广义Burgers-Huxley方程的精确解
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摘要
本文主要围绕广义Burger-Huxley方程的精确解进行了深入的研究与探讨,通过几种方法得到了该方程及其特殊形式的若干新解。主要工作包括以下几方面内容:
首先,介绍了研究工作的历史、现状、未来和本文的主要工作。
其次,介绍了与本文相关的一些基本概念、符号,给出了孤立子的定义和发生机理,探讨了孤立波和孤立子的异同,对目前所知道的孤立子按空间维数的高低进行了分类,同时对易于混淆的精确解、近似解和相似解做了必要的说明。
再次,介绍了几种求解非线性方程的方法,如计算机代数解法、齐次平衡法、Backlund变换和Auto-Backlund变换、改进的tanh-coth法等。
最后,应用上面介绍的几种求解非线性方程的方法研究广义Burger-Huxley方程及其特殊形式方程的精确解,得到了许多有意义的新解。
This paper mainly focuses on finding exact solutions of the generalized Burgers-Huxley equations and has conducted the thorough research and the discussion.Through serveral methods,which obtained a number of new solutions of this equation and its special form equations.The main content is depicted as follows:
Firstly,we study the history,current situation and future of our research, at the same time introduce the main work of this paper.
Secondly,we introduce some primary conception and denotation relate to this paper,gave the definition and befallen mechanism of solitons,discuss the similarities and differences between soliton with solitary wave,classified the solitons which we have known according to the dimension of the space. At the same time,make out some explain necessary to the exact solutions,approximately solutions and similar solutions which tend to be confued.
Thirdly,we introduce several methods to obtain solutions of the nonlinear equations.Such as,computer algebra method,homogeneous balance method,Backlund and Auto-Backlund transformation and modified tanh-coth method.
Finally,Using the several methods to obtain solutions of the nonlinear equations which introduced above,we study the exact solutions of the generalized Burgers-Huxley equation and its special form equations,which obtained a number of new solutions of this equation and its special form equations.
首先,介绍了研究工作的历史、现状、未来和本文的主要工作。
其次,介绍了与本文相关的一些基本概念、符号,给出了孤立子的定义和发生机理,探讨了孤立波和孤立子的异同,对目前所知道的孤立子按空间维数的高低进行了分类,同时对易于混淆的精确解、近似解和相似解做了必要的说明。
再次,介绍了几种求解非线性方程的方法,如计算机代数解法、齐次平衡法、Backlund变换和Auto-Backlund变换、改进的tanh-coth法等。
最后,应用上面介绍的几种求解非线性方程的方法研究广义Burger-Huxley方程及其特殊形式方程的精确解,得到了许多有意义的新解。
This paper mainly focuses on finding exact solutions of the generalized Burgers-Huxley equations and has conducted the thorough research and the discussion.Through serveral methods,which obtained a number of new solutions of this equation and its special form equations.The main content is depicted as follows:
Firstly,we study the history,current situation and future of our research, at the same time introduce the main work of this paper.
Secondly,we introduce some primary conception and denotation relate to this paper,gave the definition and befallen mechanism of solitons,discuss the similarities and differences between soliton with solitary wave,classified the solitons which we have known according to the dimension of the space. At the same time,make out some explain necessary to the exact solutions,approximately solutions and similar solutions which tend to be confued.
Thirdly,we introduce several methods to obtain solutions of the nonlinear equations.Such as,computer algebra method,homogeneous balance method,Backlund and Auto-Backlund transformation and modified tanh-coth method.
Finally,Using the several methods to obtain solutions of the nonlinear equations which introduced above,we study the exact solutions of the generalized Burgers-Huxley equation and its special form equations,which obtained a number of new solutions of this equation and its special form equations.
