摘要
非线性演化方程是描述物理现象的一类重要数学模型，也是非线性物理特别是孤立子理论最前沿的研究课题之一。非线性演化方程精确解和可积性的研究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和工程应用将起到重要作用。在非线性演化方程的研究中，寻找方程的行波解、构造多孤子解、Painlevé可积性质的检验等经常遇到复杂的符号计算和推理，有的是人力难以完成的，因此妨碍了这些问题的深入剖析。近年来，符号计算的蓬勃发展，极大地推动了非线性演化方程的研究。非线性演化方程的研究成果不断涌现，尤其是新的求解方法层出不穷。本文以非线性演化方程为研究对象，借助于符号计算这一有效研究工具，研究了多种直接代数方法在非线性演化方程精确求解中的应用、Painlevé分析及其应用，探讨了几种直接代数方法与Painlevé可积性质之间的内在联系。主要工作如下：
第一部分研究非线性演化方程的精确求解。分别从三个方面进行研究：
研究了构造非线性演化方程孤立波解的基础性方法——混合指数方法，改进了混合指数方法的关键步骤——行波约化后常微分方程及递推关系式的求解。将传统实指数方法推广到复指数情形，从而可以获得正则孤波解、奇异孤波解及周期解在内的诸多形式的行波解。
在Riccati方法、形变映射方法、“统一代数”方法的基础上，给出了构造非线性演化方程多种行波解的广义形变映射法。该方法的基本思想是利用“秩”对行波约化后的常微分方程进行分类，对同秩类型和异秩类型的方程借助于不同的一阶可解方程，从而将非线性演化方程行波解的计算问题转化为非线性代数方程组的求解问题。利用吴文俊消元法求解非线性代数方程组，即可得到非线性演化方程的多类行波解。以耦合mKdV方程组、耦合Drinfel'd-Sokolov-Wilson方程组、变形Boussinesq方程组等为例，系统地获得了包括指数解、多项式解、有理解、三角函数解、Jacobi椭圆函数解、Weierstrass椭圆函数解、孤立波解及广义孤立波解、组合形式三角函数及孤立波解在内的多种形式的行波解。由于该方法是构造性和算法化的，可以在计算机代数系统上完成解的自动推导。
采用分步确定拟解的原则，对齐次平衡法求非线性演化方程多孤子解的关键步骤作了进一步改进。以广义Boussinesq方程和bKK方程为应用实例，说明使用该方法可有效避免“中间表达式膨胀”的问题，除获得标准Hirota形式的孤子解外，还能得到其他形式的孤子解。本文获得了这两个非线性演化方程的具有双向传播特点的孤立波解和孤立子解。
第二部分研究非线性演化方程的Painlevé可积性的检验和Painlevé分析的若干应用。主要从两个方面进行研究：
由于非线性系统的Painlevé性质与可积性之间有着十分密切的联系，因此判定一个非线性系统是否具有Painlevé性质就具有非常重要的意义。本文分析了非线性演化方程Painlevé性质的几种检验方法，并利用Kruskal方法得到了广义Hirota-Satsuma耦合方程组具有Painlevé可积性质的必要条件。WTC方法和Kruskal方法是检验非线性演化方程Painlevé性质的两种重要方法，这两种方法各有所长，本文将二者结合起来并给出了检验Painlevé性质的WTC-Kruskal算法。使用WTC-Kruskal算法，不仅可以快速判定非线性演化方程的Painlevé可积性，也为寻找新的Painlevé可积系统提供了重要途径。
研究了基于Painlevé性质的若干截断展开方法在非线性演化方程可积性质及精确解研究中的若干应用。讨论了标准截断展开方法在构造不可积系统精确解、可积系统的自B(?)cklund
变换、多重孤波解和多孤子解中的应用.其次，给出了高阶截断展开法在构造非线性演化方程
新型精确解中的应用，并分析了高阶截断展开法在构造非线性可积系统Lax对、Darboux变
换的应用实例.最后将近年来发展起来的构造非线性演化方程行波解的几种直接代数方法统一
于Painle说分析的研究框架之中.
第三部分研究符号计算在广义形变映射法和Painlev乙可积性证明中的应用.主要包括:
广义形变映射法将行波解的求解间题转化为非线性代数方程组的计算问题，推演过程往往
涉及到非常繁琐的计算.本文在计算机符号系统Maple上开发了一个基于吴文俊消元法的行
波解自动求解软件包NETs.该软件包可以自动输出非线性演化方程及方程组的多类行波解.
