若干非线性发展方程的精确解
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摘要
非线性发展方程精确解的求解一直受到数学家和物理学家的热切关注.目前虽然已经建立和发展了不少非线性发展方程行之有效的求解方法,但由于非线性理论的复杂性,对于大量的非线性发展方程仍然无法求得其精确解.因此,继续寻找一些有效可行的求解方法是科学研究中一项十分重要和极具价值的工作.
本文对首次积分法的思想方法及后人对此方法的改进做出了介绍,并对一些有重要物理意义的非线性发展方程进行了求解,所得结果不仅囊括了若干已知的精确解,而且还得到了许多新的精确解.丰富及完善了这些非线性发展方程精确解的研究.
全文分为三章.第一章为绪论,介绍了非线性发展方程的基本概况和非线性发展方程精确解的研究现状,并扼要介绍了本文的主要工作.
第二章系统介绍了首次积分法及后人对此方法的改进及具体的应用操作过程.
第三章运用首次积分法对Modified Improved Boussinesq方程、(2+1)维色散长波方程组、(2+1)维Konopelchenko-Dubrovsky方程组进行了详细的求解,并获得了方程(组)新的精确解.
最后我们对本文的工作进行了总结,提出了一些自己有待研究的问题并对今后研究方向作出了展望.
Searching for exact solutions of nonlinear evolution equation has long been a popular research subject for mathematicians and physicists.At present, there has been established and developed a lot of effective method to solve nonlinear evolution equations, but a large number of nonlinear evolution equations are still unable to solve for complexity of nonlinear equations themselves.As a con-sequence,it is a very important research and valuable work to go on searching for some effective and feasible method to solve nonlinear evolution equation.
In this paper, the first integral method of thinking and successor to make improvements to this method is introduced.By using this approach,a lot of exact solutions for some nonlinear evolution equations which have important physical significance are easily presented. The results include not only a number of known exact solutions, but also a number of new exact solutions.This enriched and improved the content of exact solutions of nonlinear evolution equations.
This paper is composed of four chapters:
In the fist chapter, we introduce the basic profiles of nonlinear evolution equations and the research status of exact solutions for nonlinear evolution equa-tions.The main work of this article is introduced.
In the second chapter,we elaborate the first integral method and its appli-cation procedure in detail.
In the third chapter, we make use of first integral method to obtain a lot of new exact solutions of Modified Improved Boussinesq equation, (2+1)-dimensional dispersive long wave equations and (2+1)-dimensional Konopelchenko-Dubrovsky equations.
In the end, we summarize the work of this paper and look into the distance of research in future as well.
本文对首次积分法的思想方法及后人对此方法的改进做出了介绍,并对一些有重要物理意义的非线性发展方程进行了求解,所得结果不仅囊括了若干已知的精确解,而且还得到了许多新的精确解.丰富及完善了这些非线性发展方程精确解的研究.
全文分为三章.第一章为绪论,介绍了非线性发展方程的基本概况和非线性发展方程精确解的研究现状,并扼要介绍了本文的主要工作.
第二章系统介绍了首次积分法及后人对此方法的改进及具体的应用操作过程.
第三章运用首次积分法对Modified Improved Boussinesq方程、(2+1)维色散长波方程组、(2+1)维Konopelchenko-Dubrovsky方程组进行了详细的求解,并获得了方程(组)新的精确解.
最后我们对本文的工作进行了总结,提出了一些自己有待研究的问题并对今后研究方向作出了展望.
Searching for exact solutions of nonlinear evolution equation has long been a popular research subject for mathematicians and physicists.At present, there has been established and developed a lot of effective method to solve nonlinear evolution equations, but a large number of nonlinear evolution equations are still unable to solve for complexity of nonlinear equations themselves.As a con-sequence,it is a very important research and valuable work to go on searching for some effective and feasible method to solve nonlinear evolution equation.
In this paper, the first integral method of thinking and successor to make improvements to this method is introduced.By using this approach,a lot of exact solutions for some nonlinear evolution equations which have important physical significance are easily presented. The results include not only a number of known exact solutions, but also a number of new exact solutions.This enriched and improved the content of exact solutions of nonlinear evolution equations.
This paper is composed of four chapters:
In the fist chapter, we introduce the basic profiles of nonlinear evolution equations and the research status of exact solutions for nonlinear evolution equa-tions.The main work of this article is introduced.
In the second chapter,we elaborate the first integral method and its appli-cation procedure in detail.
In the third chapter, we make use of first integral method to obtain a lot of new exact solutions of Modified Improved Boussinesq equation, (2+1)-dimensional dispersive long wave equations and (2+1)-dimensional Konopelchenko-Dubrovsky equations.
In the end, we summarize the work of this paper and look into the distance of research in future as well.
