微分方程精确解及李对称符号计算研究
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摘要
以物理学中的问题为背景的非线性微分方程的研究是当代非线性科学的一个重要方面。创造和发展非线性微分方程新的求解方法是非线性物理最前沿的研究课题之一。目前，已经存在许多的获得非线性微分方程的精确解的方法。本文对一些求解方法进行了研究，特别是Lie对称方法，分析并改进了前人的理论和算法，并在计算机符号系统Maple上给出了相应的实现。这些理论、算法、实现对非线性微分方程的精确解构造是十分有益的。
我们对已有的求解方法，如双曲正切法、Jacobi椭圆函数展开法、假设法等进行了改进和推广，并利用改进后的方法结合我国著名数学家吴文俊的数学机械化思想，针对一些微分方程获得了一些新的精确解。这些解的发现将有助于弄清物质在非线性作用下的运动规律，对相应物理现象的科学解释起到重要作用。
但是上述方法比较分散、不系统。众所周知，Lie群方法将求解特定类型的微分方程的分散的积分方法统一到共同的概念之下。实际上，Lie无穷小变换方法为寻找常微分方程的闭合形式的解提供了广泛的应用技巧。应用到偏微分方程，Lie方法能够导出对称。找到偏微分方程的对称，可以由此获得其精确解。
目前，对称的概念在数学和物理的研究和发展中扮演着关键的角色。但具体到应用时，Lie群的方法涉及到大量的繁冗计算，因此，设计相关的计算机符号软件包非常有必要。
我们讨论了经典Lie对称和非经典Lie对称计算中的有关理论和算法，分别给出了产生经典Lie对称和非经典Lie对称决定方程组的软件包GDS和NGDS，发现了现行Maple系统上软件包liesymm的一些漏洞。
由于决定方程组是超定的、线性的或非线性的偏微分方程组，完全求解它们非常困难。通过引入对合除法的概念，将它们完备化为内嵌所有可积性条件的一种特殊形式-对合形式，这样有助于求解决定方程组。
对经典对称情形，我们分析和重新描述了计算线性偏微分方程组的最小对合基算法和Janet对合基算法，并给出了各自实现的软件包MiniIB和Janet。将软件包GDS和Janet相结合研究了广义Burgers方程的势对称，得到了其无穷参数的势对称，并利用此无穷参数势对称获得了广义Burgers方程一个新的精确解。
对于非经典对称情形，我们描述和改进了完备化非线性代数偏微分方程组到被动的对合形式的对合特征集算法。这个算法包含了已有的乘子变量法，例如基于Janet除法的Ritt算法和基于Thomas除法的Wu微分特征列算法。最近一些新的对合除法以及算法的相继提出，可明显减少Wu-Ritt特征列算法的计算步骤。基于对合特征集算法，我们给出了具体的实现软件包ICS。通过大量的计算试验，我们分析了此算法对不同等对合除法以及项序的依赖关系，获得了一些试验性的结论，这些结论对今后代数偏微分方程组的对合特征集的计算具有一定的指导和借鉴意义。
The nonlinear differential equation (NDE) based on physics is one of the important aspects in the contemporary study of nonlinear science. Exploring and developing new method to solve the NDE is one of the forefront topics in the studies of nonlinear physics. Now there are many methods for finding the exact solutions of NDE. In this paper, not only some methods are studied, especially Lie symmetry method, but also the related theory and algorithms are improved. With computer symbolic system Maple, some packages to implement the important algorithms are presented. The theory, algorithms and implementations are very instructive for constructing the exact solution of NDE.
We present and extend some methods such as the tanh-function method, the Jacobi elliptic function method, the ansatz method and so on. Based on the above methods and the theory of mathematics mechanization proposed by famous mathematician Wu Wentsiin, some type of particular exact solutions to nonlinear equations are obtained, which are helpful in clarifying the movement of matter under the nonlinear interactivities and play an important role in scientifically explaining of the corresponding physical phenomenon.
However these methods have some common shortcomings: dispersive, unsystematically. It is well known that Lie groups techniques brought diverse integration methods for solving special classes of differential equations under a common concept. Indeed, Lie's infinitesimal transformation method provides a widely applicable technique to find closed form solutions of ordinary differential equations (ODEs). Applied to partial differential equations (PDEs), Lie's method can lead to symmetries. Exploiting the symmetries of PDEs, new solutions can be derived.
Nowadays, the concept of symmetry plays a key role in the study and development of mathematics and physics. But the application of Lie group method to concrete physical systems involves tedious computations. So it is necessary to design some symbolic packages for it.
We discuss methods and algorithms used in the computation of Lie nonclassical symmetries as well as classical symmetries, and provide the symbolic packages GDS and NGDS, which are used to generate the determining system of differential equation's classical symmetry and nonclassical symmetry respectively. For GDS, some bugs are found in the package liesymm on Maple.
Because the determining systems are a linear or nonlinear overdetermined PDEs, it is very hard to solve them completely. After introducing the concept of involutive division, we can complete them to involutive forms which include all integrable conditions and maybe solve them more easily.
For the case of classical symmetry, the minimal involutive base algorithm and Janet base algorithm are analyzed and described, and the associated implementations which are named MinilB and Janet are presented and some examples are tested. With GDS and Janet, we study the
potential symmetry of a generalized Burgers equation, for which an infinite parameter potential symmetry and a new exact solution are obtained.
For the case of nonclassical symmetry, an involutive characteristic set algorithm(ICS) which reduces a nonlinear algebraic partial differential equation system to passive involution is described and improved. This algorithm converts all the existed methods using the multiplier variable approach such as Ritt's algorithm based on Janet division and Wu's algorithm based on Thomas division. Recently some new involutive divisions and algorithms are proposed, which can significantly reduce the computational steps in Wu-Ritt's characteristic set method for nonlinear algebraic PDEs. Based on the algorithm ICS, a package ICS is designed in Maple for computing involutive characteristic set. By testing many examples, we analysis the dependency of the algorithm ICS for the various orderings and involutive divisions. Some experimental results are obtained which may show a hint for computing involutive characteristic set of arbitrary algebraic partial differential equation systems there
我们对已有的求解方法，如双曲正切法、Jacobi椭圆函数展开法、假设法等进行了改进和推广，并利用改进后的方法结合我国著名数学家吴文俊的数学机械化思想，针对一些微分方程获得了一些新的精确解。这些解的发现将有助于弄清物质在非线性作用下的运动规律，对相应物理现象的科学解释起到重要作用。
但是上述方法比较分散、不系统。众所周知，Lie群方法将求解特定类型的微分方程的分散的积分方法统一到共同的概念之下。实际上，Lie无穷小变换方法为寻找常微分方程的闭合形式的解提供了广泛的应用技巧。应用到偏微分方程，Lie方法能够导出对称。找到偏微分方程的对称，可以由此获得其精确解。
目前，对称的概念在数学和物理的研究和发展中扮演着关键的角色。但具体到应用时，Lie群的方法涉及到大量的繁冗计算，因此，设计相关的计算机符号软件包非常有必要。
我们讨论了经典Lie对称和非经典Lie对称计算中的有关理论和算法，分别给出了产生经典Lie对称和非经典Lie对称决定方程组的软件包GDS和NGDS，发现了现行Maple系统上软件包liesymm的一些漏洞。
由于决定方程组是超定的、线性的或非线性的偏微分方程组，完全求解它们非常困难。通过引入对合除法的概念，将它们完备化为内嵌所有可积性条件的一种特殊形式-对合形式，这样有助于求解决定方程组。
对经典对称情形，我们分析和重新描述了计算线性偏微分方程组的最小对合基算法和Janet对合基算法，并给出了各自实现的软件包MiniIB和Janet。将软件包GDS和Janet相结合研究了广义Burgers方程的势对称，得到了其无穷参数的势对称，并利用此无穷参数势对称获得了广义Burgers方程一个新的精确解。
对于非经典对称情形，我们描述和改进了完备化非线性代数偏微分方程组到被动的对合形式的对合特征集算法。这个算法包含了已有的乘子变量法，例如基于Janet除法的Ritt算法和基于Thomas除法的Wu微分特征列算法。最近一些新的对合除法以及算法的相继提出，可明显减少Wu-Ritt特征列算法的计算步骤。基于对合特征集算法，我们给出了具体的实现软件包ICS。通过大量的计算试验，我们分析了此算法对不同等对合除法以及项序的依赖关系，获得了一些试验性的结论，这些结论对今后代数偏微分方程组的对合特征集的计算具有一定的指导和借鉴意义。
The nonlinear differential equation (NDE) based on physics is one of the important aspects in the contemporary study of nonlinear science. Exploring and developing new method to solve the NDE is one of the forefront topics in the studies of nonlinear physics. Now there are many methods for finding the exact solutions of NDE. In this paper, not only some methods are studied, especially Lie symmetry method, but also the related theory and algorithms are improved. With computer symbolic system Maple, some packages to implement the important algorithms are presented. The theory, algorithms and implementations are very instructive for constructing the exact solution of NDE.
