非线性偏微分方程的函数展开法与精确解
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摘要
求解微分方程的精确解在理论和实际中都是重要而古老的研究课题，显式解，特别是行波解可以很好地描述各种物理现象，如振动、传播波等。基于非线性方程的复杂性，至今能够求出精确解的方程很少，因此，寻求新的求解方法和拓展已有方法都是重要而有价值的工作。
首先，本文对原有“Tanh-”函数法做了新的扩展，用变系数式构造解的形式，实现常系数非线性方程和变系数非线性方程求解式的统一构造，从而同一方法可获得更多方程更多形式的精确解，也便于实现计算机上统一编程处理；为获得更广泛的解，本文对辅助Riccati方程也做了相应改进，方程中构造了任意函数项，即辅助Riccati方程改进为：φ′(ξ)=φ~2(ξ)+b(ξ)，b(ξ)为ξ的任意函数，使得构造形式更灵活，文中具体给出了b(ξ)=kξ~n的解的情形，应用该方法求解KdV方程，(2+1)维KD方程等非线性演化方程，获得了丰富的精确解簇。
其次，基于齐次平衡方法，通过引入参数，给出了求解非线性演化方程精确解的一种变换方法，对于特定方程通过变换，可以讨论方程的多孤子解，变换求解自相似解，进一步通过引入GM(Gardner-Morikawa)变换求解非线性演化方程的截断级数解。
It is an important and old research subject for obtaining the exact solution of differential equations. The explicit solution, especially the traveling wave solution, can be used to describe many physical phenomena well, such as oscillation, propagation wave etc. Up to now the exact solutions for many important equations can not still be received since the complexity of the nonlinear equations. Thus seeking new method and extending existed method, both come to be an important and valuable work.
In this paper, we generalize the Tanh-function method, exhibit the united formulae to solutions of constant coefficient equations and variable coefficient equations, by the generalized method we can get more exact solutions. Meanwhile it is convenience to be dealt with by the computer. In order to receive more solutions, we also modify the subsidiary equation to beφ'(ξ) =φ~2(ξ) + b(ξ), where b(ξ) is an arbitrary function ofξ. The arbitrariness of b(ξ) makes the solutions more flexible. Specially, the case when b(ξ) = kξ~n is discussed in this paper. And by applying the generalized method to KdV equation, (2+1) dimension KD equations and other nonlinear evolution equations, we obtain abundant exact solution families of those equations.
Based on the homogenous balance method, by introducing the parameters we give a new method of solving nonlinear partial differential equation (NLPDE)—Method of parametric transformation. For some specific NLPDE we obtain the multi-soliton solutions, self-similar solutions and trun- cation series solutions associated with the GM (Gardner-Morikawa) transformation.
首先，本文对原有“Tanh-”函数法做了新的扩展，用变系数式构造解的形式，实现常系数非线性方程和变系数非线性方程求解式的统一构造，从而同一方法可获得更多方程更多形式的精确解，也便于实现计算机上统一编程处理；为获得更广泛的解，本文对辅助Riccati方程也做了相应改进，方程中构造了任意函数项，即辅助Riccati方程改进为：φ′(ξ)=φ~2(ξ)+b(ξ)，b(ξ)为ξ的任意函数，使得构造形式更灵活，文中具体给出了b(ξ)=kξ~n的解的情形，应用该方法求解KdV方程，(2+1)维KD方程等非线性演化方程，获得了丰富的精确解簇。
其次，基于齐次平衡方法，通过引入参数，给出了求解非线性演化方程精确解的一种变换方法，对于特定方程通过变换，可以讨论方程的多孤子解，变换求解自相似解，进一步通过引入GM(Gardner-Morikawa)变换求解非线性演化方程的截断级数解。
It is an important and old research subject for obtaining the exact solution of differential equations. The explicit solution, especially the traveling wave solution, can be used to describe many physical phenomena well, such as oscillation, propagation wave etc. Up to now the exact solutions for many important equations can not still be received since the complexity of the nonlinear equations. Thus seeking new method and extending existed method, both come to be an important and valuable work.