NETS软件包对方程或方程组的维数没有限制，适用于多项式类型的非线性演化方程.除适
用于非线性偏微分方程外，NETS还适用于非线性常微分方程.某些特殊类型的非线性演化
方程，经过适当的函数变换转换为多项式类型的非线性常微分方程后，可以借助NETS实现
行波解的自动推导.
基于wTC一Kruskal算法，本文在计算机符号系统Maple上开发了wkPtest软件包.该
软件包可快速完成非线性偏微分方程及方程组Paiulev‘性质的自动检验.同时，当给定方程
不能通过Palnlev乙检验时，软件包将返回参数满足的约束条件.此外，软件包还能输出方程的
Painlev‘截断展开式.wkPtest软件包适用范围较为广泛，对方程或方程组的维数没有限制，
不仅适用于常系数方程或方程组，也适用于变系数方程或方程组以及一?
Nonlinear evolution equation is an important mathematical model for describing physical phenomenon and an important field in the contempary study of nonlinear physics, especially in the study of soliton theory. The research on the explicit solution and integrability are helpful in clarifying the movement of matter under the nonlinear interactiveties and play an important role in scientifically explaining of the corresponding physical phenomenon and engineering application. Many research topics, such as searching for exact explicit solutions, multi-soliton solution, the Painleve test et al., often involve a large amount of tedious algebra auxiliary reasoning or calculations which can become unmanageable in practice. In recent years, the development of symbolic computation accelerates the research of nonlinear evolution equation greatly. Many new methods for constructing exact solutions of nonlinear evolution equations are proposed. This dissertation mainly stduies some aspects of nonlinear evolution equations
with the aid symbolic computation, which include searching exact explicit solutions of nonlinear evolution equations by means of several direct algebraic methods proposed in recent years, the Painleve analysis and its application, the interraltions between the algebraic methods with the Painleve analysis. This dissertation consists of the following three parts.
Part I is devoted to study the explicit solutions for nonlinear evolution equations by some algebraic methods proposed in recent years.
Mixing exponential method proposed by Hereman for finding the solitary wave solutions to a nonlinear evolution equation is developed and perfected. Correspondingly, an extended mixing exponential method is obtained by expressing the solutions as an infite series of the real or complex exponential solutions of the underlying linear equations and improving the solving of recursion relations. The effectiveness of the extended approach is demonstrated by application to some nonlinear evolution equations with physical interest. Not only are steady solitary wave solutions recovered, but also the diverging and the periodic solutions are obtained.
Based on Riccati method, deformation mapping method and unified algebraic method, a generalized deformation mapping method is presented for finding travelling wave solutions to nonlinear evolution equations. According to "rank", the nonlinear evolution equations are classified into two kinds of equations. The essence of this method is to take full advantage of two different solvable first-order ordinary differential equations, and convert the problem of finding travelling wave solutions for nonlinear PDE to the problem of solving nonlinear algebraic equations. The nonlinear system is solved by using Wu elimination and a series of travelling wave solutions are then obtained. Several illustrative equations such as coupled mKdV equations, coupled Drinfel'd-Sokolov-Wilson equations, variant Boussinesq sytem etc. are investigated by this metho. A series of travelling wave solutions are obtained in a systematic way, which covered exponential function solutions, polynomial function solutions, rational function solu
tions, trigonometric function solutions, Jacobi elliptic function solutions, Weierstrass elliptic function solutions, solitary wave solution, combined-form solitary wave solutions and so on. The generalized deformation mapping method is completely algorithmic, so it can be implemented in computer algebra.
The homogeneous balance method for constructing solitary wave solutions and soliton solu-
tions is further developed on obtaining quasi-solution by step-by-step principle. The main advantage of the extended approach is to avoid the problem of "intermediate expression swell". The effectiveness of the method is demonstrated by application to the generalized Boussinesq equation and the bidirectional Kaup-Kupershmidt equation. The one-soliton, two-soliton and three-soliton solutions with multiple collisions are derived for these two equations with the assistance of Maple. Both
引文
[1] Ablowitz M J, Clarkson P A. Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1999.
[2] Newell A C. Solitons in Mathematics and Physics, SIAM: Philadephia, 1985.
[3] Faddeev L D, Takhtajan L A. Hamiltonian method in the theory of solitons, Springer-Verlag, Berlin, 1987.
[4] Matveev V B, Salle M A. Darboux transformation and solitons, Springer-Verlag, Berlin, 1991.
