关于非线性发展方程精确求解的研究
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摘要
本文主要研究了以下四方面的问题:首先介绍了修正扩展的范的偏方程方法,并以高维耦合Burgers方程为例说明了它的应用。其次应用不同于修正扩展的范的偏方程方法的常微分方程和目标函数给出了变系数mKdV方程的新的精确行波解。然后将双参数假设法进行了扩展,并应用它求出了形变Boussioesq方程的精确解。最后给出了一种新的达布变换,并由它得到了Broer-Kaup系统新的孤子型的解。本文由两章组成:第一章,绪论。在这一章中主要介绍了本文所涉及的学科的发展历史及研究现状,并简要介绍了作者的工作。第二章主要介绍了推广的tanh函数法在求非线性发展方程的孤立子解中的运用和一种新的达布变换。给出了高维耦合Burgers方程、变系数mKdV方程、形变Boussioesq方程的新的精确解。由新的达布变换,得到了Broer-Kaup系统新的孤子型的解。
This dissertation has mainly done the following four as-pects research:First,apply the modified extended Fan s sub-equationmethod to higher-dimensional coupled burgers equation.Second,educeexplicit and exact travelling wave solutions for variable coefficientmKdV equation by different Riccati equation and object function.Thethird,the generalization of Double parameter hypothesis and the ex-act solutions of variant Boussioesq equations 2.The last,give a newDarboux transformation and new soliton-like solutions for the Broer-Kaup system.The structure of this paper is as follows:Chapter 1 isconcerned with the exposition of the development and the researchsituation of several subjects which will be discussed in this paper.The main results of this dissertation are brieffy introduced in thischapter.Chapter 2 mainly apply generalized tanh function method tothe nonlinear evolution equation and give a new Darboux transforma-tion.Yield new exact solutions of higher-dimensional coupled burgersequation、variable coefficient mKdV equation、Variant BoussioesqEquations 2、the Broer-Kaup system.
This dissertation has mainly done the following four as-pects research:First,apply the modified extended Fan s sub-equationmethod to higher-dimensional coupled burgers equation.Second,educeexplicit and exact travelling wave solutions for variable coefficientmKdV equation by different Riccati equation and object function.Thethird,the generalization of Double parameter hypothesis and the ex-act solutions of variant Boussioesq equations 2.The last,give a newDarboux transformation and new soliton-like solutions for the Broer-Kaup system.The structure of this paper is as follows:Chapter 1 isconcerned with the exposition of the development and the researchsituation of several subjects which will be discussed in this paper.The main results of this dissertation are brieffy introduced in thischapter.Chapter 2 mainly apply generalized tanh function method tothe nonlinear evolution equation and give a new Darboux transforma-tion.Yield new exact solutions of higher-dimensional coupled burgersequation、variable coefficient mKdV equation、Variant BoussioesqEquations 2、the Broer-Kaup system.
引文
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[3] FermiE. ,Pasta J. R. ,and Ulam S .M .Studies of nonlinear problems.LosAlamosSci.Lab. Rep., 1955,1940-1960.
[4] GarderC .S. ,Moikawa G M.Similarity in the asymptotic behavior of collision freehydom-agn-etic waves.Courant Inst.Math.Sci.Re port,19 60,1732-1760.
[5] Peeing J. K. ,Skyme T .H R.Amodel unified field equation.Nucl.Phys.,1962,31 :550-555.
[6] ZabuskyN .,Kruskal M.Interaction of solitons in a collisionless plasma and the re-curence of initial states.Ph ys.R ev.Lett,1965,15:240-243.
[7]郭柏灵,庞小峰.孤立子.北京:科学出版社,1987.1-362.
[8] Acton J.R.,Squire P.T.Solving Equations with Physical Understand-ing.Briston:Adam Hilger Ltd, 1985. 110-118.
[9]谷超豪,郭柏灵,李翊神,胡和生等.孤立子理论与应用.杭州:浙江科学出版社,1990. 1-475.
[10]谷超豪,胡和生,周子翔.孤立子理论中的达布变换及其儿何应用.上海:上海科学技术出版社,1999.1-54.
[11] G.艾仑格.孤立子一物理学家用的数学方法.刘之景译.北京:科学出版社,1989.
[12] Bagrov V G.,Gilman D.M .Exact Solutions of Relativistic wave Equa-tons.Dordrecht: KIuwer Acdenic Publishers 1990. 1-240
[13] RemoisssenetM .Waves Called Solutions.New York:Springer-Venal,1999.1-243.
[14]李翊神.孤子与可积系统.上海:上海科技教育出版社,1999.1-79.