In this paper, we generalize the Tanh-function method, exhibit the united formulae to solutions of constant coefficient equations and variable coefficient equations, by the generalized method we can get more exact solutions. Meanwhile it is convenience to be dealt with by the computer. In order to receive more solutions, we also modify the subsidiary equation to beφ'(ξ) =φ~2(ξ) + b(ξ), where b(ξ) is an arbitrary function ofξ. The arbitrariness of b(ξ) makes the solutions more flexible. Specially, the case when b(ξ) = kξ~n is discussed in this paper. And by applying the generalized method to KdV equation, (2+1) dimension KD equations and other nonlinear evolution equations, we obtain abundant exact solution families of those equations.
Based on the homogenous balance method, by introducing the parameters we give a new method of solving nonlinear partial differential equation (NLPDE)—Method of parametric transformation. For some specific NLPDE we obtain the multi-soliton solutions, self-similar solutions and trun- cation series solutions associated with the GM (Gardner-Morikawa) transformation.
引文
[1] M. J. Ablowitz, P. A. Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering [M], Cambridge University Press. 1991.
[2] 谷超豪．孤立子理论及其应用，浙江科技出版社[M]，1990．
[3] 谷超豪，胡和生，周子翔．孤立子理论中的Darboux变换及其几何应用，上海科技出版社[M]，1999．
[4] V. B. Matveev, M. A. Salle. Darboux transformation and solitions [M], Berlin: Springer, 1991.
[5] M L Wang. Phys. Lett. A, 1995(199):169.
[6] M L Wang. Phys. Lett. A, 1996(213):279.
[7] M L Wang, Y B Zhou, Z B Li. Phys. Lett. A, 1996(216):67.
[8] 王明亮，李志斌，周宇斌．兰州大学学报(自然科学版)，1999(35)：8．
[9] 王明亮，周宇斌．兰州大学学报(自然科学版)，32(1996)1-5．
[10] E J Parkes and B R Duffy. Computer Phys Commun, 1996(98):288.
[11] B R Duffy and E J Parkes. Phys. Lett. A, 1996(214):271.
[12] E J Parkes and B R Duffy. Phys. Lett. A, 1997(229):217.
[13] Z B Li and M L Wang. J Phys A,1993(26):6027.
[14] E J Parkes. J Phys A,1994(27):L497.
[15] E G Fan, J Zhang and Y C Hon. Phys. Lett. A, 2001(291):376.
[16] V V Gudkov. J Math Phys.,1997(38):4794
[17] G X Zhang, Z B Li and Y S Duan. J science in China. A, 2001,44(3):396-401.
[18] Chaoqing Dai, Jiamin Zhu, Jiefang Zhang. Chaos Solitons & Fractals, 2006(27):881-886.
[19] Conte, R., Musette, J phys. A:Math.Gen.,1992(25):2609-2612.
[20] A.D.Polyanin, V.F.Zaltsev.Handbook of exact solutions for ordinary differential equations.CRC press[M], 1995.
[21] Milton Abramowitz, Irene A. Stegun.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.New York[M], 1972
[22] S Ghosh, A Kundu and S Nandy.J Math. Phys., 1990(40):1993.
[23] B Konopelchenko and V Dubrovsky. Phys. Lett. A, 1984(102):15.
[24] 陈登远．孤子引论．科学出版社[M]，2006 15-24．
[25] H H Chen, Y C Lee and C S Liu. Phys. Scr, 1979(20): 490.
[26] F Calogero and W Eckhaus.Inverse Problems,1987(3):229.
[27] 石玉仁，汪映海，杨红娟等．广义变系数Burgers方程的精确解．华东师范大学学报[J]．2006(5)：27-33．
[28] C Z Qu and A Q Wang.Commun Theor Phys,1996(26): 369.
[29] M L Wang. J Phys. Lett A,1996(213): 279.