[5] Roy Chowdhury A. The Painlevé Analysis and its Applications, Chapman & Hall/CRC, Baton Rouge, Florida, 2000.
[6] Steeb W H, Euler W. Nonlinear evolution equations and Painlevé test, World Scientific, Singapore, 1988.
[7] Conte R Ed. The Painlevé Property, One Century Later, Springer Verlag, New York, 1999.
[8] 郭柏灵，庞小峰．孤立子．科学出版社，北京．1987．
[9] 王明亮．非线性发展方程与孤立子，兰州大学出版社，兰州．1990．
[10] 谷超豪等．孤立子理论及其应用，浙江科学技术出版社．杭州，1990．
[11] 谷超豪，胡和生，周子翔．孤立于理论中的Darboux变换及其几何应用，上海科技出版社，上海．1999．
[12] 李翊神．非线性科学选讲，中国科学技术大学出版社，合肥．1994．
[13] 李翊神．孤子与可积系统．上海科技出版社，上海．1999．
[14] Miura R M. The Korteweg-de Vries equations: a survey of results, SIAM rev.,(1967), 18: 412.
[15] Lax P D. Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math., (1968), 21: 467.
[16] Ablowitz M J, Kaup D J, Newell A C and Segur H. The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math., (1974), 53: 249.
[17] Ablowitz M J, Ramani A, Segur H. A connection between nonlinear evolution equations and ordinary differential equations of P-type, Ⅰ, Ⅱ, J. Math. Phys., (1980), 21: 715; 21: 1006.
[18] Kodama Y, Ablowitz M J and Satsuma J. Direct and inverse scattering problems of the nonlinear intermediate long wave equation, J. Math. Phys., (1982), 23: 564.
[19] Fokas A S. On the inverse scattering transform of multidimensional nonlinear equations related to first-order systems in the plane, J. Math. Phys.,(1984), 25: 2494.
[20] Rogers C, Shadwick W R. Backlund transformation and their application, Academic Press, New York, 1982.
[21] Wahlquist H D, Estabrook F B. Backlund transformations for solitons of the Korteweg-de Vries equation, Phys. Rev. Lett., (1973), 31: 1386.
[22] Wahlquist H D, Estabrook F B. Prolongation Structures and nonlinear evolution equations, J. Math. Phys., (1976), 17: 1293.
[23] 胡星标，李勇．DJKM方程的Bcklund变换及非线性叠加公式，数学物理学报，(1991)，11：164．
[24] Hu X B, Li Y. Superposition formulae of a fifth-order KdV equation and its modified equation, Acta Math. Appl. Sin., (1988), 4: 46.
[25] Hu X B, Zhu Z N. A Bcklund transformation and nonlinear superposition formula for BelovChaltikian lattice, J Phys A: Math. Gen., (1998), 31: 4755.
[26] Matveev V B, Sklyanin E K. On Bcklund transformations for many-body systems, J Phys A: Math. Gen.,(1998), 31: 2241.
[27] Choudhury A G, Chowhury A R. Canonical and Bcklund transformations for discrete integrable systems and classical r-matrix, Phys. Left. A,(2001), 280: 280.
[28] Zeng Y B. et al., Canonical explicit Bicklund transformations with spectrality for constrained flows of soliton hierachies, Physica A, (2002), 303: 321.
[29] Weiss J, Tabor M and Carnevale G. The Painlevsé property of partial differential equations, J. Math. Phys., (1983), 24: 522.
[30] Li Z B, Wang M L. Travelling wave solutions to the two-dimensional KdV-Burgers equation, J. Phys. A: Math.Gen., (1993), 26: 6027.
[31] Wang M L. Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A, (1995), 199: 169; Exact solutions for a compound KdV-Burgers equation, (1996), 213: 279.
[32] Wang M L, Zhou Y B, Li Z B. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A, (1996), 216: 67.
[33] Wang M L, Wang X M, Zhou Y B. An auto-Bcklund transformation and exact solutions to a generalized KdV equation with variable coefficients and their applications, Phys. Lett. A, (2002), 303: 45.
[34] 张解放．长水波近似方程的多孤子解，物理学报，(1998)，47：1190．
[35] 张解放．变更Boussinesq方程和Kupershmidt方程的多孤子解，应用数学与力学，(2000)，21：171．
[36] 闫振亚．非线性波与可积系统，博士学位论文，大连理工大学，2002．
Yan Z Y, Zhang H Q. Multiple soliton-like and periodic-like solutions to the generalization of integrable of (2+1)-dimensional dispersive long-wave equations, J. Phys. Soc. Jpn., (2002), 71: 437.
