liu Shida. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations.Phys Lett A,2001, 289(12):69-74
[40]Wang Mingliang, Zhou Yubin, Li Zhibin. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics.Phys Lett A,1996,216(15):67-75
[41]Fu Z T, Liu S K,Liu S D. Elliptic equation and new solutions to nonlinear wave equations.Comm Theoret Phys,2004(42):343-346.
[42]Fu Z T, Liu S D,Liu S K. Solving Nonlinear Wave Equations by Elliptic Equation.Commun. Theor. Phys,2003(39):531-536.
[43]Fu Z T,Liu S K,Liu S D. A New Approach to Solve Nonlinear Wave Equations.Commun Theor Phys.2003(39):27-30.
[44]套格图桑,斯仁道尔吉.双曲函数型辅助方程构造具5次强非线性项的波方程的新精确孤波解.物理学报.2006(01).
[45]套格图桑,斯仁道尔吉.BBM方程和修正的BBM方程新的精确孤立波解.物理学报.2004(12).
[46]Mingliang Wang, XiangzhengLi, JinliangZhang. The (G'/G)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A 2008 (372):417-423
[47]Mingliang Wang, Jinliang Zhang, Xiangzheng Li. Application of the (G'/G)-expansion to travelling wave solutions of the Broer-Kaup and the approximate long water wave equations, Appl. Math. Comput..2008(206): 321-326
[48]Xiangzheng Li, Mingliang Wang. The (G'/G)-expansion method and traveling wave solutions for a higher-order nonlinear schrodinger equation, Appl. Math. Comput. (2009), doi:10.1016/j.amc.2008.12.005
[49]Bekir A,Cevikel AC. New exact travelling wave solutions of nonlinear physical models, Chaos, Solitons & Fractals (2008),doi:10.1016/j.chaos.2008.07.017
[50]Sheng Zhang, Ling Dong, Jin-Mei Ba, Ying-Na Sun. The (G'/G)-expansion method for nonlinear differential-difference equations, Phys. Lett. A 2009 (373): 905-910
[51]R. Hirota. Exact solutions of the Sine-Gordon equation for multiple collisions of solitons, J.Phys.Soc.Jpn.33(5) (1972) 1459-1463
[52]Biswas A. Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion, Commun Nonlinear Sci Numer Simulat, 2009 (14):3503-3506
[53]Wazwaz AM. The tanh-coth and the sine-cosine methods for kinks, solitons, and periodic solutions for the Pochhammer-Chree equations, Appl. Math. Comput 195 (2008) 24-33.
[54]Wazwaz A.M. Solitary wave solutions of the generalized shallow water wave (GSWW) equation by Hirota's method, tanh-coth method and Exp-function method. Appl. Math. Comput.,2008.202(1):275-286.
[55]Wazwaz A.M. Multiple-soliton solutions for the generalized (1+1)-dimensional and the generalized (2+1)-dimensional Ito equations. Appl. Math. Comput.,2008. 202(2):840-849.
[56]Wazwaz A.M. Multiple-soliton solutions for the Lax-Kadomtsev-Petviashvili (Lax-KP) equation. Appl. Math. Comput.,2008.201(1-2):168-174.
[57]Wazwaz A.M. The Hirota's direct method for multiple-soliton solutions for three model equations of shallow water waves. Appl. Math. Comput.,2008.201(1-2): 489-503.
[58]Wazwaz A.M., Multiple-soliton solutions of two extended model equations for shallow water waves. Appl. Math. Comput.,2008.201(1-2):790-799.
[59]Wazwaz A.M. The Hirota's bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equation. Appl. Math. Comput.,2008.200(1):160-166.
[60]Wazwaz A.M. Multiple-front solutions for the Burgers-Kadomtsev-Petviashvili equation. Appl. Math. Comput.,2008.200(1):437-443.
[61]Wazwaz A.M. The Hirota's direct method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Ito seventh-order equation. Appl. Math. Comput.,2008.199(1):133-138.
[62]Wazwaz A.M. Multiple-soliton solutions for the fifth order Caudrey-Dodd-Gibbon (CDG) equation. Appl. Math. Comput.,2008.197(2): 719-724.
[63]Wazwaz A.M. Multiple-soliton solutions for the Lax seventh-order equation. Appl. Math. Comput.,2008.198(2):877-881.
[64]Wazwaz A.M. Four (2+1)-dimensional integrable extensions of the KdV equation: Multiple-soliton and multiple singular soliton solutions. Appl. Math. Comput., 2009.215(4):1463-1476.