[30] 纪峰波，求解非线性演化方程函数展式法的统一．兰州大学[M]，2006：16-26．
[31] 王明亮，李志斌，周宇斌．齐次平衡原则及其应用[J]，兰州大学学报(自然科学版)，35 (1999)8-16．
[32] 王明亮，周宇斌．一个非线性波动方程的精确解[J]，兰州大学学报(自然科学版)，32 (1996)1-5．
[33] 黄定江，张鸿庆．扩展的双曲函数法和Zakharov方程组的新精确孤立波解[J]，物理学报，53(2004)2434-1438．
[34] Engui Fan. Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics [J], Chaos Solitons & Fractals, 16 (2003) 819-839.
[35] 张解放，陈芳跃．截断展开方法和广义变系数KdV方程新的精确类孤子解[J]，物理学报，50(2001)1648-1650．
[36] Masaaki Ito. Symmetries and conservation laws of a coupled nonlinear wave equation [J], Phys. Lett. A, 91 (1982) 335-338.
[37] Zuntao Fu, Lin Zhang, Shida Liu, Shikno Lin. Fractional transformation and new solutions to mKdV equation [J], Phys. Lett. A, 325 (2004) 363-369.
[38] M.Remoissenet. Waves Called Solitons [M], Berlin: Springer, 1996.
[39] E. E. Tzirtzilakis, Ch. Skokos, T. C. Bountis. A numerical study of solitions of the Bonssinesq equation using spectral methods [J], Math. Phys., 12(2002)6161-6165.
[40] 吴文俊．科学通报，1986(31)：535
[41] 吴文俊．数学机械化．北京：科学出版社[M]，1999．
[42] B Buchberger.Grobner Bases: an algorithmic method in polynomial idea theory, renct trends in multidimensional sysytem theory.Ed N K Bose.Riedal Publishing Company, 1985.
[43] Q Cox,J Little and D O'shea. Ideas,varieties and algorithms: An introduction to computational algebraic geometry and commutative algebra. Springer,UT M, 1992.
[44] W Malfliet. Am J Phys,1992(60):650.
[45] E J Parkes. J Phys A,1994(27): L497.
[46] E G Fan, J Zhang and Y C Hon. Phys Lett A,2001(291):376.
[47] V V Gudkov.J Math Phys,1997(38):4794.
[48] Whitham G B. Linear and nonlinear wave. New York: A Wiley-Interscience Publication,1974.
[49] Y B Zeng. Chin Ann of Math,1991(24): 78.
[50] N Nirmala and M Vedan. J Math Phys,1984(27): 2644.
[51] E. S. Cheb-Terrab. A Computational Approach for the Exact Solving of Systems of Partial Differential Equations. Computer Physics Communications, 2001
[52] H D Wahlquist and F B Estabrook. J Math Phys,1976(17):1293.
[53] D G Zhang.Phys Lett,1996(223):436.
[54] R Hirota."Direct Methods of Finding Exact Solution of Nonlinear Evolution Equations", in "Backlund Tansformations",Ed. by R M Miura, Lecture Notes in Mathematics,Springer,Berlin, Heidelberg,New York Vol. 515,(1976).
[55] J Weiss, M Tabor and G Carnevale. J Math Phys,1983(24): 522-526.
[56] W Hereman and M Takaoka.J Phys A,1990(23):4805.
[57] J Weiss.J Math Phys,1983(24):1405-1413.
[58] J Weiss.J Math Phys,1984(25):13-24.
[59] C. S. Gardner, J.M. Greene, M. D. Kruskal and R. M. Miura.Method for solving the Korteweg-de-Vries equation. Phys. Rev. Lett.,1967(19):1095-1097.
[60] R. Hirota. Exact solution of the Korteweg-de-Vries equation for multiple collisions of silitons.Phys. Rev.Lett.,1971 (27):1192-1194.
[61] G. W. Bluman and J. D. Cole. The general similarity solution of the heat equation. J Math. Mech.,1969(18): 1025-1042.