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[16] Dodd R .K,EilbeckJ .C .,Gibbon J .D .,MomsH .C. Soliton and nonlinearwaves.New York:Academic Press,1982.1-287.
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[18] W,艾克霍思,A.范哈顿.逆散射变换和孤立子理论.黄迅成译.上海:上海科学技术文献出版社,1984.1-207.
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[20] Lax P. D .Integrals of nonlinear equations of evolution and solitary waves.CommunPure Appl.Math,1968,21:461490.
[21] Ablowitz M .J. ,Kaup D .J. ,Newell A. C and Segur H Method for solving thesine-gordon equation.Phys.Rev.Let.,1973,30:1262-1264.
[22] AblowitzM .J. ,KaupD .J. ,Newell A .C and Segur H .Nonlineare volution equa-tions of Physical significance.Phys.Rev.Let,1973,31:125-127.
[23] AblowitzM .J. ,KaupD .J. ,NewellA .C and Segur H .The inverse scateringtransform Fourier Analysis for Non-linear problems.Studies in Appl. Math., 1974,53: 249-315.
[24] HirotaR .Exact soliton solutions of a Korteweg-de-Vries equation for multiple col-liesions of solitons.Phys.Rev.Lett, 19 71,27 :11 92-1194.
[25] HirotaR .Exact envelope soliton solutions of a nonlinear w ave equation.J. Math.Phys.,1973, 14: 805-809.
[26] HirotaR.Direct methods of finding exact solutions of nonlinear evolution equations,in B a..cklund Transformation and its Relation to the Inverse Scattering Problem.Berlin:Springer,1976. 1-321.
[27] Wahlquist H .D. ,E stabmok F .B. B.B a..cklund Transformation for solitons of theKorteweg-deVries equation.Phy. Rev.Let,1973,31:1386-1390.
[28] Rogers C.,Shadwick W. R. B.B a..cklund Transformation and their applica-tion.New York: academic Press, 1982.1-216.
[29] C .H .G u,Z .X .Zhou.on the Darboux matrices ofB a..cklund Transformation forthe ASKN system.MathPhys.Lets,1987,10:179-184.
[30] C .H .G u,H .S. H u.Aunfied explicit form for B a..cklund Transformations forgeneralized hierarchies of KdV equation.Math Phys Lett.,1986,11 ,32 5-335.
[31] C. H. Gu. On the Darboux form of B a..cklund Transformations. Tianjing: Proc.Nankai symposium on integrable system, 1987. 1-60.
[32] M .L. W ang.Exact solution of acompound KdV Burgers equation.Phys.LettA,1996,213: 279-287.
[33] M .L. Wang,Y B Z hou.Application of a homogeneous balance method to exactsolutions of nonlinear equations in mathematical physics.Phys. Lett A, 1996,216:67-75.
[34] M .L W ang.Solitary wave solutions for variant Boussineaq equations.Phys Let A,1995, 199:169-172.
[35]乔志军.非线性波动方程的数值解.科学通报,1998,43:1149-1153.
[36] E .G Fan.Traveling wave solutions for nonlinear equations using symbolic compu-tation. Com.Math.App,2002,43:671-680.
[37] E. G Fan. Travelling wave solutions of nonlinear evolution equations by using sym-bolic computation Appl.Math.J. ChineseUniv.Ser.B, 2001,16 (2):149-155.
[38]范恩贵,张鸿庆.非线性波动方程的孤波解.物理学报,1997,46(7):1254-1258.
[39] E .G. Fan.Solution solutions for the new complex version of a coupled KdV equationand a coupled MKdV equation.Phys.Lett,20 01,28 5:37 3-376.
[40]范恩贵.可积系统与计算机代数.北京:科学出版社,2003.1-160.
[41] Engui Fan, Ran Zhang. Applications of Jacobi elliptic function expansion methodto special-type nonlinear equations.Phys.Let.A, 2002,305:38 3-392.
[42] EnguiFan,jian Zhang,Benny YC.Hon.A new complex line soliton for the two di-mensional Kdv-Burgers equation.Phy.Lett.A, 2001,29 1,37 6-380.
[43] OlverP J .Application of Lie group to Diferential Equation.Berlin:Springer,1986.1-178.
[44] CarksonP A .,Kruskal M .D .New similarity reduction of the Boussinesq equa-tion.J.Math. Phys.,1989,30,2201-2207.
[45] Mal?iet W .Solitary wave solutions of nonlinear wave equations.Am .J. Phys,1992,60(7): 650.654.