[65]Wazwaz A.M. A (3+1)-dimensional nonlinear evolution equation with multiple soliton solutions and multiple singular soliton solutions. Appl. Math. Comput., 2009.215(4):1548-1552.
[66]Wazwaz A.M. Multiple soliton solutions and multiple singular soliton solutions for (2+1)-dimensional shallow water wave equations. Phys. Lett. A,2009. 373(33):2927-2930.
[67]Wazwaz A.M. Multiple kink solutions and multiple singular kink solutions for (2+1)-dimensional nonlinear models generated by the Jaulent-Miodek hierarchy. Phys. Lett. A,2009.373(21):1844-1846.
[68]Wazwaz A.M. Multiple-soliton solutions and multiple-singular soliton solutions for two higher-dimensional shallow water wave equations. Appl. Math. Comput., 2009.211(2):495-501.
[69]Wazwaz A.M. Multiple kink solutions and multiple singular kink solutions for two systems of coupled Burgers-type equations. Commun. Nonlinear. Sci. Numer. Simulat.,2009.14(7):2962-2970.
[70]Monro.S, Parkes EJ. The derivation of a modified Zakharov-Kuznetsov equation and the stability of it ssolutions, J. Plasma Phys1999;62(3):305-17
[71]Zakharov VE, Kuznetsov EA. On three-dimensional solitons. Sov Phys 1974:39:285-8
[72]Biswas A, Zerrad E.1-soliton solution of the Zakharov-Kuznetsov equation with dual-power law nonlinearity, Commun Nonlinear Sci Numer Simulat,14(2009), 3574-3577.
[73]Wazwaz A.M. Nonlinear dispersive special type of the Zakharov-Kuznetsov equation ZK(n,n) with Compact and noncompact structures, Appl. Math. Comput, 161 (2005) 577-590.
[74]王飞.Potential Kadomstev-Petviashvili方程的精确孤子解.郑州大学,2007
[75]H. C.Ma. The auxiliary equation method for solving the Zakharov-Kuznetsov (ZK) equation, Computers and Mathematics with Applications (2009), doi:10.1016/j.camwa.2009.03.036
[76]A Bekir,A Boz. Application of Exp-function method for (2+1)-dimensional nonlinear evolution equations, Chaos, Solitons and Fractals 40 (2009) 458-465
[77]S Zhang. A further improved extended Fan sub-equation method for (2+1)-dimensional breaking soliton equations, Appl. Math. Comput 199(2008)259-267
[78]Ryogo Hitota. The direct method in soliton theory. Cambridge University Press
[79]陈登远.孤子理论.科学出版社.2006
[80]钱贤民,张叶.双线性算子和非线性方程的双线性形式.绍兴文理学院学报(自然科学版).22(4),2002(12)：10-15
[81]Wenxiu Ma, Yuncheng You, Solving the Korteweg-de Vries Equation by Its Bilinear Form:Wronskian Solutions, Trans. Amer. Math. Soc.,357 (2005) 5: 1753-1778
[82]Wenxiu Ma. Wronskians, Generalized Wronskians and Solutions to the Korteweg-de Vries Equation. arXiv:nlin/0303068
[83]Wenxiu Ma. Complexiton solutions to integrable equations, arXiv:nlin/0502035
[84]Wenxiu Ma, Ruguang Zhou, Liang Gao. Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in 2+1 dimensions. arXiv:0812.4316
[85]张大军,邓淑芳.孤子解的Wronskian表示.上海大学学报(自然科学版).2002(3),232-242
[86]张大军,邓淑芳,陈登远.mKdV-SineGordon方程的多孤子解.数学物理学报.2004,24A(3)：257-264
[87]林麦麦,段文山,吕克璞.Hirota方法求解KP方程的多孤子解.西北师范大学学报(自然科学版).2004(03)
[88]陈登远,张大军,毕金钵.AKNS方程的新双Wronski解.中国科学(A辑：数学).2007(11)
[89]赵俊霄.Hirota双线性方法和Pfaff式在孤立子理论中的应用.中国科学院数学与系统科学研究院.博士论文.2005年
[90]S.Kuru. Compactons and kink-like solutions of BBM-like equations by means of factorization. Chaos, Solitons & Fractals.2009,42(1):626-633
[91]P.G. Estevez, S. Kuru, J. Negro, L.M. Nieto.Travelling wave solutions of the generalized Benjamin-Bona-Mahony equation.Chaos, Solitons & Fractals,2009, 40(4):2031-2040
[92]S. Kuru.Traveling wave solutions of the BBM-like equations.J. Phys. A:Math. Theor.42 (2009) 375203 (12pp)
[93]Dengshan Wang, Hongbo Li. Single and multi-solitary wave solutions to a class of nonlinear evolution equations.J. Math. Anal. Appl.343 (2008) 273-298
[94]H.C. Rosu, O. Cornejo-Perez. Supersymmetric pairing of kinks for polynomial nonlinearities. Phys. Rev. E 71 (2005) 046607
[95]P.G. Estevez, S. Kuru, J. Negro, L.M. Negro.Travelling wave solutions of two-dimensional Korteweg-de Vries-Burgers and Kadomtsev-Petviashvili equations, J. Phys. A:Math. Gen.39 (2006) 11441-11452.