[62] P. A. Clarkson and M. D. Kruskal. New similarity solutions of the Boussinesq equation. J Math. Phys.,1989(30):2201-2213.
[63] 范恩贵．可积系统与计算机代数．北京：科学出版社[M]，1999．
[64] E G Fan and H Q Zhang. Appl Math Mech.,1998(19): 645.
[65] E G Fan. Commun Theor Phys., 2001(35): 523.
[66] 曹策问．科学通报，1989(34)：723
[67] 乔志军．应用数学，1995(18)：287．
[68] Z J Qiao. J Phys. A,1993(26): 4407.
[2] 谷超豪．孤立子理论及其应用，浙江科技出版社[M]，1990．
[3] 谷超豪，胡和生，周子翔．孤立子理论中的Darboux变换及其几何应用，上海科技出版社[M]，1999．
[4] V. B. Matveev, M. A. Salle. Darboux transformation and solitions [M], Berlin: Springer, 1991.
[5] M L Wang. Phys. Lett. A, 1995(199):169.
[6] M L Wang. Phys. Lett. A, 1996(213):279.
[7] M L Wang, Y B Zhou, Z B Li. Phys. Lett. A, 1996(216):67.
[8] 王明亮，李志斌，周宇斌．兰州大学学报(自然科学版)，1999(35)：8．
[9] 王明亮，周宇斌．兰州大学学报(自然科学版)，32(1996)1-5．
[10] E J Parkes and B R Duffy. Computer Phys Commun, 1996(98):288.
[11] B R Duffy and E J Parkes. Phys. Lett. A, 1996(214):271.
[12] E J Parkes and B R Duffy. Phys. Lett. A, 1997(229):217.
[13] Z B Li and M L Wang. J Phys A,1993(26):6027.
[14] E J Parkes. J Phys A,1994(27):L497.
[15] E G Fan, J Zhang and Y C Hon. Phys. Lett. A, 2001(291):376.
[16] V V Gudkov. J Math Phys.,1997(38):4794
[17] G X Zhang, Z B Li and Y S Duan. J science in China. A, 2001,44(3):396-401.
[18] Chaoqing Dai, Jiamin Zhu, Jiefang Zhang. Chaos Solitons & Fractals, 2006(27):881-886.
[19] Conte, R., Musette, J phys. A:Math.Gen.,1992(25):2609-2612.
[20] A.D.Polyanin, V.F.Zaltsev.Handbook of exact solutions for ordinary differential equations.CRC press[M], 1995.
[21] Milton Abramowitz, Irene A. Stegun.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.New York[M], 1972
[22] S Ghosh, A Kundu and S Nandy.J Math. Phys., 1990(40):1993.
[23] B Konopelchenko and V Dubrovsky. Phys. Lett. A, 1984(102):15.
[24] 陈登远．孤子引论．科学出版社[M]，2006 15-24．
[25] H H Chen, Y C Lee and C S Liu. Phys. Scr, 1979(20): 490.
[26] F Calogero and W Eckhaus.Inverse Problems,1987(3):229.
[27] 石玉仁，汪映海，杨红娟等．广义变系数Burgers方程的精确解．华东师范大学学报[J]．2006(5)：27-33．
[28] C Z Qu and A Q Wang.Commun Theor Phys,1996(26): 369.
[29] M L Wang. J Phys. Lett A,1996(213): 279.
[30] 纪峰波，求解非线性演化方程函数展式法的统一．兰州大学[M]，2006：16-26．
[31] 王明亮，李志斌，周宇斌．齐次平衡原则及其应用[J]，兰州大学学报(自然科学版)，35 (1999)8-16．
[32] 王明亮，周宇斌．一个非线性波动方程的精确解[J]，兰州大学学报(自然科学版)，32 (1996)1-5．
[33] 黄定江，张鸿庆．扩展的双曲函数法和Zakharov方程组的新精确孤立波解[J]，物理学报，53(2004)2434-1438．
[34] Engui Fan. Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics [J], Chaos Solitons & Fractals, 16 (2003) 819-839.