引文
[1]范恩贵.《可积系统与计算机代数》,科学出版社,2004
[2]郭柏灵,旁小峰.《孤立子》,科学出版社,1987
[3]刘式达,刘式适.《物理学中的非线性方程》,北京大学出版社,2000
[4]李翊神.《孤子与可积系统》,上海科技教育出版社,1999
[5]旁小峰.《孤子物理学》,四川科学技术出版社,2003
[6]陈陆君,梁昌洪.《孤立子理论及其应用》,西安电子科技大学出版社,1997
[7]刘正荣.关于孤立子的研究[J],云南大学学报,2003,25(3):207-211
[8]张解放,陈芳跃.截断展开法和广义KdV变系数方程新的类孤子解[J].物理学报.2001,50(9):1648-1650
[9]黄正洪.非线性耗散方程的变换及对称约化[J].兰州大学学报.1992,2(35):1-6
[10]闫振亚,张鸿庆.(2+1)维非线性长波方程的相似约化与解析解[J].数学物理学报.2001,21A(3):384-390
[11]杜先云.(2+1)维Burgers方程的新的精确解[J].西南师范大学学报(自然科学版).2006,31(4):29-31
[12]杨先林,唐驾时.两类变系数kdV方程的新精确孤波解[J].湖南大学学报(自然科学版).2007,34(12):72-75
[13]夏鸿鸣,何万生,温志贤.Tanh函数法的推广及应用[J].甘肃科学学报.2006,12(4):10-12
[14]刘妍丽,张健.具有高阶非线性项的广义kdV方程的精确孤波解[J].四川师范大学学报(自然科学版).2005,28(2):142-145
[15]赵水标.KdV-Burgers方程新的复合孤波解[J].宁波大学学报(理工版).2004,17(1):32-34
[16]石玉仁,吕克璞,段文山等.组合kdV方程的显式精确解[J].物理学报.2003,52(02):0267-0270
[17]沈水金.Jacobi椭圆函数展开法的应用[J].绍兴文理学院学报.2006,26(10):13-16
[18]王石瑛,周国中,郭冠平.Jacobi椭圆函数新的展开法对非线性耦合标量场方程双周期解的求解[J].山西师范大学学报(自然科学版).2005,19(2):25-30
[19]吴国将,史马良,张苗.一般变换下双Jacobi椭圆函数展开法及应用.物理学报.2006,55(8):3858-3863
[20]吴国将,史马良,张苗.扩展的Jacobi椭圆函数展开法和Zakharov方程组的新的精确周期解.物理学报.2007,56(9):5054-5059
[21]Yan Z Y and Zhang HQ.New explicit solitary wave solutions and periodic wave solutions for WhithamCBorer Kaup equation in shallow water.Phys.Lett A.2001(285):355
[22]CHEN Y,LI B.Travelling wave solutions for generalized symmetric regularized long-wave equations with high-order nonlinear terms.Chin.Phys.2004:13(03),302-306
[23]王明亮.用F-展开法解Sine-Gordon方程[J].河南科技大学学报.2005,26(1):79-82
[24]Mansifield EL and Clarkson.Darboux Transformation and Soliton.Math.Comput.Simul.1997(43):39-43
[25]石玉仁,吕克璞,段文山等.组合kdv方程的显式精确解[J].物理学报,2003,52(02):0267-0270
[26]B.Dey.Compacton solutions for a class of two parameter generalized odd-order KdV equations.Phys.Rev.E.1998,57(4):4733-4738
[27]梅建琴,张鸿庆(2+1)维广义浅水波方程的类孤子解与周期解.数学物理学报.2005,25A(6):784-788
[28]Chen Y and Li B.New similarity reduction of the Bossinesq equation Chaos,Solitons and Fractals.2004(19):977-989
[29]HUANG D J,ZHANG H Q.Variable-coefficient projective Riccati equation method and its application to a new(2+1) dimensional simplified generalized Broer-Kaup system.Chaos,Solitons and Fractals.2005(23):601-607
[30]Z.Y.Yan,G.Bluman.New compacton soliton solutions and solitary patterns solutions of nonlinearly dispersive Boussinesq equations.[J].Computer Physics Communications.2002(149):11-18
[31]Z.Y.Yan,G.Bluman.New families of solitons with compact support for Boussinesq-like B(m,n)equations with fully nonlinear dispersion[J].Chaos,Solitons and Fractals.2002(14):1151-1158