[46] Z .X .Zh ou.A method to obtain explicit solutions of 1+ 2dimensional AKNS sys-tem.Phys. Lett,,1992,A 1 68:370-375.
[47] Wood J .C .Harmonic maps into symmetric spaces and integrable syatem. Harmonicmaps and integrable systems.Vieweg,1994,29 -44.
[48] Takassaki K .A new approach to the self-dual Yang-Mills equations.Comm.Math.Phys, 1984,94:35-39.
[49]李翊神.孤子与可积系统.上海:上海科技教育出版社,1999.12.
[50]董焕河,张玉峰.几类三阶,四阶非线性系统的不稳定性.洛阳大学学报-1999:14(4),11-14.
[51] leble SB,Ustinov NV.Darboux transformation,deep reductions and solitons.J Phys.A 1993;26;5007-16.
[52] Gu CH,Hu HS,Zhou ZX.Darboux transformation in soliton theory and its geometricapplications.Shanghai Sci.Tech.publ;1999.
[53] Mal?iet W.Solitary wave solutions of nonlinear wave equations.Am J Phys.1992;60:650-4.
[54] EnguiFan. 2003.Phys.A.7009-7026.
[55] Liu Chunping Double parameter hypothesis and the exact solutions of variantshallow water wave equations.Acta Mathematica Sinica.2000,13(1):15-18.
[56] Y.S.Li,J.E.Zhang.Phys.Lett.A.284(2001)253-258.
[57]张鸿庆,张玉峰. Benjamin方程的B a..cklund变换、非线性叠加公式及无穷守恒律[J]应用数学和力学,2001,(10) .
[58]张鸿庆,张玉峰.组合KdV与MKdV方程B a..cklund变换及其一类精确解.《大连理工大学学报》,2001年04期.
[59]张鸿庆张玉峰.Benjamin方程的B a..cklund变换、非线性叠加公式及无穷守恒律.应用数学和力学,2001年第10期.
[60]董焕河,龚新波,张玉峰.演化方程的Darboux变换的一类求法. [J];工科数学,2001年06期; 80-82.
[61]张玉峰,张鸿庆.一族Liouville可积系及其约束流的Lax表示、Darboux变换.应用数学和力学,2002:23(1),23-30.
[62]李志斌,张善卿.非线性波方程准确孤立波解的符号计算.数学物理学报,1997, 17(1):81-89.
[63]范恩贵.可积系统与计算机代数.北京:科技出版社,2004.
[64]张鸿庆.弹性力学方程组一般解的统一理论.大连理工大学学报,1978, 18:23-47.
[65]张鸿庆,王震宇.胡海昌.解的完备性和逼近性.科学通报,1985, 30(5):342-344.
[66]张鸿庆.吴方向.一类偏微分方程组的一般解及其在壳体理论中的应用.力学学报,1992,24(6):700-707.
[67]张鸿庆,冯红.构造弹性力学位移函数的机械化算法.应用数学与力学,1995, 16(4):315-322.
[68]张鸿庆.偏微分方程组的一般解与完备性.现代数学与力学,1991, IV.
[69]张玉峰.孤立子方程求解与可积系统. [D];大连理工大学, 2002年.
[70]张玉峰,孔令臣,杨耕文.广义变系数KdV,mKdV方程的精确类孤子解. [J];甘肃工业大学学报, 2002年03期.
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[72]赵熙强,张玉峰,闫庆友.具有变系数的广义Burgers-KdV方程新精确解.New explicitexact solutions to a type of generalized Burgers-KdV equations with variant coef-ficients.《大连理工大学学报Journal of Dalian University of Technology》.
[73]张玉峰,张鸿庆等.Darboux变换和带附加项的Du?ng型方程的精确解.工程数学学报,2001年04期.
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[75]谢福鼎.Wu-Ritt消元法在偏微分代数方程中的应用:(博士学位论文).大连:大连理工大学,2003.
[2] Korteweg D.J. ,DeVries G .On the change of form of long waves advancing channeland on a new type o f long stationary waves.Ph il.M ag.1895,39(5):442-443.
[3] FermiE. ,Pasta J. R. ,and Ulam S .M .Studies of nonlinear problems.LosAlamosSci.Lab. Rep., 1955,1940-1960.
[4] GarderC .S. ,Moikawa G M.Similarity in the asymptotic behavior of collision freehydom-agn-etic waves.Courant Inst.Math.Sci.Re port,19 60,1732-1760.
[5] Peeing J. K. ,Skyme T .H R.Amodel unified field equation.Nucl.Phys.,1962,31 :550-555.