We present and extend some methods such as the tanh-function method, the Jacobi elliptic function method, the ansatz method and so on. Based on the above methods and the theory of mathematics mechanization proposed by famous mathematician Wu Wentsiin, some type of particular exact solutions to nonlinear equations are obtained, which are helpful in clarifying the movement of matter under the nonlinear interactivities and play an important role in scientifically explaining of the corresponding physical phenomenon.
However these methods have some common shortcomings: dispersive, unsystematically. It is well known that Lie groups techniques brought diverse integration methods for solving special classes of differential equations under a common concept. Indeed, Lie's infinitesimal transformation method provides a widely applicable technique to find closed form solutions of ordinary differential equations (ODEs). Applied to partial differential equations (PDEs), Lie's method can lead to symmetries. Exploiting the symmetries of PDEs, new solutions can be derived.
Nowadays, the concept of symmetry plays a key role in the study and development of mathematics and physics. But the application of Lie group method to concrete physical systems involves tedious computations. So it is necessary to design some symbolic packages for it.
We discuss methods and algorithms used in the computation of Lie nonclassical symmetries as well as classical symmetries, and provide the symbolic packages GDS and NGDS, which are used to generate the determining system of differential equation's classical symmetry and nonclassical symmetry respectively. For GDS, some bugs are found in the package liesymm on Maple.
Because the determining systems are a linear or nonlinear overdetermined PDEs, it is very hard to solve them completely. After introducing the concept of involutive division, we can complete them to involutive forms which include all integrable conditions and maybe solve them more easily.
For the case of classical symmetry, the minimal involutive base algorithm and Janet base algorithm are analyzed and described, and the associated implementations which are named MinilB and Janet are presented and some examples are tested. With GDS and Janet, we study the
potential symmetry of a generalized Burgers equation, for which an infinite parameter potential symmetry and a new exact solution are obtained.
For the case of nonclassical symmetry, an involutive characteristic set algorithm(ICS) which reduces a nonlinear algebraic partial differential equation system to passive involution is described and improved. This algorithm converts all the existed methods using the multiplier variable approach such as Ritt's algorithm based on Janet division and Wu's algorithm based on Thomas division. Recently some new involutive divisions and algorithms are proposed, which can significantly reduce the computational steps in Wu-Ritt's characteristic set method for nonlinear algebraic PDEs. Based on the algorithm ICS, a package ICS is designed in Maple for computing involutive characteristic set. By testing many examples, we analysis the dependency of the algorithm ICS for the various orderings and involutive divisions. Some experimental results are obtained which may show a hint for computing involutive characteristic set of arbitrary algebraic partial differential equation systems there
引文
[1] Garder C S, Greene J M and Kruskal M D and Mirura R M. Method for solving the Korteweg-de Vries equation. Phys.Rev.Lett. 19(1967):1095.
[2] Garder C S, Greene J M and Kruskal M D and Mirura B. M. Korteweg-de Vries equation and generalizations.Ⅵ.Methods for exact solution. Comm.Pure Appl.Math. 27(1974):97.
[3] Lax P D. Integrals of nonlinear equations of evolution and solitary waves. Comm.Pure and Appl.Math. 21(1968):467.
[4] Ablowitz M J and Clarkson P A. Nonlinear Evolution and Inverse Scattering. New York: Cambridge University Press, (1991):47.
[5] Hirota R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys.Rev.Lett. 27(1971):1192.
[6] Miura M R. Bcklund Transformation. Berlin: Springer-Verlag, 1978.
[7] Li Z B and Wang M L. Travelling wave solutions to the two-dimensional KdV-Burgers equation. J.Phys.A : Math. Gen. 26(1993):6027.
[8] Wang M L, Zhou Y B and Li Z B. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys.Lett.A. 216(1996):67.
[9] Wang M L. Solitary wave solutions for variant Boussinesq equations. Phys. Lett.A. 199(1995):169.
[10] Wang M L. Exact solutions for a compound KdV-Burgers equation. Phys. Lett.A. 213(1996):279.
[11] 范恩贵，张鸿庆．非线性孤子方程的齐次平衡法．物理学报．47(1998)：353．
[12] Hereman W, Banerjee P P, Korpel A, Assanto G, van Immerzeele A and Meerpoel A. Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method. J.Phys.A :Math. Gen. 19(1986):607.
[13] 徐桂琼，李志斌．构造非线性发展方程孤波解的混合指数方法．物理学报．51(2002)：946．
[14] 徐桂琼，李志斌．扩展的混合指数方法及其应用．物理学报．51(2002)：1424．
[15] Lan H B and Wang K L. Exact solutions for some coupled nonlinear equations Ⅱ. J.Phys.A:Math.Gen. 23(1990): 4097.
[16] Malfliet W. Solitary wave solutions of nonlinear wave equations. Am.J.Phys. 60(1992):650.
[17] Duffy B R and Parkes E J. Travelling solitary wave solutions to a seventh-order generalized KdV equation. Phys.Lett.A. 214(1996):271.
[18] 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．17(1997)：81．
[19] Parkes E J and Duffy B R. Travelling solitary wave solutions to a compound KdV-Burgers equation. Phys.Lett.A. 229(1997):217.
[20] Fan E G and Zhang H Q. A note on the homogeneous balance method. Phys.Lett.A. 246(1998):403.
[21] Fan E G. Extended tanh-function method and its applications to nonlinear equations. Phys.Lett.A. 277(2000):212.
[22] 张桂戌．李志斌．段一士．非线性波方程的精确孤立波解．中国科学(A辑)．30(2000)：1103．
[23] 李志斌．姚若侠．非线性耦合微分方程组的精确解析解．物理学报．50(2001)：2062．
[24] 柳银萍．基于吴方法的孤波自动求解软件包及其应用．硕士学位论文，华东师范大学，2001．
[25] Li Z B and Liu Y P. Rath:A Maple package for finding travelling solitary wave solutions to nonlinear evolution equations. Comput.Phys.Commun. 148(2002):256.
[26] Liu Y P and Li Z B. An Automated Algebraic Method for finding a series of exact travelling wave
solutions of nonlinear evolution equations. Chin.Phys.Lett. 20(2003):317.
[27] 刘式适，傅遵涛．刘式达．赵强．Jacobi椭圆函数展开法及其在求解非线性波动方程中的应用．物理学报．50(2001)：2068．
[28] 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，51(2002)：10．
[29] Shikuo Liu, Zuntao Fu, Shida Liu and Qiang Zhao. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett.A. 289(2001):69.
[30] Zuntao Fu, Shikuo Liu, Shida Liu and Qiang Zhao. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys. Lett.A. 290(2001):72.