[32]何进春.关于Kdv方程孤子解的研究[J].应用数学.2007,20(1):145-150
[33]何进春.mKdv方程N-孤子解的验证[J].应用数学.2007,20(4):808-813
[34]殷久利,田立新.一类非线性方程的Backlund变换以及紧孤立波解的线性稳定性[J].数学物理学报,2007,27A(1):27-36
[35]陈登远.Backlund变换和N-孤子解.数学研究与评论.2005,20(03):479-488
[36]曾昕,张鸿庆.(2+1)维Boussinesq方程的Backlund变换与精确解[J].物理学报.2005,54(04):1476-1480
[37]A.M.Wazwaz.Compactons structures for fifth-order KdV like equations in higher dimensions [J].Appli.Math.and Comput.2002(130):425-440
[38]Matveev VA and Salle MA.Darboux transformations and solitons.Berlin:Heidelberg Springer-Verlag.1991(05):305-307
[39]Lixin Tian,Xiuying Song.New peaked solitary wave solutions of the generalized Camassa-Holm equation[J].Chaos,Solitons and Fractals.2004,19(3):621-639
[40]P.Rosenau,J.M.Hyman.Compactons:solitons with finite wavelengths.[J]Phys.Rev.Lett.70(5) (1993):564-567
[41]P.Rosenau.On nonanalytic solitary waves due to nonlinear dispersion.Phys.Lett.A;1997,(230):305-318
[42]P.T.Dinda,M.Remoissent.Breather compactons in nonlinear Klein-Gordon systems.Phys.Rev.E.1999,(60):6218-6221
[43]P.Rosenau.Nonlinear dispersion and compact structures.[J]Phys.Rev.Lett.1994,73:1737 -1741
[44]P.Rosenau,D.Levy.Compactons In a Class Of Nonlinearly Quintic Equations.[J]Phys.Lett.A.1999,(252):297-306
[45]P.Rosenau.Compact and noncompact dispersive patterns.Phys.Lett.A.2000,275:193-203
[46]He ping,Chen Zheng,Fu Jun.Solution of Kdv equation by computer algebra[J].Physics Department,Hainan Teachers University,Haikou 571158,China.2002:505-509
[47]Hereman W,Banerjee PP,Korpel A,Assanto G,Immerzeele A,vanand A and Meepeel.Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method.Phs,A:math,Gen[J].1986,(19):607-628
[48]王彬炜,尚亚东.扩展齐次平衡法与BBM方程的精确解.广州大学学报(自然科学版),2007(01):17-19
[49]J.Satsuma,in:M.Ablowitz,B.Fuchssteiner,M.Kruskal(Eds.).Topics in Soliton Theory and Exactly Solvable Nonlinear Equations.World Scientific,Singapore,1987
[50]B.Batiha,M.S.M,I.Hashim.A note on the Adomian decomposition method for the generalized Burgers-Huxley equation.Applized Mathematics and Computation,2006
[51]Xijun Deng.Travelling wave solutions for the generalized Burgers-Huxley equation.Applized Mathematics and Computation,2008
[52]M.Hassan.Exact solitary wave solutions for a generalized KdV- Burgers equation.Chaos[J],Solitons and Fractals 19 2004;1201-1206
[53]M.T.Darvishi,S.Kheybari and F.Khani.Spectral collocation method and Darvishi's preconditionings to solve the generalized Burgers-Huxley equation.Communications in Nonlinear Science and Numerical Simulation,December 2008;2091-2103