[37] 卢竟，颜家壬．非线性偏微分方程的多孤子解，物理学报．(2002)，51：1428．
[38] 徐桂琼，李志斌．两个非线性发展方程的双向孤波解和孤子解，物理学报，(2003)，52：1848．
[39] Darboux G. Compts Rendus Hebdomadaires des Seances de 1'Academie des Sciences, Paris, (1882), 94: 1456.
[40] Wadati M. et al., Simple derivation of Bcklund transformation from Riccati form of inverse method, Prog. Theor. Phys., (1975), 53: 418.
[41] Gu C H. A unified explicit form of Bcklund transformations for generalized hierarchies of KdV equations, Lett. Math. Phys., (1986), 11: 325.
Gu C H, Zhou Z X. On the Darboux matrices of Bcklund transformations for AKNS systems, Lett. Math. Phys., (1987), 13: 179.
Gu C H, Zhou Z X. On Darboux transformations for soliton equations in high dimensional spacetime, Lett. Math. Phys., (1994), 32: 1.
Gu C H. Integrable evolution systems based on generalized selfdual Yang-Mills equations and their soliton-like solutions, Lett. Math. Phys., (1995), 35: 61.
[42] Hirota R. Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., (1971), 27: 1192.
[43] Hu X B, Wang D L, Tam H W and Xue W M. Soliton solutions to the Jimbo-Miwa equations and the Fordy-Gibbons-Jimbo-Miwa equation, Phys. Lett. A,(1999), 262: 310.
[44] Hu X B, Wang D L and Qian X M. Soliton solutions and symmetries of the 2+1 dimensional Kanp-Kupershmidt equation, Phys. Lett. A, (1999), 262: 409.
[45] Chen D Y, Zhang D J, Deng S F. The novel multisoliton solutions of the mKdV sinegordon equation, J. Phys. Soc. Japan, (2002), 71: 658.
[46] Dent S F, Chen D Y. The novel multisoliton solutions of the KP equation, J. Phys. Soc. Japan, (2001), 70: 2174.
[47] 陈登远，孤立子引论，预印本，2002．
[48] Gilson C R, Hu X B, Ma W X, Tam H W. Two integrable differential-difference equations derived from the discrete BKP equation and their related equations, Physica D, (2003), 175: 177.
[49] 范恩贵．孤立子和可积系统，博士学位论文，大连理工大学，1998．
范恩贵，张鸿庆．非线性孤子方程的齐次平衡法，物理学报，(1998)，47：353．
[50] 吴斌．上海交通大学硕士论文，2003．
[51] Lou S Y. On the coherent structures of Nizhnik-Novikov-Veselov equation, Phys. Lett. A, (2000), 277: 94.
Lou S Y, Lin J and Tang X Y. Painlev é integrability and multi-dromion solutions of the 2+1 dimensional AKNS system, Eur. Phys. J. B, (2001), 22: 473.
Tang X Y, Lou S Y and Zhang Y. Localized exicitations in (2+1)-dimensional systems, Phys. Rev. E., (2002), 66: 046601.
Tang X Y, Lou S Y. Extended multilinear variable separation approach and multivalued localized excitations for some (2+1)-dimensional integrable systems, J. Math. Phys., (2003), 44: 4000.
[52] Rosales R. Exact solutions of some nonlinear evolution equations, Stud. Appl. Math., (1978), 59: 117.
[53] Korpel A. Solitary wave formation through nonlinear coupling of finite exponential waves, Phys. Left. A, (1978), 68: 179.
[54] Hereman W, Banerjee P P, Korpel A et al. Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method, J. Phys. A:Math. Gen., (1986), 19:607.
[55] Hereman W, Takaoka M. Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA, J. Phys. A: Math. Gen., (1990), 23: 4805.
[56] Panigrahy M, Dash P C. Soliton solutions of a coupled field using the mixing exponential method, Phys. Lett. A, (1999), 261: 284.
[57] Lan H B, Wang K L. Exact solution for two nonlinear equations, J.Phys.A: Math.Gen., (1990), 23: 4097.
[58] Huang G X, Lou S Y, Dai X X, Exact and explicit solitary wave solutions to a model equation for water waves, Phys. Lett. A, (1989), 139: 373.
Huang G X, Lou S Y, Dai X X and Yan J R, A model equation and its exact two-dimensional solitary wave solution for wave dynamics, Chin. Phys. Left, (1989), 6: 393.