[6] ZabuskyN .,Kruskal M.Interaction of solitons in a collisionless plasma and the re-curence of initial states.Ph ys.R ev.Lett,1965,15:240-243.
[7]郭柏灵,庞小峰.孤立子.北京:科学出版社,1987.1-362.
[8] Acton J.R.,Squire P.T.Solving Equations with Physical Understand-ing.Briston:Adam Hilger Ltd, 1985. 110-118.
[9]谷超豪,郭柏灵,李翊神,胡和生等.孤立子理论与应用.杭州:浙江科学出版社,1990. 1-475.
[10]谷超豪,胡和生,周子翔.孤立子理论中的达布变换及其儿何应用.上海:上海科学技术出版社,1999.1-54.
[11] G.艾仑格.孤立子一物理学家用的数学方法.刘之景译.北京:科学出版社,1989.
[12] Bagrov V G.,Gilman D.M .Exact Solutions of Relativistic wave Equa-tons.Dordrecht: KIuwer Acdenic Publishers 1990. 1-240
[13] RemoisssenetM .Waves Called Solutions.New York:Springer-Venal,1999.1-243.
[14]李翊神.孤子与可积系统.上海:上海科技教育出版社,1999.1-79.
[15] Ablowitz M. J.,Clarbkson P A. Solitons, Nolinear Evohrtion Equations and In-verse Scattering.NewYork:Cambridge University Press,1991.1- 438.
[16] Dodd R .K,EilbeckJ .C .,Gibbon J .D .,MomsH .C. Soliton and nonlinearwaves.New York:Academic Press,1982.1-287.
[17] B .E.扎哈罗夫,C.B.马纳利夫,C.N.诺维科夫等.孤子理论.彭启才译,北京:科学出版社,1985.1-239.
[18] W,艾克霍思,A.范哈顿.逆散射变换和孤立子理论.黄迅成译.上海:上海科学技术文献出版社,1984.1-207.
[19] GarderC .S. ,GreenJ .M ,KruskalM .D.,MiuraRM.Method for solving the Ko-rteweg DeVries equation.Phys.Rev.Lett,1967,19:1095-1097.
[20] Lax P. D .Integrals of nonlinear equations of evolution and solitary waves.CommunPure Appl.Math,1968,21:461490.
[21] Ablowitz M .J. ,Kaup D .J. ,Newell A. C and Segur H Method for solving thesine-gordon equation.Phys.Rev.Let.,1973,30:1262-1264.
[22] AblowitzM .J. ,KaupD .J. ,Newell A .C and Segur H .Nonlineare volution equa-tions of Physical significance.Phys.Rev.Let,1973,31:125-127.
[23] AblowitzM .J. ,KaupD .J. ,NewellA .C and Segur H .The inverse scateringtransform Fourier Analysis for Non-linear problems.Studies in Appl. Math., 1974,53: 249-315.
[24] HirotaR .Exact soliton solutions of a Korteweg-de-Vries equation for multiple col-liesions of solitons.Phys.Rev.Lett, 19 71,27 :11 92-1194.
[25] HirotaR .Exact envelope soliton solutions of a nonlinear w ave equation.J. Math.Phys.,1973, 14: 805-809.
[26] HirotaR.Direct methods of finding exact solutions of nonlinear evolution equations,in B a..cklund Transformation and its Relation to the Inverse Scattering Problem.Berlin:Springer,1976. 1-321.
[27] Wahlquist H .D. ,E stabmok F .B. B.B a..cklund Transformation for solitons of theKorteweg-deVries equation.Phy. Rev.Let,1973,31:1386-1390.
[28] Rogers C.,Shadwick W. R. B.B a..cklund Transformation and their applica-tion.New York: academic Press, 1982.1-216.
[29] C .H .G u,Z .X .Zhou.on the Darboux matrices ofB a..cklund Transformation forthe ASKN system.MathPhys.Lets,1987,10:179-184.
[30] C .H .G u,H .S. H u.Aunfied explicit form for B a..cklund Transformations forgeneralized hierarchies of KdV equation.Math Phys Lett.,1986,11 ,32 5-335.
[31] C. H. Gu. On the Darboux form of B a..cklund Transformations. Tianjing: Proc.Nankai symposium on integrable system, 1987. 1-60.
[32] M .L. W ang.Exact solution of acompound KdV Burgers equation.Phys.LettA,1996,213: 279-287.
[33] M .L. Wang,Y B Z hou.Application of a homogeneous balance method to exactsolutions of nonlinear equations in mathematical physics.Phys. Lett A, 1996,216:67-75.
[34] M .L W ang.Solitary wave solutions for variant Boussineaq equations.Phys Let A,1995, 199:169-172.