引文
[1]李翊神.非线性科学研究选讲[M].中国科学技术大学出版社,1994
[2]郭柏林.非线性演化方程.上海：上海科技教育出版社,1995：1-401
[3]李大潜,秦铁虎.物理学与偏微分方程(上册)[M].北京：高等教育出版社,1997
[4]李大潜,秦铁虎.物理学与偏微分方程(下册)[M].北京：高等教育出版社,1997
[5]刘式适,刘世达.物理学中的非线性方程.北京：北京大学出版社,2000：1-390
[6]Qu C Z,He W L,Dou J H.Separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source [J]. Physics D.,2000, 144:97-123
[7]Qu C Z,He W L,Dou J H.Separation of variables and exact solutions of generalized nonlinear Klein-Gordon equations [J]. Prog.Theor.Phys.,2001, 105:379-398
[8]Estevez P G,Qu C Z, Zhang S L. Separation of variables of general-ized porous mediun equation with nonlinear source[J]. J.Math.Anal.Appl., 2002,275:44-59
[9]Estevez P G,Qu C Z.Separation of variables in nonlinear wave equation with a variable wave speed[J],Theor.Math.Phys.,2002,133:1490-1497
[10]Ablowitz M J,Kaup D J,Newell A C et al.The inverse scattering transform-Fburier analysis for nonlinear problems[J].Stud Appl Math,1966,53:249-315
[11]Gardner C S,Greene J M,Kruskal M D.Method for solvong the Kortweg-de Vries equation[J].Phys.Rev.Lett,1967,19:1095-1097
[12]Lax P D.Integrals of nonlinear equations of and solitary waves[J]. Com-mun.Pure Appl.Math.,1968,21:467-490
[13]Zakharov V E,Shabat A B.Exact theory of two-dimensional self-focusing and one-dimensional waves in nonlinear media[J].Sov.Phys.JETP,1972,34:62-69
[14]Ablowitz M J,Kaup D J,Newell A C,et al.Method for solving the sine-Gordon equation[J].Phys.Rev.Lett.,1973,30:1262-1264
[15]Ablowitz M J,Kaup D J,Newell A C,et al.Nonlinear evolution equations of physical sigificance[J].Phys.Rev.Lett.,1974,31:125-127
[16]Ablowitz M J,Kaup D J,Newell A C,et al.The inverse scattering transform Fourier analysis for nonlinear problem[J].Study.in Appl.Math.,1974,53:249-315
[17]Hirota R.Exact solutions of the Kortweg-de Vries equation for multiple collisious of solitous[J].Phya.Rev.Lett.,1971,27:1192-1194
[18]Malfliet W.Solitary wave solutions of nonlinear wave equation [J]. Am J Phys,1992,60:650-654
[19]Wang Mingliang.The solitary wave solutions for variant Boussinesq equa-tions[J].Physics Letter A,1995,199:169-172
[20]Wang Mingliang.Exact silutions for the RLW-Burgers equa-tion[J].Mathematica Applicata,1995,8:51-55
[21]Wang Mingliang. Exact silutions for a compoud KdV-Burgers equa-tion [J]. Physics Letters A,1996,213:279-287
[22]Wang Mingliang,Wang Y M,Zhang J L.The peridic wave solutions for twosystems of nonlinear wave equationg[J].Chin.Phys.,2003,12(12):1341-1348
[23]Fu Z T,Liu S K,Liu S D.New Jacobi elliptic fuction expansion and new peri-odic wave solutions of nonlinear wave equations[J].Phys.Lett.A,2001,290:72-76
[24]Fu Z T,Liu S D.New rational form solutions to mKdV equation[J]. Com-mun.Theor.Phys.,2005,43:423-426
[25]Zhang H Q.New exact travelling wave solutions for some nonlinear evolution equations[J].Chaos Solitions and Fractals,2005,26:921-925
[26]Zhang J L,Wang Y M,Wang M L.New applications of the homogeneous balance principle[J].Chin.Phys.,2003,12(3):245-250.