[59] Malfiet W. Solitary wave solutions of nonlinear wave equations, Am J. Phys., (1992), 60: 650.
[60] Li Z B. Wu method and solitons, Proc. of ASCM'95 eds Shi H. and Kobayashi H.,(Tokyo:Scientists Incorporated) (1995):157.
[61] Li Z B, Shi H. Exact solutions for Belousov-Zhabotinskii reaction diffusion system, Appl.Math.JCU,(1996), 118: 1.
[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)，17：81．
[63] Fan E G, Zhang H Q. New exact solutions for a system of coupled KdV equations, Phys. Lett. A (1998), 245: 389.
[64] Fan E G. Extended tanh-function method and its applications to nonlinear equations, Phys. Lett.
A, (2000), 277: 212.
[65] Ma W X et al. Explicit and exact solutions to a Kolmogorov-Petroskii-Piskunov equation, Int. J. Nonlinear Mech., (1996), 31: 329.
[66] Gao Y T, Tian B. Generalized hyperbolic-function method with computerized symbolic computation to construct the solitonic solutions to nonlinear equations of mathematical physics, Comput. Phys. Comm., 2001, 133: 158.
[67] Gao Y T, Tian B. Some two-dimensional and non-travelling wave observable effects of the shallow water waves, Phys. Left. A, (2002), 301: 74.
[68] 姚若侠．非线性耦合标量方程组孤波解的符号计算，硕士学位论文．华东师范大学，2001．
[69] 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，(2001)，50：2062．
[70] Yao R X, Li Z B. New exact solutions for three evolution equations, Phys. Lett. A, 2002, 297: 196.
[71] 闫振亚，张鸿庆．具有三个任意函数的变系数KdV-mKdV方程的精确类孤子解．物理学报．(1999)，48：1957．
[72] Yan Z Y. New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations, Phys. Lett. A, (2001), 292: 100.
[73] Chen Y, Li B, Zhang H Q. Auto-Bcklund transformation and exact solutions for modified nonlinear dispersive mK(m,n) equations, Chaos, Solitons and Fractals, (2003), 17: 693.
[74] Lu Z S, Zhang H Q. On a further extended tanh method, Phys. Lett. A, (2003), 307: 269.
[75] 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，(2001)，50：2068．
Liu S K, Fu Z T, Liu S D, Zhao Q. Jacobi elliptic expansion function method and periodic wave solutions of nonlinear wave solutions, Phys. Lett. A, (2001), 289: 69.
Fu Z T, Liu S K, Liu S D, Zhao Q. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave solutions, Phys. Lett. A, (2001), 290: 72.
[76] Lou S Y, Ni G J. The relations among a special type of solutions in some (D+1)-dimensional nonlinear equations, J. Math. Phys., (1989), 30: 1614.
[77] Fan E G. Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method, J. Phys. A: Math. Gen., (2002), 35: 6853.
Fan E G. An algebraic method for finding a series of exact solutions to integrable and non-integrable nonlinear evolution equations, J. Phys. A: Math. Gen., (2003), 36: 7009.
Fan E G. A new algebraic method for finding the line soliton solutions and doubly periodic wave solutions to a two-dimensional perturbed KdV equalton, Chaos, Solitons and Fractals, (2003), 15: 567.
Fan E G, Dai H H. A direct approach with computerized symbolic computation for finding a series of travelling waves to nonlinear equations, Comput. Phys. Commun., (2003), 153: 17.
[78] Lie S. Arch for Math., (1881), 6: 328; Gesamm elte Abhandlungen, Vol. Ⅲ, Teubner B G, Leipzig, (1922),492.
[79] Ovsiannikov L V. Group properties of differential equations, Novosibirsl, 1962.
[80] Ovsiannikov L V. Group analysis of differential equations, Academic Press, New York, 1982.
[81] Ibragimov N H. Transformation groups applied to mathematical physics, Reidel Boston, 1985.
[82] Bluman G W, Cole J D. Similarity method for differential equation, Springer-Verlag, New York, 1974.
[83] Olver P J. Evolution equations possessing infinitely many symmetries, J. Math. Phys., (1977), 18:
1212.
[84] Olver P J. Application of Lie groups to differential equations, GTM, No.107, Springer-Verlag, New York, 1986.
[85] Chen H H et al. Advance in nonlinear wave, ed L. Debnath, 1982, 233.
[86] Li Y S, Zhu G C. New set of symmetries of the integrable equations, Lie algebra and non-isospectral evolution equations. Ⅱ. AKNS system, J. Phys.A: Math.Gen., (1986), 19: 3713.