[31] Liu Y P and Li Z B. An Automated Jacobi Elliptic Function Method for finding Periodic Wave solutions to Nonlinear Evolution Equations. Chin.Phys.Lett. 19(2002):1228.
[32] 李世奇．杜惠琴．Maple计算机代数系统应用及程序设计．重庆大学出版社，1999．
[33] 张韵华．符号计算系统Mathematica教程．科学出版社，2001．
[34] 吴文俊．几何定理机器证明的基本原理．科学出版社，北京，1984．
[35] Olver P J. Applications of Lie Groups to Differential Equations. New York: Springer-Verlag. 1993.
[36] Bluman G W and Kumei S. Symmetries and Differential Equations. New York: Springer-Verlag. 1989.
[37] Schwarz F. Symmetries of differential equations: From Sophns Lie to Computer Algebra. SIAM Review 30(1988):450.
[38] Schwarz F. Programming with abstract data types:The symmetry packages SODE and SPDE. Lecture Notes in Comput.Sci. 296(1988):167.
[39] Schwarz F and Augnstin S. An algorithm for determining the size of symmetry groups. Computing. 49(1992):95.
[40] Champagne B, Hereman W and Winternitz P. The computer calculation of Lie point symmetries of large systems of differential equations. Comp.Phys.Comm. 66(1991):319.
[41] Hereman W. SYMMGRP.MAX:A symbolic program for symmetry analysis of partial differential equations, In Exploiting Symmetry in Applied and Numerical Analysis, Proc. AMS-SIAM Summer Seminar, Fort Collins, Colorado, 1992, Lectures in Applied Mathematics 29, (Edited by E.Allgower et al.), pp.241-257, American Mathematical Society, Providence, Rhode Island(1993).
[42] Hereman W. Review of symbolic software for the computation of Lie symmetries of differential equations. Euro.Math.Bulletin. 1(1994):45.
[43] Hereman W. Symbolic Software for Lie Symmetry Analysis. In CRC Handbook of Lie group Analysis of Differential Equations. Volume 3: New Trends in Theoretical Developments and Computational Methods. (Edited by Ibragimov N H). Chapter 13, CRC Press, Boca Raton, Florida, 1995.
[44] Baumann G. Lie Symmetries of Deferential Equations: A Mathematics program to determine Lie symmetries, MathSource 0202-622. Illinois: Wolfram Research Inc, Champaign ,1992.
[45] Carminati J, J S Devitt and G J Fee.Isogroups of differential equations using algebraic computing. J.Sym. Comp. 14(1992):103.
[46] Reid G J. Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating Its Taylor sires solution. Euro.J.Appl.Math. 2(1991):293.
[47] Reid G J, Wittkopf A D and Boulton A. Reduction of systems of nonlinear partial differential equations to simplified involutive forms. Euro.J.Appl.Math. 7(1996):635.
[48] Reid G J and Wittkopf. The Long Guid to The Standard Form Package, Department of Mathe
matics, University of British Columbia, Vancouver, Canada, 1993.
[49] Mansfield E L. Differential GrSbner Bases. PH.D Thesis, Department of Computer Science, University of Sydney, Sydney, Australia, 1991. ~
[50] Mansfield E L. DIFFGROB2: A Symbolic Algebra Package for Analyzing Systems of PDE using Maple. User's Manual for Release 2. Department of Mathematics, University of Exeter, Exeter, UK, 1993.
[51] Wu Wentsün. On the Foundation of Algebraic differential Geometry. Mathematics Mechanization Research Preprints. 3(1989):1.
[52] 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，1999．
[53] Wu Wentsün. On the decision problem and mechanization of theorem proving in elementary geometry. Scientia Sinica. 21(1978):157.
[54] Wu Wentsün. Basic Principles of Mechanical Theorem Proving in Geometry, J.Sys.Sci.Math.Sci. 4(1984):207.
[55] Wu Wentsün. On Zeros of Algebraic Equations-An Applications of Ritt's Principle. Kexue Tongbao 31(1986):1.
[56] Wu Wentsün. A Zero Structure Theorem for Polynomial Equations Solving. Mathematics Mechanization Research Preprints. 1(1987):1.
[57] 石赫．数学机械化引论．湖南教育出版社，1998．
[58] Fan E G. An algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinear evolution equations. J. Phys. A: Math. Gen. 36(2003):7009-7026.
[59] Fan E G. A new algebraic method for finding the line soliton solutions and doubly periodic wave solution to a two-dimensional perturbed KdV equation. Chaos.Solitons and Fractals. 15(2003):567.
[60] Yan Z Y. New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations. Phys.Lett.A. 292(2001):100.
[61] 姚若侠．非线性耦合标量方程组孤立波解的符号计算．硕士学位论文，华东师范大学，2001．
[62] Lie S. Zur allgemeinen Theorie der partielle Differentialgleichungen beliebeger Ordnung. Leipzig Berich. 1(1895):53.
[63] Birkhoff G. Hydrodynamics-A Study in Logic, Fact and Similitude. 1st ed., Princeton University Press, Princeton, 1950.
[64] Ovsiannikov L V. Groups and group-invariant solutions of differential equations.Dokl.Akad.Nauk USSR. 118(1958):439(in Russian).
[65] Ovsiannikov L V. Group Properties of Differential Equations. Novosibirsk, Moscow,1962(In Rnssian).
[66] Ovsiannikov L V. Group Analysis of Differential Equations. Academic Press, New York, 1982.
[67] Bluman G W and Cole J D. The general similarity solution of the heat equation. J.Math.Mech. 18(1969): 1025.
[68] Bluman G W, Kumei S and Reid G J. New Classes of symmetries for partial differential equations. J.Math.Phys. 29(1988): 806. Erratum, J.Math.Phys. 29:2320.
[69] Bluman G W and Reid G J. New symmetries for ordinary differential equations. IMA J.Appl.Math. 40(1988):87.
[70] Krasil'shchik I S and Vinogradov A M. Nonlocal Trends in the Geometry of Differential Equations, Symmetries, Conservation Laws, and Bklund Transformations. Aeta Appl.Math. 15(1989):161.
[71] Akhatov I S, Gazizov R K and Ibragimov N H. J.Soviet.Math. 55(1991):1401.
[72] Noether E. Invariant Variations Problems. Nachr, Kunig, Gesell, wis-sen, Gottingen, Math.Phys. K1(1918):235.
[73] Ol(?)er P J. Evolution equations possessing infinitely many symmetries. J.Math.Phys. 18(1977):1212.
[74] Fokas A S and Liu Q M. Nonlinear interaction of travelling waves of nonintegrable equations. Phys.Rev.Lett. 72(1994):3293.
[75] Liu Q M and Fokas A S. Exact interaction of solitary waves for certain nonintegrable equations. J.Math.Phys. 37(1996):324.
[76] Qu C Z. Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method. IMA J.Appl.math. 62(1999):283.
[77] Qu C Z, Zhang S L and Liu R C. Separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source. Physica D 144(2000):97.
[78] Chou K S and Qu C Z. Generalized conditional symmetries of nonlinear differential-difference equations. Phys.Lett.A. 280(2001):303.
[79] Gandarias M L. Nonclassical Potential Symmetries of the Burgers Equation. Symmetry in nonlinear Mathematical Physics. 1(1997):130.
[80] Gandarias M L. New symmetries for a model of fast diffusion. Phys.Lett.A. 286(2001):153.
[81] Kowalevskaya S. Zur Theorie der partiellen Differential gleichungen. J.Reine Angew.Math. 80(1875):1.
[82] Bourlet M C. Sur les équations aux dérivées partielles simultanées. Ann.Sci.cole Norm.Sup. 8(3)(1891), Suppl.S.3-S.63.
[83] Lewy H. An example of a smooth linear partial differential equation without solution. Ann.Math. 66(1957):155.