[27]Zhang J L,Ren D M,Wang M L,et al.The periodic wave solutions for the gen-eralized Nizhnik-Novikov-Veselov equation[J].Chin.Phys.,2003,12(8):825-830
[28]Feng Zhao sheng.On explicit exact solutions to the compoubd Burgers-KdV equation [J]. Physics Letters A,2002,293:57-66
[29]Feng Zhao sheng.Exact lolution to an approximate sine-Gorden equation in (n+1)-dimensional space [J]. Physics Letters A,2002,302:64-76
[30]Feng Zhao sheng,X.H.Wang.The first integral method to the two-dimensional Burgers-Korteweg-de Vries equation[J]. Physics Letters A, 2003,308:173-178
[31]刘希强.非线性发展方程显示解的研究[D].北京：中国工程物理研究院,2002
[32]马正义.非线性波动方程的精确解及其孤子结构[D].上海：上海大学,2007
[33]徐淑奖.几类非线性偏微分方程解法及解的性质的探讨[D].北京：北京邮电大学.2006
[34]S.Abbasbandy, A.Shirzadi. The firet integral method for modified Benjamin-Bona-Mahony equation[J]. Commun Nonlinear Sci Numer Simulat,2010, 15:1759-1764
[35]Filiz Tascan,Ahmet Bekir.Travelling wave solutions of the Cahn-Allen equation by using first integral method [J].Applied Mathematics and Computation,2009,207,279-282
[36]Filiz Tascan,Ahmet Bekir,Murat Koparan.Travelling wave solutions of non-linear evolution equation by using the first integral method [J]. Commun Non-linear Sci Numer Simulat,2009,14:1810-1815
[37]N.Taghizadeh, M.Mirzazadeh, F.Farahrooz. Exact solutions of the nonlinear Schrodinger equation by the first integral method[J]. J.Math.Anal.Appl. 2011,374:549-553
[38]Lu Bin,Zhang Hongqing,Xie Fuding.Travelling wave solutions of nonlinear partial equations by using the first intergral method [J]. Applied Mathematics and Computation,2010,216:1329-1336
[39]Li Huaying,Guo Yucui.New exact solutions to the Fitzhugh-Nagumo equa-tion[J].Applied Mathematics and Commputation,2006,180:524-528
[40]N.Taghizadeh,M.Mirzazadeh,F.Farahrooz.Exact solutions of the modified KdV-KP equation and the Burgers-KP equation by using the first inter-gral method [J].2011, x xx:xxx - xxx
[41]Deng Xijun.Exact peaked wave solution of CHγ equation by the first-integral method [J].Applied Mathematics and Computation,2008,206:806-809
[42]M J Ablowitz,P.A.Clarkson.Solitons,Nonlinear Evolution Equations and In-verse Scatting[M].New York:Cambridge University Press,1991
[43]姜东梅,张鸿庆Modified Improved Boussinesq方程的显式精确解[J].甘肃工业大学学报,2002,28(3)：104-106
[44]Abdul Majid Wazwaz. New kinks and solitons solutions to the (2+1)-dimensional Konopelchenko-Dubrovsky equation [J]. Mathematical and Computer Modelling,2007,45:473-479
[2]郭柏林.非线性演化方程.上海：上海科技教育出版社,1995：1-401
[3]李大潜,秦铁虎.物理学与偏微分方程(上册)[M].北京：高等教育出版社,1997
[4]李大潜,秦铁虎.物理学与偏微分方程(下册)[M].北京：高等教育出版社,1997
[5]刘式适,刘世达.物理学中的非线性方程.北京：北京大学出版社,2000：1-390
[6]Qu C Z,He W L,Dou J H.Separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source [J]. Physics D.,2000, 144:97-123
[7]Qu C Z,He W L,Dou J H.Separation of variables and exact solutions of generalized nonlinear Klein-Gordon equations [J]. Prog.Theor.Phys.,2001, 105:379-398
[8]Estevez P G,Qu C Z, Zhang S L. Separation of variables of general-ized porous mediun equation with nonlinear source[J]. J.Math.Anal.Appl., 2002,275:44-59
[9]Estevez P G,Qu C Z.Separation of variables in nonlinear wave equation with a variable wave speed[J],Theor.Math.Phys.,2002,133:1490-1497
[10]Ablowitz M J,Kaup D J,Newell A C et al.The inverse scattering transform-Fburier analysis for nonlinear problems[J].Stud Appl Math,1966,53:249-315
[11]Gardner C S,Greene J M,Kruskal M D.Method for solvong the Kortweg-de Vries equation[J].Phys.Rev.Lett,1967,19:1095-1097
[12]Lax P D.Integrals of nonlinear equations of and solitary waves[J]. Com-mun.Pure Appl.Math.,1968,21:467-490
[13]Zakharov V E,Shabat A B.Exact theory of two-dimensional self-focusing and one-dimensional waves in nonlinear media[J].Sov.Phys.JETP,1972,34:62-69
[14]Ablowitz M J,Kaup D J,Newell A C,et al.Method for solving the sine-Gordon equation[J].Phys.Rev.Lett.,1973,30:1262-1264