[87] 田畴．Burgers方程新的强对称、对称和李代数，中国科学，A辑．(1987)，1009．
[88] Clarkson P A, Kruskal M D. New similarity reductions of the Boussinesq equation, J. Math. Phys., (1989), 30: 2201.
[89] Levi D, Winternitz P. Nonclassical symmetry reduction:example of the Boussinesq equation, J. Phys.A: Math. Gen., (1989), 22: 2915.
[90] Lou S Y. A note on the similarity reductions of the Boussinesq equation, Phys. Lett. A, (1990), 151: 133.
[91] Lou S Y. Similarity reductions of the KP equation by a direct method, J. Phys. A: Math.Gen., (1990), 23: L694.
Lou S Y. Generalized Boussinesq equation and KdV equation-Painlevé properties, Bcklund transformation and Lax pairs, Sci. China A, (1991), 34: 1098.
[92] Lou S Y, Ruan H Y. Nonclassical analysis and Painlevé property for the Kupershmidt equations, J. Phys. A:Math.Gen., (1993), 26: 4679.
[93] Nucci M C, Clarkson P A. The nonclassical method is more general than the direct method for symmetry reductions, An example pf the FitzHugh-Nagumo eqution, Phys. Lett. A, (1992), 164.
[94] Lou S Y. Symmetries of the Kadomtsev-Petviashvili equation, J. Phys. A:Math. Gen., (1993), 26: 4387; Generalized symmetries and W_∞ algebras in three dimensiònal Toda field theory, Phys. Rev. Left., (1993), 71: 4099.
[95] Lou S Y, Qian X M. Generalized symmetries and algebras of the two-dimensional differential-difference Toda equation, J. Phys. A: Math. Gen., (1994), 27: L641;
[96] Lou S Y, Tang X Y, Lin J. Similarity and conditional similarity reductions of a (2+1)-dimensional KdV equation via a direct method, J. Math. Phys., (2000), 41: 8286.
[97] 李翊神．程艺．新KdV方程族的对称和守恒量，中国科学．(1988)，31A：1．
[98] Bluman G W, Kumei S. Symmetries and differential equations, AMS, No.81, Springer-Verlag, New York, 1989.
[99] 程艺，汪启存，田畴，李翊神．Burgers方程族的对称、Bcklund变换和Painlevé性质之间的联系，应用数学学报．(1991)，14：180．
[100] 曾云波．数学学报，(1992)，35：454；递推算子与Painlevé性质．数学年刊．(1991)，124：78．
[101] Russell J S. Report on waves, Fourteen meeting of the British association for the advancement of science, John Murray, London, 1844, 311.
[102] Whiham G B. Nonlinear dispersive waves, Proc. Roy. Soc. London A, (1965), 283: 238.
[103] Ablowitz M J, Segur h. Solitons and the inverse scattering transformation, SIAM: Philadephia, 1981.
[104] Kruskal M D, Zabusky N J. Princeton Plasma Physics Lab. Ann. Rep., MATT-Q-21, Princeton, 1963, 301.
[105] Miura R M. Korteweg-de Vries equation and generalizations I. A remarkable explicit nonlinear transformation, J. Math. Phys., (1968), 9: 1202.
[106] Kruskal M D. Korteweg-de Vries equation and generalizations V. Uniqueness and nonexistence of polynomial conservation laws, J.Math. Phys., (1970),11: 952.
[107] 屠规彰，秦孟兆．非线性演化方程的不变群与守恒律，中国科学，A辑．(1980)，5：421．
[108] 屠规彰，秦孟兆．非线性演化方程的对称和守恒律之间的关系，科学通报，(1979)，29：913．
[109] Ablowitz M J, Ramani A, Segur H. Nonlinear evolution equations and ordinary differential equations of Painlevé type, Left. Nuovo Cimento., (1978), 23: 333.
[110] Jimbo M, Kruskal M D, Miwa T. Painleve test for the self-dual Yang-Mills equation, Phys. Lett. A, (1982), 92: 59.
[111] Fordy A P, Pickering A. Analysing negative resonances in the Painlevé test, Phys. Lett. A, (1991), 160: 347.
[112] Newell A C, Ratiu T, Tabor M and Zeng Y B. Proc. Int. Conf. Nonlinear Dynamical Systems, Montreal, 1986.
[113] Weiss J. The Painleve property for partial differential equations. Ⅱ. Backlund transformation, Lax pairs, and the Schwarzian derivative, J. Math. Phys., (1983), 24: 1405.