[84] cartan E. Les Systèmes Diffèrentielles Extèrieurs et leurs Applications Gèomètriques. Hermann, Paris, 1945.
[85] Khler. Einführung in die Theorie der Systeme von Differentialgleichungen. Teubner, Leipzig, 1934.
[86] Bryant R L, Chern S S, Gardner R B, Goldschmidt H L and Griffiths P A. Exterior Differential Systems. Mathematical Science Research Institute Publications 18. Springer-Verlag, New York, 1991.
[87] Hartley D H. Involutive analysis for non-linear exterior differential systems. Math.Comp.Model., 25(1997):51.
[88] Hartley D H and Tucker R W. A constructive implemeatation of the Cartan-Khler theory of exterior differential systems. J.Symb. Comp. 12(1991):655.
[89] Seiler W M, Indices and Solvability for General Systems of Differential Equations. Computer Algebra in Scientific Computing CASC' 99, V.G. Ghanza, E.W. Mayr, E.V. Vorozhtsov (eds.), Springer, Berlin 1999, pp. 365.
[90] Pommaret J.F. System of Partial Differential Equations and Lie Pseudogroups. Gordon & Breach, New York, 1978.
[91] Pommaret J.F. Partial Differential Equations and Group Theory. New Perspectives for Applications, Kluwer, Dordrecht, 1994.
[92] 刘木兰．Grbner基理论及其应用，科学出版社，2000．
[93] Rosenfeld A. Specializations in differential algebra. Trans.Amer.Math.Soc. 90(1958):394.
[94] Kolchin E. Differential Algebra and Algebraic Groups. New York: Academic Press, 1973.
[95] Boulier F, Lazard D,Pllivier F, et al. Representation for the radical of a finitely generated differ-
ential ideal. In:Levelt A H M, ed. Proc of ISAC'95. Canada: ACM Press, 1999, 158.
[96] Li Z and Wang D M. Coherent,regular and simple system in zero decompositions of partial differential system. J.Sys.math.Sci. 12(5)(1999):43.
[97] Rust C J, Reid G J and Wittkopf A D. Existence and uniqueness theorems for formal power series solutions of analytic differential systems. In:Proc of ISSAC'99. New York: ACM Press,1999, 105.
[98] Hubert E. Factorization-free decomposition algorithms in differential algebra. J.Symb.Comp. 29(2000):641.
[99] Riquier C. Les Systèmes d'Equations aux Dérivées Partielles. Paris:Gauthier-Villars,1910.
[100] Janet M. Sur les Systèmes d'Equationa aux Dérivées Partielles. J.Math.Pure.Appl. 3(1920):65.
[101] Thomas J. Differential systems. New York: American Mathematical Society, 1937.
[102] Ritt J F. Differential Algebra. New York: American Mathematical Society, 1950.
[103] Gerdt V P. Blinkov Y.A. Involutive Bases of Polynomial Ideals. Math.Comp.Simul. 45(1998): 519.
[104] Gerdt V P. Blinkov Y.A. Minimal Involutive Bases. Math. Comp. Simul. 45(1998): 543.
[105] Gerdt V P. Completion of Linear Differential Systems to Involution. In: Ganzha V.G. et al., Editors, CASC'99: Computer Algebra in scientific Computing. Proceedings of the Second Workshop on Computer Algebra in Scientific Computing, Springer-Verlag, Berlin Heidelberg, 1999, pp.115-137.
[106] Chen Y F, Gao X.S. Involutive directions and new involutive divisions. Comp.Math.Appl. 41(2001): 945.
[107] 陈玉福，高小山．微分多项式系统的对合特征集．中国科学(A辑)．33(2)(2003)：97．
[108] Yoshinaga T, Wakamiya M, Kakutani T. Recurrence and chaotic behavior resulting from nonlinear interaction between long and short waves Phys.Fluids. 3(1991):83.
[109] Funakoshi M, Oikawa M. The Resonant Interaction between a Long Internal Gravity Wave and a Surface Gravity Wave Packet. J.Phys.Soc.Jpn. 52(1983):1982.
[110] Yoshinaga T, Kakutani T. Solitary and E-Shock Waves in a Resonant System between Long and Short Waves. J.Phys.Soc.Jpn., 63(1994): 445.
[111] Porubov A V. Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid layer~(*1). Phys. Lett.A. 221(1996): 391.
[112] Porubov A V and Velarde M G. Exact periodic solutions of the complex GinzburgLandau equation. J. Math. Phys. 40(1999): 884.
[113] Feng X. Exploratory approach to explicit solution of nonlinear equati.ons. Int. J. Theor. Phys. 39(2000): 207.
[114] Seyler C E and Fenstermacher D L. A symmetric regularized-long-wave equation. Phys. Fluids. 27(1984): 4.
[115] Korpel A and Banerjee P. A Heuristic Guide to Nonlinear Dispersive Wave Equations and Soliton Type Solutions. Proc. IEEE. 72(1984): 1109.
[116] Han P and Lou S Y. Abundant Symmetry Structure of Kaup-Kupershmidt Hierarchies. Chinese Phys.Lett.10(1993)：257；韩平．楼森岳．Kaup．Kupershmidt方程的对称代数．物理学报．43(1994)：1041．
[117] Wu T Y, Zhang J E, Ccok L P, Roythurd V and Tulin M (Eels.) Math is for Solving Problems, SIAM 1996: 233-241.
[118] Li Y S, Ma W X and Zhang J E. Darboux transformations of classical Boussinesq system and its new solutions Phys.Lett.A. 27(2000):60.
[119]周振江．李志斌．Broer-Kaup系统的达布变换和新的精确解．物理学报．52(2)(2003)：262．
[120]文双春，徐文成，郭旗，刘颂豪．变系数非线性Schrdinger方程孤子的演化．中国科学(A辑)．27(1997)：949．
[121]M. Remoissenet. Waves called Solitons: Concepts and Experiments. Berlin Heidelberg:Springer-Verlag,, 1994, p166.
[122]阮航宇，陈一新．寻找变系数非线性方程精确解的新方法．物理学报．49(2000)：177．
[123]Reda G. Abd EI-Rahman. Exact solution of the particle-cluster dynamic equation. Appl.Math. Comput. 132(1)(2002):105.
[124]P.R. Gordoa. Symmetries, exact solutions, and nonlinear superposition formulas for two integrable partial differential equations. J. Math. Phys. 41(7)(2000):4713.
[125]Conte R. The Palnlevè property. New York: Springer-Verlag, 1999.
[126]Becker T. Grbner Bases: A Computional Approach to Commutative Algebra. New York: Springer-Verlag, 1993.
[127]朝鲁．吴-微分特征列法及其在微分方程对称和力学中的应用．博士学位论文，大连理工大学，1997．
[128]陈玉福．微分特征列法理论研究及应用．博士学位论文，大连理工大学．1999．
[129]潘素起．线性齐次偏微分方程组的吴微分特征列与Janet基，硕士学位论文．兰州大学，2000．
[130]徐桂琼．微分方程对称群的符号计算及其在Mathmatica系统上的实现．硕士学位论文，兰州大学，1998．
[131]Cartan E. Sur certaines exoressions différentielles a le problèmede Pfaff. Annales Ecole Normale, 3-e serie,16(1899):239.
[132]Arais E A, Shapeev V P and Yanenko N N. Realizatíon of Cartan's Method of Exterior Differential Forms on an Electronic Computer. Sov.Math.Dokl. 15(1)(1974):203.
[133]Zharkov A Yu and Blinkov Yu A. Involutive Approach to Investigating Polynomial Systems. Math. Comp.Simul. 42(1996):323.
[134]Gerdt V P. On the relation between Pommaret and Janet Bases. http://arxiv.org/PS_cache/math/pdf/0004/0004100.pdf, 2000-04-15.