[15]Ablowitz M J,Kaup D J,Newell A C,et al.Nonlinear evolution equations of physical sigificance[J].Phys.Rev.Lett.,1974,31:125-127
[16]Ablowitz M J,Kaup D J,Newell A C,et al.The inverse scattering transform Fourier analysis for nonlinear problem[J].Study.in Appl.Math.,1974,53:249-315
[17]Hirota R.Exact solutions of the Kortweg-de Vries equation for multiple collisious of solitous[J].Phya.Rev.Lett.,1971,27:1192-1194
[18]Malfliet W.Solitary wave solutions of nonlinear wave equation [J]. Am J Phys,1992,60:650-654
[19]Wang Mingliang.The solitary wave solutions for variant Boussinesq equa-tions[J].Physics Letter A,1995,199:169-172
[20]Wang Mingliang.Exact silutions for the RLW-Burgers equa-tion[J].Mathematica Applicata,1995,8:51-55
[21]Wang Mingliang. Exact silutions for a compoud KdV-Burgers equa-tion [J]. Physics Letters A,1996,213:279-287
[22]Wang Mingliang,Wang Y M,Zhang J L.The peridic wave solutions for twosystems of nonlinear wave equationg[J].Chin.Phys.,2003,12(12):1341-1348
[23]Fu Z T,Liu S K,Liu S D.New Jacobi elliptic fuction expansion and new peri-odic wave solutions of nonlinear wave equations[J].Phys.Lett.A,2001,290:72-76
[24]Fu Z T,Liu S D.New rational form solutions to mKdV equation[J]. Com-mun.Theor.Phys.,2005,43:423-426
[25]Zhang H Q.New exact travelling wave solutions for some nonlinear evolution equations[J].Chaos Solitions and Fractals,2005,26:921-925
[26]Zhang J L,Wang Y M,Wang M L.New applications of the homogeneous balance principle[J].Chin.Phys.,2003,12(3):245-250.
[27]Zhang J L,Ren D M,Wang M L,et al.The periodic wave solutions for the gen-eralized Nizhnik-Novikov-Veselov equation[J].Chin.Phys.,2003,12(8):825-830
[28]Feng Zhao sheng.On explicit exact solutions to the compoubd Burgers-KdV equation [J]. Physics Letters A,2002,293:57-66
[29]Feng Zhao sheng.Exact lolution to an approximate sine-Gorden equation in (n+1)-dimensional space [J]. Physics Letters A,2002,302:64-76
[30]Feng Zhao sheng,X.H.Wang.The first integral method to the two-dimensional Burgers-Korteweg-de Vries equation[J]. Physics Letters A, 2003,308:173-178
[31]刘希强.非线性发展方程显示解的研究[D].北京：中国工程物理研究院,2002
[32]马正义.非线性波动方程的精确解及其孤子结构[D].上海：上海大学,2007
[33]徐淑奖.几类非线性偏微分方程解法及解的性质的探讨[D].北京：北京邮电大学.2006
[34]S.Abbasbandy, A.Shirzadi. The firet integral method for modified Benjamin-Bona-Mahony equation[J]. Commun Nonlinear Sci Numer Simulat,2010, 15:1759-1764
[35]Filiz Tascan,Ahmet Bekir.Travelling wave solutions of the Cahn-Allen equation by using first integral method [J].Applied Mathematics and Computation,2009,207,279-282
[36]Filiz Tascan,Ahmet Bekir,Murat Koparan.Travelling wave solutions of non-linear evolution equation by using the first integral method [J]. Commun Non-linear Sci Numer Simulat,2009,14:1810-1815
[37]N.Taghizadeh, M.Mirzazadeh, F.Farahrooz. Exact solutions of the nonlinear Schrodinger equation by the first integral method[J]. J.Math.Anal.Appl. 2011,374:549-553
[38]Lu Bin,Zhang Hongqing,Xie Fuding.Travelling wave solutions of nonlinear partial equations by using the first intergral method [J]. Applied Mathematics and Computation,2010,216:1329-1336
[39]Li Huaying,Guo Yucui.New exact solutions to the Fitzhugh-Nagumo equa-tion[J].Applied Mathematics and Commputation,2006,180:524-528
[40]N.Taghizadeh,M.Mirzazadeh,F.Farahrooz.Exact solutions of the modified KdV-KP equation and the Burgers-KP equation by using the first inter-gral method [J].2011, x xx:xxx - xxx
[41]Deng Xijun.Exact peaked wave solution of CHγ equation by the first-integral method [J].Applied Mathematics and Computation,2008,206:806-809
[42]M J Ablowitz,P.A.Clarkson.Solitons,Nonlinear Evolution Equations and In-verse Scatting[M].New York:Cambridge University Press,1991
[43]姜东梅,张鸿庆Modified Improved Boussinesq方程的显式精确解[J].甘肃工业大学学报,2002,28(3)：104-106
[44]Abdul Majid Wazwaz. New kinks and solitons solutions to the (2+1)-dimensional Konopelchenko-Dubrovsky equation [J]. Mathematical and Computer Modelling,2007,45:473-479