[114] Strampp W, Renner F. Painlevé series, recursion operator and Backlünd transformations, Topic in Soliton Theory and Exact Solvable Nonlinear Equations, Eds. M.J. Ablowitz, B. Fuchssteiner and M. Kruskal, World Scientific, Singerpore, (1987), 284.
[115] Weiss J. On classes of integrable systems and the Painlevé property, J. Math. Phys., (1984), 25: 13; The Sine-Gordon equations: Complete and partial integrability, 25: 2226; Modified equations, rational solutions and the Painlevé property for the KP and Hirota-Satsuma equations, (1985), 26: 258.
[116] Chen Z X, Guo B Y. Analytic solutions of carrier flow equations via the Painleve analysis approach, J. Phys. A:Math. Gen., (1989), 22:5187.
[117] Cariello F, Tabor M. Nonintegrable evolution equations, Physica D, (1989), 39: 77.
[118] Estévez P G, Gordon P R. Nonclassical symmetries and the singular manifold method: theory and six examples, Stud. Appl. Math., (1995), 21: 73.
Estévez P G, Conde E, Gordon P R. Unified approach to Miura, Bcklund and Darboux transformations for nonlinear partial differential equations, J. Nonlinear Math. Phys., (1998), 5: 82.
Estévez P G. A nonisospectral problem in (2+1) dimensions derived from KP, Inverse Problems, (2001), 17: 1043.
[119] Lou S Y. Conformal invariance and integrable models, J. Phys. A: Math. Gen.,(1997), 20: 4803; KdV extensions with Palnlevé property, J. Math. Math., (1998), 39: 2112.
[120] Chen L L, Lou S Y. Higher dimensional integrable models with Palnlevé property obtained from 1+1-dimensional Schwarz KdV equation, Z. Naturforsch, (1998), 53a: 689.
[121] 楼森岳．高维可积模型探索，中国科学．A辑，(1997)，27：941．
[122] 楼森岳．共形不变的Painlevé分析法和高维可积模型，中国科学，A辑，(1999)，29：177．
[123] 王东明等．符号计算选讲，科学出版社，北京：2003．
[124] 吴文俊．几何定理机械证明的基本原理，北京，科学出版社，1984．
[125] 关蔼雯．吴文俊消元法讲义，北京理工大学，1994．
[126] Ablowitz M J, Kaup D J, Newell A C and Segur H. Method for solving the sine-Gordon equation, Phys. Rev. Lett., (1973), 30: 1262; 31: 125.
[127] 李志斌，张善卿．吴方法在偏微分方程中应用的一个实例．兰州大学学报，(1996)，32：22．
[128] Parkes E J, Duffy B R. An automated tanh-function method for finding solitary wave solutions to
nonlinear evolution equation, Comput. Phys. Commun., (1996), 98: 288.
[129] 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．
Li Z B, Liu Y P. RATH: A Maple package for finding traveling solitary wave solutions to nonlinear evolution equations. Cornput. Phys. Comm., (2002), 148: 244.
Liu Y P, Li Z B. A Maple package for finding exact solitary wave solutions of coupled nonlinear evolution equations. Cornput. Phys. Comm., (2002), 155: 65.
[130] Liu Y P, Li Z B, An automated algebraic method for finding a series of exact traveling wave solutions of nonlinear evolution equation. Chin. Phys. Left., (2003), 20: 317.
[131] Hereman W, Zhuang W. Symbolic Computation of solitons with Macsyma, in "Computational and applied mathematics Ⅱ:Differential equation", W.F. Ames and P. J. van der Houwen, eds. North Holland, Amsterdam, The Netherlands, 1992.
[132] Geng X G, Tam H W. Darboux transformation and soliton solutions for generalized nonlinear Schrrdinger equation, J. Phys. Soc. Jpn., (1998), 68: 1508.
[133] Fan E G. A unified and explicit construction of N-soliton solutions for the nonlinear Schrdinger equation, Commun. Theor. Phys., (2001), 36: 401.
[134] 周振江．1+1维非线性发展方程的N重Darboux变换及其多孤子解的符号计算，硕士学位论文，华东师范大学，2002．
[135] 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，(2003)，52：262．
[136] Hereman W. Review of symbolic software for the computation of Lie symmetries of differential equations, Euromath Bulletin, (1994), 1: 45.