[35]乔志军.非线性波动方程的数值解.科学通报,1998,43:1149-1153.
[36] E .G Fan.Traveling wave solutions for nonlinear equations using symbolic compu-tation. Com.Math.App,2002,43:671-680.
[37] E. G Fan. Travelling wave solutions of nonlinear evolution equations by using sym-bolic computation Appl.Math.J. ChineseUniv.Ser.B, 2001,16 (2):149-155.
[38]范恩贵,张鸿庆.非线性波动方程的孤波解.物理学报,1997,46(7):1254-1258.
[39] E .G. Fan.Solution solutions for the new complex version of a coupled KdV equationand a coupled MKdV equation.Phys.Lett,20 01,28 5:37 3-376.
[40]范恩贵.可积系统与计算机代数.北京:科学出版社,2003.1-160.
[41] Engui Fan, Ran Zhang. Applications of Jacobi elliptic function expansion methodto special-type nonlinear equations.Phys.Let.A, 2002,305:38 3-392.
[42] EnguiFan,jian Zhang,Benny YC.Hon.A new complex line soliton for the two di-mensional Kdv-Burgers equation.Phy.Lett.A, 2001,29 1,37 6-380.
[43] OlverP J .Application of Lie group to Diferential Equation.Berlin:Springer,1986.1-178.
[44] CarksonP A .,Kruskal M .D .New similarity reduction of the Boussinesq equa-tion.J.Math. Phys.,1989,30,2201-2207.
[45] Mal?iet W .Solitary wave solutions of nonlinear wave equations.Am .J. Phys,1992,60(7): 650.654.
[46] Z .X .Zh ou.A method to obtain explicit solutions of 1+ 2dimensional AKNS sys-tem.Phys. Lett,,1992,A 1 68:370-375.
[47] Wood J .C .Harmonic maps into symmetric spaces and integrable syatem. Harmonicmaps and integrable systems.Vieweg,1994,29 -44.
[48] Takassaki K .A new approach to the self-dual Yang-Mills equations.Comm.Math.Phys, 1984,94:35-39.
[49]李翊神.孤子与可积系统.上海:上海科技教育出版社,1999.12.
[50]董焕河,张玉峰.几类三阶,四阶非线性系统的不稳定性.洛阳大学学报-1999:14(4),11-14.
[51] leble SB,Ustinov NV.Darboux transformation,deep reductions and solitons.J Phys.A 1993;26;5007-16.
[52] Gu CH,Hu HS,Zhou ZX.Darboux transformation in soliton theory and its geometricapplications.Shanghai Sci.Tech.publ;1999.
[53] Mal?iet W.Solitary wave solutions of nonlinear wave equations.Am J Phys.1992;60:650-4.
[54] EnguiFan. 2003.Phys.A.7009-7026.
[55] Liu Chunping Double parameter hypothesis and the exact solutions of variantshallow water wave equations.Acta Mathematica Sinica.2000,13(1):15-18.
[56] Y.S.Li,J.E.Zhang.Phys.Lett.A.284(2001)253-258.
[57]张鸿庆,张玉峰. Benjamin方程的B a..cklund变换、非线性叠加公式及无穷守恒律[J]应用数学和力学,2001,(10) .
[58]张鸿庆,张玉峰.组合KdV与MKdV方程B a..cklund变换及其一类精确解.《大连理工大学学报》,2001年04期.
[59]张鸿庆张玉峰.Benjamin方程的B a..cklund变换、非线性叠加公式及无穷守恒律.应用数学和力学,2001年第10期.
[60]董焕河,龚新波,张玉峰.演化方程的Darboux变换的一类求法. [J];工科数学,2001年06期; 80-82.
[61]张玉峰,张鸿庆.一族Liouville可积系及其约束流的Lax表示、Darboux变换.应用数学和力学,2002:23(1),23-30.
[62]李志斌,张善卿.非线性波方程准确孤立波解的符号计算.数学物理学报,1997, 17(1):81-89.
[63]范恩贵.可积系统与计算机代数.北京:科技出版社,2004.
[64]张鸿庆.弹性力学方程组一般解的统一理论.大连理工大学学报,1978, 18:23-47.
[65]张鸿庆,王震宇.胡海昌.解的完备性和逼近性.科学通报,1985, 30(5):342-344.
[66]张鸿庆.吴方向.一类偏微分方程组的一般解及其在壳体理论中的应用.力学学报,1992,24(6):700-707.
[67]张鸿庆,冯红.构造弹性力学位移函数的机械化算法.应用数学与力学,1995, 16(4):315-322.
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