[35] 张解放，陈芳跃．截断展开方法和广义变系数KdV方程新的精确类孤子解[J]，物理学报，50(2001)1648-1650．
[36] Masaaki Ito. Symmetries and conservation laws of a coupled nonlinear wave equation [J], Phys. Lett. A, 91 (1982) 335-338.
[37] Zuntao Fu, Lin Zhang, Shida Liu, Shikno Lin. Fractional transformation and new solutions to mKdV equation [J], Phys. Lett. A, 325 (2004) 363-369.
[38] M.Remoissenet. Waves Called Solitons [M], Berlin: Springer, 1996.
[39] E. E. Tzirtzilakis, Ch. Skokos, T. C. Bountis. A numerical study of solitions of the Bonssinesq equation using spectral methods [J], Math. Phys., 12(2002)6161-6165.
[40] 吴文俊．科学通报，1986(31)：535
[41] 吴文俊．数学机械化．北京：科学出版社[M]，1999．
[42] B Buchberger.Grobner Bases: an algorithmic method in polynomial idea theory, renct trends in multidimensional sysytem theory.Ed N K Bose.Riedal Publishing Company, 1985.
[43] Q Cox,J Little and D O'shea. Ideas,varieties and algorithms: An introduction to computational algebraic geometry and commutative algebra. Springer,UT M, 1992.
[44] W Malfliet. Am J Phys,1992(60):650.
[45] E J Parkes. J Phys A,1994(27): L497.
[46] E G Fan, J Zhang and Y C Hon. Phys Lett A,2001(291):376.
[47] V V Gudkov.J Math Phys,1997(38):4794.
[48] Whitham G B. Linear and nonlinear wave. New York: A Wiley-Interscience Publication,1974.
[49] Y B Zeng. Chin Ann of Math,1991(24): 78.
[50] N Nirmala and M Vedan. J Math Phys,1984(27): 2644.
[51] E. S. Cheb-Terrab. A Computational Approach for the Exact Solving of Systems of Partial Differential Equations. Computer Physics Communications, 2001
[52] H D Wahlquist and F B Estabrook. J Math Phys,1976(17):1293.
[53] D G Zhang.Phys Lett,1996(223):436.
[54] R Hirota."Direct Methods of Finding Exact Solution of Nonlinear Evolution Equations", in "Backlund Tansformations",Ed. by R M Miura, Lecture Notes in Mathematics,Springer,Berlin, Heidelberg,New York Vol. 515,(1976).
[55] J Weiss, M Tabor and G Carnevale. J Math Phys,1983(24): 522-526.
[56] W Hereman and M Takaoka.J Phys A,1990(23):4805.
[57] J Weiss.J Math Phys,1983(24):1405-1413.
[58] J Weiss.J Math Phys,1984(25):13-24.
[59] C. S. Gardner, J.M. Greene, M. D. Kruskal and R. M. Miura.Method for solving the Korteweg-de-Vries equation. Phys. Rev. Lett.,1967(19):1095-1097.
[60] R. Hirota. Exact solution of the Korteweg-de-Vries equation for multiple collisions of silitons.Phys. Rev.Lett.,1971 (27):1192-1194.
[61] G. W. Bluman and J. D. Cole. The general similarity solution of the heat equation. J Math. Mech.,1969(18): 1025-1042.
[62] P. A. Clarkson and M. D. Kruskal. New similarity solutions of the Boussinesq equation. J Math. Phys.,1989(30):2201-2213.
[63] 范恩贵．可积系统与计算机代数．北京：科学出版社[M]，1999．
[64] E G Fan and H Q Zhang. Appl Math Mech.,1998(19): 645.
[65] E G Fan. Commun Theor Phys., 2001(35): 523.
[66] 曹策问．科学通报，1989(34)：723
[67] 乔志军．应用数学，1995(18)：287．
[68] Z J Qiao. J Phys. A,1993(26): 4407.