[135]Apel J. Theory of Involutive Divisions and an Application to Hilbert Function. J.Symb.Comp. 25(1998):683
[136]Mall D. On the relation Between Grbner and Pommaret Bases. AAECC 9(1998): 117.
[137]Zharkov A Yu. Solving Zero-Dimensional Iavolutive Systems. In: Algorithms in Algebraic Geometry and Applications, Gonzales-Vega L, Recio T(eds). Progress in Mathematics, Vol.143, Birkhuser, Basel, 1996, pp.389-399.
[138]Zharkov A Yu. Involutive Polynomial Bases: General cases. Preprint JINR E5-94-244, Dubna: JINR, 1994.
[139]Seiler W M. A Combinatorial Approach to Involution and δ-Regularity. Preprint, University of Mannheim, 2002.
[140]Topunov V.L. Reducing Systems of Linear Differential Equations to a Passive Form. Acta Appl.Math. 16(1989): 191.
[141]Pucci E and Saccomandi G. Potential symmetries and solutions by reduction of partial differential equations. J.Phys.A: Math. Gen. 26(1993): 681.
[142]Khater A H, Moussa M H M and Abdul-Aziz S F. Potential symmetries and invariant solutions for the generalized one-dimensional Fokker-Planck equation. Physica A. 304(2002):395.
[143]Khater A H, Moussa M H M and Abdul-Aziz S F. Potential symmetries and invariant solutions
for the inhomogeneous nonlinear diffusion equation. Physica A. 312(2002): 99.
[144] Slebodzinski W. Exterior Forms and Their Application. Warsaw: PWN,1970.
[145] Logan J D. Invariant variational principles. New York: Academic press, 1977.
[146] Bluman G W and Doran-Wu P. The Use of Factors to Discover Potential Systems or Linearizations. Acta Appl.Math. 41(1995): 21.
[147] Anco S C and Bluman G W. Direct Construction of Conservation Laws from Field Equations Phys.Rev.Lett. 78(1997): 2869.
[148] Burgers J M. The nonlinear diffusion equation asymptotic solutions and statistical problems. Reidel: Dordrecht, 1974.
[149] Kumei S and Bluman G W. When nonlinear differential equations are equivalent to linear differential equations. SIAM J.Appl.Math. 42(1982): 1157.
[150] Fushchich W I and Zhdanov R Z. Symmetry and exact solutions of nonlinear spinor equations Physics Reports. 172(1989):123.
[151] Gaeta G. On the conditional symmetries of Levi and Winternitz. J.Phys.A:Math.Gen. 23(1990):3643.
[152] Levi D and Winternitz P. Nonclassical symmetry reduction: example of the Boussinesq equation. J.Phys.A:Math.Gen. 22(1989):2915.
[153] Olver P J and Rosenan P. Group-invariant solutions of differential equations. SIAM J.Appl.Math. 47(1987):263.
[154] Pucci E. Similarity reductions of partial differential equations. J.Phys.A: Math. Gen. 25(1992):2631.
[155] Olver P J and Rosenau P. The construction of special solutions to partial differential equations. Phys.Lett.A. 114(1986):107.
[156] Mansfield E L. The Nonclassical group analysis of the heat equation. J.Math.Anal.Appl. 231(1999):526.
[157] Clarkson P A and Kruskal M D. New similarity reductions of the Boussinesq equation. J.Math.Phys. 30(1989):2201.
[158] Nucci M C and Clarkson P A. The nonclassical method is more general than the direct method for symmetry reductions: an examples of the Fitzhugh-Nagumo equations. Phys.Lett. A. 164(1992):49.
[159] Olver P J. Direct reduction and differential constraints. Proc.R.Soc.Lond. A 444(1994):509.
[160] Clarkson P A. Nonclassical symmetry reductions of the Boussinesq equation. Chaos, Solitons Fractals 5(1995):2261.
[161] Clarkson P A and Priestley T J. Symmetries of a Class of Nonlinear Fourth Order Partial Differential Equations. Journal of Nonlinear Mathematical Physics. 6(1)(1999):66.
[162] Clarkson P A and Mansfield E L. Symmetry Reductions and Exact solutions of a class of Nonlinear Heat Equations. Physica D. 70(1993): 250.
[163] Clarkson P A and Mansfield E L. Algorithms For the Nonclassical Method of Symmetry Reductions. SIAM J.Appl.Math.54(6)(1994): 1693.
[164] Delft P, Tomei C, and Trubowitz E. Inverse scattering and the Boussinesq equation. Commun. Pure Appl.Math. 35(1982):567.
[165] Bonssinesq J. Théorie de l'intumescence appelée onde solitaire ou de translation se propageant dans un canal rectangulaire, Comptes Rendus. 72(1871):755.
[166] Ursell F. The long-wave paradox in the theory of gravity waves. Proc. Camb.Phil.Soc. 49(1953):685.
[167] Toda M. Studies of a nonlinear lattice. Phys.Rep. 8(1975):1.
[168] Scott A C. The application of Bcklund transforms to physical problems, Bcklund Transformations, Miura R M,ed. (Lect. Notes in Math., 515, Springer,Berlin,1975), 80-105.
[169] Nishitani T and Tajiri N. On similarity solutions of the Boussinesq equation. Phys.Lett.A 89(1982):379.
[170] Rosenau P and Schwarzmeier J L. On similarity solutions of Boussinesq type equations. Phys.Lett.A . 115(1986):75.
[171] Whitham G B. Linear and Nonlinear Waves. New York: Wiley,1974.
[172] Weiss J, Tabor M and Carnevale G. The Painlevé property for partial differential equations. J.Math.Phys. 24(1983):522.
[173] Boulier F, Lazard D, Ollivier F and Petitot M. Representation for the radical of a finitely generated differential ideal. Research Report LIFL IT-306 (1997).
[174] Hubert E. Essential Components of an Algebraic Differential Equation. J. Symb.Comp. 28(1999):45.
[2] Garder C S, Greene J M and Kruskal M D and Mirura B. M. Korteweg-de Vries equation and generalizations.Ⅵ.Methods for exact solution. Comm.Pure Appl.Math. 27(1974):97.
[3] Lax P D. Integrals of nonlinear equations of evolution and solitary waves. Comm.Pure and Appl.Math. 21(1968):467.
[4] Ablowitz M J and Clarkson P A. Nonlinear Evolution and Inverse Scattering. New York: Cambridge University Press, (1991):47.
[5] Hirota R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys.Rev.Lett. 27(1971):1192.
[6] Miura M R. Bcklund Transformation. Berlin: Springer-Verlag, 1978.
[7] Li Z B and Wang M L. Travelling wave solutions to the two-dimensional KdV-Burgers equation. J.Phys.A : Math. Gen. 26(1993):6027.
[8] Wang M L, Zhou Y B and Li Z B. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys.Lett.A. 216(1996):67.
[9] Wang M L. Solitary wave solutions for variant Boussinesq equations. Phys. Lett.A. 199(1995):169.
[10] Wang M L. Exact solutions for a compound KdV-Burgers equation. Phys. Lett.A. 213(1996):279.
[11] 范恩贵，张鸿庆．非线性孤子方程的齐次平衡法．物理学报．47(1998)：353．
[12] Hereman W, Banerjee P P, Korpel A, Assanto G, van Immerzeele A and Meerpoel A. Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method. J.Phys.A :Math. Gen. 19(1986):607.
[13] 徐桂琼，李志斌．构造非线性发展方程孤波解的混合指数方法．物理学报．51(2002)：946．
[14] 徐桂琼，李志斌．扩展的混合指数方法及其应用．物理学报．51(2002)：1424．
[15] Lan H B and Wang K L. Exact solutions for some coupled nonlinear equations Ⅱ. J.Phys.A:Math.Gen. 23(1990): 4097.