[2]郭柏灵,旁小峰.《孤立子》,科学出版社,1987
[3]刘式达,刘式适.《物理学中的非线性方程》,北京大学出版社,2000
[4]李翊神.《孤子与可积系统》,上海科技教育出版社,1999
[5]旁小峰.《孤子物理学》,四川科学技术出版社,2003
[6]陈陆君,梁昌洪.《孤立子理论及其应用》,西安电子科技大学出版社,1997
[7]刘正荣.关于孤立子的研究[J],云南大学学报,2003,25(3):207-211
[8]张解放,陈芳跃.截断展开法和广义KdV变系数方程新的类孤子解[J].物理学报.2001,50(9):1648-1650
[9]黄正洪.非线性耗散方程的变换及对称约化[J].兰州大学学报.1992,2(35):1-6
[10]闫振亚,张鸿庆.(2+1)维非线性长波方程的相似约化与解析解[J].数学物理学报.2001,21A(3):384-390
[11]杜先云.(2+1)维Burgers方程的新的精确解[J].西南师范大学学报(自然科学版).2006,31(4):29-31
[12]杨先林,唐驾时.两类变系数kdV方程的新精确孤波解[J].湖南大学学报(自然科学版).2007,34(12):72-75
[13]夏鸿鸣,何万生,温志贤.Tanh函数法的推广及应用[J].甘肃科学学报.2006,12(4):10-12
[14]刘妍丽,张健.具有高阶非线性项的广义kdV方程的精确孤波解[J].四川师范大学学报(自然科学版).2005,28(2):142-145
[15]赵水标.KdV-Burgers方程新的复合孤波解[J].宁波大学学报(理工版).2004,17(1):32-34
[16]石玉仁,吕克璞,段文山等.组合kdV方程的显式精确解[J].物理学报.2003,52(02):0267-0270
[17]沈水金.Jacobi椭圆函数展开法的应用[J].绍兴文理学院学报.2006,26(10):13-16
[18]王石瑛,周国中,郭冠平.Jacobi椭圆函数新的展开法对非线性耦合标量场方程双周期解的求解[J].山西师范大学学报(自然科学版).2005,19(2):25-30
[19]吴国将,史马良,张苗.一般变换下双Jacobi椭圆函数展开法及应用.物理学报.2006,55(8):3858-3863
[20]吴国将,史马良,张苗.扩展的Jacobi椭圆函数展开法和Zakharov方程组的新的精确周期解.物理学报.2007,56(9):5054-5059
[21]Yan Z Y and Zhang HQ.New explicit solitary wave solutions and periodic wave solutions for WhithamCBorer Kaup equation in shallow water.Phys.Lett A.2001(285):355
[22]CHEN Y,LI B.Travelling wave solutions for generalized symmetric regularized long-wave equations with high-order nonlinear terms.Chin.Phys.2004:13(03),302-306
[23]王明亮.用F-展开法解Sine-Gordon方程[J].河南科技大学学报.2005,26(1):79-82
[24]Mansifield EL and Clarkson.Darboux Transformation and Soliton.Math.Comput.Simul.1997(43):39-43
[25]石玉仁,吕克璞,段文山等.组合kdv方程的显式精确解[J].物理学报,2003,52(02):0267-0270
[26]B.Dey.Compacton solutions for a class of two parameter generalized odd-order KdV equations.Phys.Rev.E.1998,57(4):4733-4738
[27]梅建琴,张鸿庆(2+1)维广义浅水波方程的类孤子解与周期解.数学物理学报.2005,25A(6):784-788
[28]Chen Y and Li B.New similarity reduction of the Bossinesq equation Chaos,Solitons and Fractals.2004(19):977-989
[29]HUANG D J,ZHANG H Q.Variable-coefficient projective Riccati equation method and its application to a new(2+1) dimensional simplified generalized Broer-Kaup system.Chaos,Solitons and Fractals.2005(23):601-607
[30]Z.Y.Yan,G.Bluman.New compacton soliton solutions and solitary patterns solutions of nonlinearly dispersive Boussinesq equations.[J].Computer Physics Communications.2002(149):11-18
[31]Z.Y.Yan,G.Bluman.New families of solitons with compact support for Boussinesq-like B(m,n)equations with fully nonlinear dispersion[J].Chaos,Solitons and Fractals.2002(14):1151-1158