[96]刘式适,刘式达.特殊函数.气象出版社.1988
[97]Taogetusang, Sirendaoreji. New exact solitary wave solutions to the BBM and mBBM equations. Acta Physica Sinica,2004(53),4052-4060
[98]J.P.Boyd. Peakons and cashoidal waves:traveling wave solutions of the Camassa-Holm Equation. Appl.Math.Comput.81(2-3)(1997)173-187
[99]Wazwaz A.M. New compact and noncompact solutions for two variants of a modified Camassa-Holm equation. Appl.Math.Comput.163 (3) (2005) 1165-1179.
[100]Teng Xiaoyan, Zhang Weiguo, Feng Liping. Qualitative Analysis and Exact Solutions of Klein-Gordon Equation. Mathematica Applicata.21(1),2008,44-48
[101]Wazwaz A.M. The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation.Appl. Math. Comput.,167(2),2005,1179-1195
[102]M.F. El-Sabbagh, A.T. Ali. New generalized Jacobi elliptic function expansion method, Commun. Nonlinear. Sci. Numer. Simulat..13(9),2008,1758-1766
[103]Calogero F, Degasperis A. Nuovo Cimento B.1977:39-54.
[104]M. Senthilvelan. On the extended applications of Homogenous Balance Method. Appl. Math. Comput.123(2001)381-388
[105]Ibrahim E. Inan, Dogan Kaya. Some exact solutions to the potential Kadomtsev-Petviashvili equation and to a system of shallow water wave equations. Phys. Lett. A.355 (2006) 314-318
[106]Dogan Kaya, Salah M. El-Sayed. Numerical soliton-like solutions of the potential Kadomtsev-Petviashvili equation by the decomposition method, Phys. Lett. A.320 (2003) 192-199
[107]D. Xian, H.Chen, Symmetry reduced and new exact non-traveling wave solutions of potential Kadomtsev-Petviashvili equation with p-power, Appl. Math. Comput In Press
[108]Wazwaz A.M.Exact solutions with compact and noncompactstructures for the one-dimensional generalized Benjamin-Bona-Mahony equation, Commun. Nonlinear. Sci. Numer. Simulat.2005 (10) 855-867
[109]Abdul-Majid Wazwaz.New travelling wave solutions of different physical structures to generalized BBM equation, Phys. Lett. A 2006 (355):358-362
[110]Yaning Tang, Wei Xu, Liang Gao, Jianwei Shen. An algebraic method with computerized symbolic computation for the one-dimensional generalized BBM equation of any order. Chaos, Solitons and Fractals 2007 (32):1846-1852
[111]H.A. Abdusalam, E.S. Fahmy. Exact solution for the generalized Telegraph Fisher's equation, Chaos, Solitons and Fractals.2009 (41):1550-1556
[112]E.S.Fahmy. Travelling wave solutions for some time-delayed equations through factorizations. Chaos, Solitons and Fractals.2008 (38):1209-1216
[113]Whittaker, E.T., G.N.Watson. A course of modern analysis.北京：世界图书出版公司.2008
[114]Zhenya Yan, George Bluman. New compacton soliton solutions and solitary patterns solutions of nonlinearly dispersive Boussinesq equations, Computer Physics Communications,2002,149(3):11-18
[115]Mustafa Inc. New solitary wave solutions with compact support and Jacobi elliptic function solutions for the nonlinearly dispersive Boussinesq equations. Chaos, Solitons & Fractals.2008,37(3):792-798
[116]Yonggui Zhu. Exact special solutions with solitary patterns for Boussinesq-like B(m,n) equations with fully nonlinear dispersion. Chaos, Solitons & Fractals. 2004,22(1):213-220
[117]Lijun Zhang, Liqun Chen, Xuwen Huo. New exact compacton, peakon and solitary solutions of the generalized Boussinesq-like B(m,n) equations with nonlinear dispersion. Nonlinear Analysis:Theory, Methods & Applications,2007, 67(12):3276-3282