[16] Malfliet W. Solitary wave solutions of nonlinear wave equations. Am.J.Phys. 60(1992):650.
[17] Duffy B R and Parkes E J. Travelling solitary wave solutions to a seventh-order generalized KdV equation. Phys.Lett.A. 214(1996):271.
[18] 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．17(1997)：81．
[19] Parkes E J and Duffy B R. Travelling solitary wave solutions to a compound KdV-Burgers equation. Phys.Lett.A. 229(1997):217.
[20] Fan E G and Zhang H Q. A note on the homogeneous balance method. Phys.Lett.A. 246(1998):403.
[21] Fan E G. Extended tanh-function method and its applications to nonlinear equations. Phys.Lett.A. 277(2000):212.
[22] 张桂戌．李志斌．段一士．非线性波方程的精确孤立波解．中国科学(A辑)．30(2000)：1103．
[23] 李志斌．姚若侠．非线性耦合微分方程组的精确解析解．物理学报．50(2001)：2062．
[24] 柳银萍．基于吴方法的孤波自动求解软件包及其应用．硕士学位论文，华东师范大学，2001．
[25] Li Z B and Liu Y P. Rath:A Maple package for finding travelling solitary wave solutions to nonlinear evolution equations. Comput.Phys.Commun. 148(2002):256.
[26] Liu Y P and Li Z B. An Automated Algebraic Method for finding a series of exact travelling wave
solutions of nonlinear evolution equations. Chin.Phys.Lett. 20(2003):317.
[27] 刘式适，傅遵涛．刘式达．赵强．Jacobi椭圆函数展开法及其在求解非线性波动方程中的应用．物理学报．50(2001)：2068．
[28] 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，51(2002)：10．
[29] Shikuo Liu, Zuntao Fu, Shida Liu and Qiang Zhao. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett.A. 289(2001):69.
[30] Zuntao Fu, Shikuo Liu, Shida Liu and Qiang Zhao. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys. Lett.A. 290(2001):72.
[31] Liu Y P and Li Z B. An Automated Jacobi Elliptic Function Method for finding Periodic Wave solutions to Nonlinear Evolution Equations. Chin.Phys.Lett. 19(2002):1228.
[32] 李世奇．杜惠琴．Maple计算机代数系统应用及程序设计．重庆大学出版社，1999．
[33] 张韵华．符号计算系统Mathematica教程．科学出版社，2001．
[34] 吴文俊．几何定理机器证明的基本原理．科学出版社，北京，1984．
[35] Olver P J. Applications of Lie Groups to Differential Equations. New York: Springer-Verlag. 1993.
[36] Bluman G W and Kumei S. Symmetries and Differential Equations. New York: Springer-Verlag. 1989.
[37] Schwarz F. Symmetries of differential equations: From Sophns Lie to Computer Algebra. SIAM Review 30(1988):450.
[38] Schwarz F. Programming with abstract data types:The symmetry packages SODE and SPDE. Lecture Notes in Comput.Sci. 296(1988):167.
[39] Schwarz F and Augnstin S. An algorithm for determining the size of symmetry groups. Computing. 49(1992):95.
[40] Champagne B, Hereman W and Winternitz P. The computer calculation of Lie point symmetries of large systems of differential equations. Comp.Phys.Comm. 66(1991):319.
[41] Hereman W. SYMMGRP.MAX:A symbolic program for symmetry analysis of partial differential equations, In Exploiting Symmetry in Applied and Numerical Analysis, Proc. AMS-SIAM Summer Seminar, Fort Collins, Colorado, 1992, Lectures in Applied Mathematics 29, (Edited by E.Allgower et al.), pp.241-257, American Mathematical Society, Providence, Rhode Island(1993).
[42] Hereman W. Review of symbolic software for the computation of Lie symmetries of differential equations. Euro.Math.Bulletin. 1(1994):45.
[43] Hereman W. Symbolic Software for Lie Symmetry Analysis. In CRC Handbook of Lie group Analysis of Differential Equations. Volume 3: New Trends in Theoretical Developments and Computational Methods. (Edited by Ibragimov N H). Chapter 13, CRC Press, Boca Raton, Florida, 1995.
[44] Baumann G. Lie Symmetries of Deferential Equations: A Mathematics program to determine Lie symmetries, MathSource 0202-622. Illinois: Wolfram Research Inc, Champaign ,1992.
[45] Carminati J, J S Devitt and G J Fee.Isogroups of differential equations using algebraic computing. J.Sym. Comp. 14(1992):103.
[46] Reid G J. Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating Its Taylor sires solution. Euro.J.Appl.Math. 2(1991):293.
[47] Reid G J, Wittkopf A D and Boulton A. Reduction of systems of nonlinear partial differential equations to simplified involutive forms. Euro.J.Appl.Math. 7(1996):635.
[48] Reid G J and Wittkopf. The Long Guid to The Standard Form Package, Department of Mathe
matics, University of British Columbia, Vancouver, Canada, 1993.
[49] Mansfield E L. Differential GrSbner Bases. PH.D Thesis, Department of Computer Science, University of Sydney, Sydney, Australia, 1991. ~
[50] Mansfield E L. DIFFGROB2: A Symbolic Algebra Package for Analyzing Systems of PDE using Maple. User's Manual for Release 2. Department of Mathematics, University of Exeter, Exeter, UK, 1993.
[51] Wu Wentsün. On the Foundation of Algebraic differential Geometry. Mathematics Mechanization Research Preprints. 3(1989):1.
[52] 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，1999．
[53] Wu Wentsün. On the decision problem and mechanization of theorem proving in elementary geometry. Scientia Sinica. 21(1978):157.
[54] Wu Wentsün. Basic Principles of Mechanical Theorem Proving in Geometry, J.Sys.Sci.Math.Sci. 4(1984):207.
[55] Wu Wentsün. On Zeros of Algebraic Equations-An Applications of Ritt's Principle. Kexue Tongbao 31(1986):1.
[56] Wu Wentsün. A Zero Structure Theorem for Polynomial Equations Solving. Mathematics Mechanization Research Preprints. 1(1987):1.
[57] 石赫．数学机械化引论．湖南教育出版社，1998．
[58] Fan E G. An algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinear evolution equations. J. Phys. A: Math. Gen. 36(2003):7009-7026.
[59] Fan E G. A new algebraic method for finding the line soliton solutions and doubly periodic wave solution to a two-dimensional perturbed KdV equation. Chaos.Solitons and Fractals. 15(2003):567.
[60] Yan Z Y. New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations. Phys.Lett.A. 292(2001):100.
[61] 姚若侠．非线性耦合标量方程组孤立波解的符号计算．硕士学位论文，华东师范大学，2001．
[62] Lie S. Zur allgemeinen Theorie der partielle Differentialgleichungen beliebeger Ordnung. Leipzig Berich. 1(1895):53.
[63] Birkhoff G. Hydrodynamics-A Study in Logic, Fact and Similitude. 1st ed., Princeton University Press, Princeton, 1950.
[64] Ovsiannikov L V. Groups and group-invariant solutions of differential equations.Dokl.Akad.Nauk USSR. 118(1958):439(in Russian).
[65] Ovsiannikov L V. Group Properties of Differential Equations. Novosibirsk, Moscow,1962(In Rnssian).
[66] Ovsiannikov L V. Group Analysis of Differential Equations. Academic Press, New York, 1982.
[67] Bluman G W and Cole J D. The general similarity solution of the heat equation. J.Math.Mech. 18(1969): 1025.
[68] Bluman G W, Kumei S and Reid G J. New Classes of symmetries for partial differential equations. J.Math.Phys. 29(1988): 806. Erratum, J.Math.Phys. 29:2320.
[69] Bluman G W and Reid G J. New symmetries for ordinary differential equations. IMA J.Appl.Math. 40(1988):87.