[32]何进春.关于Kdv方程孤子解的研究[J].应用数学.2007,20(1):145-150
[33]何进春.mKdv方程N-孤子解的验证[J].应用数学.2007,20(4):808-813
[34]殷久利,田立新.一类非线性方程的Backlund变换以及紧孤立波解的线性稳定性[J].数学物理学报,2007,27A(1):27-36
[35]陈登远.Backlund变换和N-孤子解.数学研究与评论.2005,20(03):479-488
[36]曾昕,张鸿庆.(2+1)维Boussinesq方程的Backlund变换与精确解[J].物理学报.2005,54(04):1476-1480
[37]A.M.Wazwaz.Compactons structures for fifth-order KdV like equations in higher dimensions [J].Appli.Math.and Comput.2002(130):425-440
[38]Matveev VA and Salle MA.Darboux transformations and solitons.Berlin:Heidelberg Springer-Verlag.1991(05):305-307
[39]Lixin Tian,Xiuying Song.New peaked solitary wave solutions of the generalized Camassa-Holm equation[J].Chaos,Solitons and Fractals.2004,19(3):621-639
[40]P.Rosenau,J.M.Hyman.Compactons:solitons with finite wavelengths.[J]Phys.Rev.Lett.70(5) (1993):564-567
[41]P.Rosenau.On nonanalytic solitary waves due to nonlinear dispersion.Phys.Lett.A;1997,(230):305-318
[42]P.T.Dinda,M.Remoissent.Breather compactons in nonlinear Klein-Gordon systems.Phys.Rev.E.1999,(60):6218-6221
[43]P.Rosenau.Nonlinear dispersion and compact structures.[J]Phys.Rev.Lett.1994,73:1737 -1741
[44]P.Rosenau,D.Levy.Compactons In a Class Of Nonlinearly Quintic Equations.[J]Phys.Lett.A.1999,(252):297-306
[45]P.Rosenau.Compact and noncompact dispersive patterns.Phys.Lett.A.2000,275:193-203
[46]He ping,Chen Zheng,Fu Jun.Solution of Kdv equation by computer algebra[J].Physics Department,Hainan Teachers University,Haikou 571158,China.2002:505-509
[47]Hereman W,Banerjee PP,Korpel A,Assanto G,Immerzeele A,vanand A and Meepeel.Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method.Phs,A:math,Gen[J].1986,(19):607-628
[48]王彬炜,尚亚东.扩展齐次平衡法与BBM方程的精确解.广州大学学报(自然科学版),2007(01):17-19
[49]J.Satsuma,in:M.Ablowitz,B.Fuchssteiner,M.Kruskal(Eds.).Topics in Soliton Theory and Exactly Solvable Nonlinear Equations.World Scientific,Singapore,1987
[50]B.Batiha,M.S.M,I.Hashim.A note on the Adomian decomposition method for the generalized Burgers-Huxley equation.Applized Mathematics and Computation,2006
[51]Xijun Deng.Travelling wave solutions for the generalized Burgers-Huxley equation.Applized Mathematics and Computation,2008
[52]M.Hassan.Exact solitary wave solutions for a generalized KdV- Burgers equation.Chaos[J],Solitons and Fractals 19 2004;1201-1206
[53]M.T.Darvishi,S.Kheybari and F.Khani.Spectral collocation method and Darvishi's preconditionings to solve the generalized Burgers-Huxley equation.Communications in Nonlinear Science and Numerical Simulation,December 2008;2091-2103