[70] Krasil'shchik I S and Vinogradov A M. Nonlocal Trends in the Geometry of Differential Equations, Symmetries, Conservation Laws, and Bklund Transformations. Aeta Appl.Math. 15(1989):161.
[71] Akhatov I S, Gazizov R K and Ibragimov N H. J.Soviet.Math. 55(1991):1401.
[72] Noether E. Invariant Variations Problems. Nachr, Kunig, Gesell, wis-sen, Gottingen, Math.Phys. K1(1918):235.
[73] Ol(?)er P J. Evolution equations possessing infinitely many symmetries. J.Math.Phys. 18(1977):1212.
[74] Fokas A S and Liu Q M. Nonlinear interaction of travelling waves of nonintegrable equations. Phys.Rev.Lett. 72(1994):3293.
[75] Liu Q M and Fokas A S. Exact interaction of solitary waves for certain nonintegrable equations. J.Math.Phys. 37(1996):324.
[76] Qu C Z. Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method. IMA J.Appl.math. 62(1999):283.
[77] Qu C Z, Zhang S L and Liu R C. Separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source. Physica D 144(2000):97.
[78] Chou K S and Qu C Z. Generalized conditional symmetries of nonlinear differential-difference equations. Phys.Lett.A. 280(2001):303.
[79] Gandarias M L. Nonclassical Potential Symmetries of the Burgers Equation. Symmetry in nonlinear Mathematical Physics. 1(1997):130.
[80] Gandarias M L. New symmetries for a model of fast diffusion. Phys.Lett.A. 286(2001):153.
[81] Kowalevskaya S. Zur Theorie der partiellen Differential gleichungen. J.Reine Angew.Math. 80(1875):1.
[82] Bourlet M C. Sur les équations aux dérivées partielles simultanées. Ann.Sci.cole Norm.Sup. 8(3)(1891), Suppl.S.3-S.63.
[83] Lewy H. An example of a smooth linear partial differential equation without solution. Ann.Math. 66(1957):155.
[84] cartan E. Les Systèmes Diffèrentielles Extèrieurs et leurs Applications Gèomètriques. Hermann, Paris, 1945.
[85] Khler. Einführung in die Theorie der Systeme von Differentialgleichungen. Teubner, Leipzig, 1934.
[86] Bryant R L, Chern S S, Gardner R B, Goldschmidt H L and Griffiths P A. Exterior Differential Systems. Mathematical Science Research Institute Publications 18. Springer-Verlag, New York, 1991.
[87] Hartley D H. Involutive analysis for non-linear exterior differential systems. Math.Comp.Model., 25(1997):51.
[88] Hartley D H and Tucker R W. A constructive implemeatation of the Cartan-Khler theory of exterior differential systems. J.Symb. Comp. 12(1991):655.
[89] Seiler W M, Indices and Solvability for General Systems of Differential Equations. Computer Algebra in Scientific Computing CASC' 99, V.G. Ghanza, E.W. Mayr, E.V. Vorozhtsov (eds.), Springer, Berlin 1999, pp. 365.
[90] Pommaret J.F. System of Partial Differential Equations and Lie Pseudogroups. Gordon & Breach, New York, 1978.
[91] Pommaret J.F. Partial Differential Equations and Group Theory. New Perspectives for Applications, Kluwer, Dordrecht, 1994.
[92] 刘木兰．Grbner基理论及其应用，科学出版社，2000．
[93] Rosenfeld A. Specializations in differential algebra. Trans.Amer.Math.Soc. 90(1958):394.
[94] Kolchin E. Differential Algebra and Algebraic Groups. New York: Academic Press, 1973.
[95] Boulier F, Lazard D,Pllivier F, et al. Representation for the radical of a finitely generated differ-
ential ideal. In:Levelt A H M, ed. Proc of ISAC'95. Canada: ACM Press, 1999, 158.
[96] Li Z and Wang D M. Coherent,regular and simple system in zero decompositions of partial differential system. J.Sys.math.Sci. 12(5)(1999):43.
[97] Rust C J, Reid G J and Wittkopf A D. Existence and uniqueness theorems for formal power series solutions of analytic differential systems. In:Proc of ISSAC'99. New York: ACM Press,1999, 105.
[98] Hubert E. Factorization-free decomposition algorithms in differential algebra. J.Symb.Comp. 29(2000):641.
[99] Riquier C. Les Systèmes d'Equations aux Dérivées Partielles. Paris:Gauthier-Villars,1910.
[100] Janet M. Sur les Systèmes d'Equationa aux Dérivées Partielles. J.Math.Pure.Appl. 3(1920):65.
[101] Thomas J. Differential systems. New York: American Mathematical Society, 1937.
[102] Ritt J F. Differential Algebra. New York: American Mathematical Society, 1950.
[103] Gerdt V P. Blinkov Y.A. Involutive Bases of Polynomial Ideals. Math.Comp.Simul. 45(1998): 519.
[104] Gerdt V P. Blinkov Y.A. Minimal Involutive Bases. Math. Comp. Simul. 45(1998): 543.
[105] Gerdt V P. Completion of Linear Differential Systems to Involution. In: Ganzha V.G. et al., Editors, CASC'99: Computer Algebra in scientific Computing. Proceedings of the Second Workshop on Computer Algebra in Scientific Computing, Springer-Verlag, Berlin Heidelberg, 1999, pp.115-137.
[106] Chen Y F, Gao X.S. Involutive directions and new involutive divisions. Comp.Math.Appl. 41(2001): 945.
[107] 陈玉福，高小山．微分多项式系统的对合特征集．中国科学(A辑)．33(2)(2003)：97．
[108] Yoshinaga T, Wakamiya M, Kakutani T. Recurrence and chaotic behavior resulting from nonlinear interaction between long and short waves Phys.Fluids. 3(1991):83.
[109] Funakoshi M, Oikawa M. The Resonant Interaction between a Long Internal Gravity Wave and a Surface Gravity Wave Packet. J.Phys.Soc.Jpn. 52(1983):1982.
[110] Yoshinaga T, Kakutani T. Solitary and E-Shock Waves in a Resonant System between Long and Short Waves. J.Phys.Soc.Jpn., 63(1994): 445.
[111] Porubov A V. Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid layer~(*1). Phys. Lett.A. 221(1996): 391.
[112] Porubov A V and Velarde M G. Exact periodic solutions of the complex GinzburgLandau equation. J. Math. Phys. 40(1999): 884.
[113] Feng X. Exploratory approach to explicit solution of nonlinear equati.ons. Int. J. Theor. Phys. 39(2000): 207.
[114] Seyler C E and Fenstermacher D L. A symmetric regularized-long-wave equation. Phys. Fluids. 27(1984): 4.
[115] Korpel A and Banerjee P. A Heuristic Guide to Nonlinear Dispersive Wave Equations and Soliton Type Solutions. Proc. IEEE. 72(1984): 1109.
[116] Han P and Lou S Y. Abundant Symmetry Structure of Kaup-Kupershmidt Hierarchies. Chinese Phys.Lett.10(1993)：257；韩平．楼森岳．Kaup．Kupershmidt方程的对称代数．物理学报．43(1994)：1041．
[117] Wu T Y, Zhang J E, Ccok L P, Roythurd V and Tulin M (Eels.) Math is for Solving Problems, SIAM 1996: 233-241.
[118] Li Y S, Ma W X and Zhang J E. Darboux transformations of classical Boussinesq system and its new solutions Phys.Lett.A. 27(2000):60.
[119]周振江．李志斌．Broer-Kaup系统的达布变换和新的精确解．物理学报．52(2)(2003)：262．
[120]文双春，徐文成，郭旗，刘颂豪．变系数非线性Schrdinger方程孤子的演化．中国科学(A辑)．27(1997)：949．
[121]M. Remoissenet. Waves called Solitons: Concepts and Experiments. Berlin Heidelberg:Springer-Verlag,, 1994, p166.
[122]阮航宇，陈一新．寻找变系数非线性方程精确解的新方法．物理学报．49(2000)：177．
[123]Reda G. Abd EI-Rahman. Exact solution of the particle-cluster dynamic equation. Appl.Math. Comput. 132(1)(2002):105.
[124]P.R. Gordoa. Symmetries, exact solutions, and nonlinear superposition formulas for two integrable partial differential equations. J. Math. Phys. 41(7)(2000):4713.
[125]Conte R. The Palnlevè property. New York: Springer-Verlag, 1999.
[126]Becker T. Grbner Bases: A Computional Approach to Commutative Algebra. New York: Springer-Verlag, 1993.
[127]朝鲁．吴-微分特征列法及其在微分方程对称和力学中的应用．博士学位论文，大连理工大学，1997．
[128]陈玉福．微分特征列法理论研究及应用．博士学位论文，大连理工大学．1999．
[129]潘素起．线性齐次偏微分方程组的吴微分特征列与Janet基，硕士学位论文．兰州大学，2000．
[130]徐桂琼．微分方程对称群的符号计算及其在Mathmatica系统上的实现．硕士学位论文，兰州大学，1998．
[131]Cartan E. Sur certaines exoressions différentielles a le problèmede Pfaff. Annales Ecole Normale, 3-e serie,16(1899):239.
[132]Arais E A, Shapeev V P and Yanenko N N. Realizatíon of Cartan's Method of Exterior Differential Forms on an Electronic Computer. Sov.Math.Dokl. 15(1)(1974):203.
[133]Zharkov A Yu and Blinkov Yu A. Involutive Approach to Investigating Polynomial Systems. Math. Comp.Simul. 42(1996):323.
[134]Gerdt V P. On the relation between Pommaret and Janet Bases. http://arxiv.org/PS_cache/math/pdf/0004/0004100.pdf, 2000-04-15.
[135]Apel J. Theory of Involutive Divisions and an Application to Hilbert Function. J.Symb.Comp. 25(1998):683
[136]Mall D. On the relation Between Grbner and Pommaret Bases. AAECC 9(1998): 117.
[137]Zharkov A Yu. Solving Zero-Dimensional Iavolutive Systems. In: Algorithms in Algebraic Geometry and Applications, Gonzales-Vega L, Recio T(eds). Progress in Mathematics, Vol.143, Birkhuser, Basel, 1996, pp.389-399.
[138]Zharkov A Yu. Involutive Polynomial Bases: General cases. Preprint JINR E5-94-244, Dubna: JINR, 1994.
[139]Seiler W M. A Combinatorial Approach to Involution and δ-Regularity. Preprint, University of Mannheim, 2002.
[140]Topunov V.L. Reducing Systems of Linear Differential Equations to a Passive Form. Acta Appl.Math. 16(1989): 191.
[141]Pucci E and Saccomandi G. Potential symmetries and solutions by reduction of partial differential equations. J.Phys.A: Math. Gen. 26(1993): 681.
[142]Khater A H, Moussa M H M and Abdul-Aziz S F. Potential symmetries and invariant solutions for the generalized one-dimensional Fokker-Planck equation. Physica A. 304(2002):395.
[143]Khater A H, Moussa M H M and Abdul-Aziz S F. Potential symmetries and invariant solutions
for the inhomogeneous nonlinear diffusion equation. Physica A. 312(2002): 99.
[144] Slebodzinski W. Exterior Forms and Their Application. Warsaw: PWN,1970.
[145] Logan J D. Invariant variational principles. New York: Academic press, 1977.
[146] Bluman G W and Doran-Wu P. The Use of Factors to Discover Potential Systems or Linearizations. Acta Appl.Math. 41(1995): 21.
[147] Anco S C and Bluman G W. Direct Construction of Conservation Laws from Field Equations Phys.Rev.Lett. 78(1997): 2869.
[148] Burgers J M. The nonlinear diffusion equation asymptotic solutions and statistical problems. Reidel: Dordrecht, 1974.
[149] Kumei S and Bluman G W. When nonlinear differential equations are equivalent to linear differential equations. SIAM J.Appl.Math. 42(1982): 1157.
[150] Fushchich W I and Zhdanov R Z. Symmetry and exact solutions of nonlinear spinor equations Physics Reports. 172(1989):123.
[151] Gaeta G. On the conditional symmetries of Levi and Winternitz. J.Phys.A:Math.Gen. 23(1990):3643.
[152] Levi D and Winternitz P. Nonclassical symmetry reduction: example of the Boussinesq equation. J.Phys.A:Math.Gen. 22(1989):2915.
[153] Olver P J and Rosenan P. Group-invariant solutions of differential equations. SIAM J.Appl.Math. 47(1987):263.
[154] Pucci E. Similarity reductions of partial differential equations. J.Phys.A: Math. Gen. 25(1992):2631.
[155] Olver P J and Rosenau P. The construction of special solutions to partial differential equations. Phys.Lett.A. 114(1986):107.
[156] Mansfield E L. The Nonclassical group analysis of the heat equation. J.Math.Anal.Appl. 231(1999):526.
[157] Clarkson P A and Kruskal M D. New similarity reductions of the Boussinesq equation. J.Math.Phys. 30(1989):2201.
[158] Nucci M C and Clarkson P A. The nonclassical method is more general than the direct method for symmetry reductions: an examples of the Fitzhugh-Nagumo equations. Phys.Lett. A. 164(1992):49.
[159] Olver P J. Direct reduction and differential constraints. Proc.R.Soc.Lond. A 444(1994):509.
[160] Clarkson P A. Nonclassical symmetry reductions of the Boussinesq equation. Chaos, Solitons Fractals 5(1995):2261.
[161] Clarkson P A and Priestley T J. Symmetries of a Class of Nonlinear Fourth Order Partial Differential Equations. Journal of Nonlinear Mathematical Physics. 6(1)(1999):66.
[162] Clarkson P A and Mansfield E L. Symmetry Reductions and Exact solutions of a class of Nonlinear Heat Equations. Physica D. 70(1993): 250.
[163] Clarkson P A and Mansfield E L. Algorithms For the Nonclassical Method of Symmetry Reductions. SIAM J.Appl.Math.54(6)(1994): 1693.
[164] Delft P, Tomei C, and Trubowitz E. Inverse scattering and the Boussinesq equation. Commun. Pure Appl.Math. 35(1982):567.
[165] Bonssinesq J. Théorie de l'intumescence appelée onde solitaire ou de translation se propageant dans un canal rectangulaire, Comptes Rendus. 72(1871):755.
[166] Ursell F. The long-wave paradox in the theory of gravity waves. Proc. Camb.Phil.Soc. 49(1953):685.
[167] Toda M. Studies of a nonlinear lattice. Phys.Rep. 8(1975):1.
[168] Scott A C. The application of Bcklund transforms to physical problems, Bcklund Transformations, Miura R M,ed. (Lect. Notes in Math., 515, Springer,Berlin,1975), 80-105.
[169] Nishitani T and Tajiri N. On similarity solutions of the Boussinesq equation. Phys.Lett.A 89(1982):379.
[170] Rosenau P and Schwarzmeier J L. On similarity solutions of Boussinesq type equations. Phys.Lett.A . 115(1986):75.
[171] Whitham G B. Linear and Nonlinear Waves. New York: Wiley,1974.
[172] Weiss J, Tabor M and Carnevale G. The Painlevé property for partial differential equations. J.Math.Phys. 24(1983):522.
[173] Boulier F, Lazard D, Ollivier F and Petitot M. Representation for the radical of a finitely generated differential ideal. Research Report LIFL IT-306 (1997).
[174] Hubert E. Essential Components of an Algebraic Differential Equation. J. Symb.Comp. 28(1999):45.