[137] Schwarz F. Automatically determing symmetries of ordinary differential equations, in:Proc. Eufocal'83, Londen, England, 1983, Ed.:J.A. van Hulzen, Lecture Notes in computer science 162, Springer Verlag, New York, 1982, 45.
[138] Schwarz F. The package SPDE for determing symmetries of partial differential equations. User's Mannual. Distributed with Reduce 3.3 Rand Corporation, Santa Monica, California, 1987.
[139] Nucci M C. Interactive Reduce programs for calculating classical, non-classical, and Lie-Bcklund symmetries of differential equations. Preprint GT Math. 062090-051, School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, 1990.
[140] Champagne B, Hereman W and Winternitz P. The computer calculation of Lie point symmetries of large systems.of differential equations, Comput. Phys. Commun., (1991), 66: 319.
[141] 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, Eds.:E.Allgower, K. Georg and R. Miranda, Lectures in applied mathematics AMS, Providence, Rhode Island, 1993.
[142] Baumann G. Lie symmetries of differential equations:, A Mathematica program to dertermine Lie symmetries. Wolfram Research Inc., Champaign, Illinois, MathSource 0202-622, 1992.
[143] Rocha Filho T W, Figueiredo A, Brenig L. [QPSI] A Maple package for the determination of quasi-polynomial symmetries and invariants, Comput. Phys. Commun., (1999), 117: 263.
[144] 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．
[145] Ito M. Symmetries and conservation laws of a coupled nonlinear wave equation, Phys. Lett. A, (1982), 91: 335.
[146] Ito M. SYMCD-A REDUCE package for finding symmetries and conserved densities of systems of nonlinear evolution equations, Comput. Phys. Commun.,(1994), 79: 547.
[147] Gerdt V P, Zharkov A Y. Computer generation of necessary integrability conditions for polynomialnonlinear evolution systems. In. S. Watenabe and M. Nagata(eds) Proc.ISSAC'90, Tokyo, Japan, Academic Press, 1990, 250.
[148] Sanders J A, Roelofs M. An algorithmic approach to conservation laws using the 3-dimensional Heisenberg algebra, Tech. Rep. 2, RIACA, Amesterdam, 1994.
[149] Sanders J A, Wang J P. On the (non)existence of conservation laws for evolution equations, Tech. Rep. WS-445, Vrije Universiteit, Amesterdam, 1995.
[150] Gktas , Hereman W. Symbolic computation of conserved densities for systems of nonlinear evolution equations, J. Symbolic Computation, (1997), 24: 591.
[151] Gktas , Hereman W, Colagrosso M, Miller A. Computation of conserved densities for systems of nonlinear differential-difference equations, Phys. Lett. A, (1997), 236: 30.
[152] Yao R X, Li Z B. On x/t-dependant conservation laws of the generalized nth-order KdV equation, Chinese Journal of Physics, (2004), (in Press).
[153] Conte R. Singularities of differential equations and integrability, in:D.Benest, C. Froeschlé(Eds.), Introduction to methods of complex analysis and geometry for classical mechanics and nonlinear waves, Editions Frontiéres, Gif-sur-Yvette, 1994, 49.
[154] Conte R, Fordy A P, Pickering A. A perturbative approach to nonlinear differential equations, Physica D, (1993), 69: 33.
[155] Scheen C. Implementation of the Painlevé test for ordinary differential equations, Theor. Comp. Sci., (1997), 187: 87.
[156] Hajee G. The investigation of movable critical points of ordinary differential equations and systems of ordinary differential equations using formula manipulation, Memorandem Nr.386 Department of Applied Mathematics Twentc University of Technology 7500 AE Enschede, 1982.
[157] Rand D W, Winternitz P. ODEPAINLEV -A MACSYMA package for painlevé analysis of ordinary differential equations, Comput. Phys. Comm., (1986), 42: 359.
[158] Hlavat L. Test of resonances in the Painlevé analysis, Comput. Phys. Comm., (1986), 42: 427.
[159] Renner F. A constructive REDUCE package based upon the Painlevé analysis of nonlinear evolution equations in Hamiltonian and/or normal form, Comput. Phys. Commun., (1992), 70: 409.
[160] Hereman W, Angenent S. The Painlevé test for nonlinear ordinary and partial differential equations, Macsyma Newsletter, (1989), 6: 11.
[161] Hereman W. Algorithmic integrability tests of nonlinear differential and lattice equations, Comput. Phys. Commun., (1998), 115: 428.
[162] 朱思铭，施齐焉，现代数学和力学(MMM-Ⅶ)，上海大学出版社，1997，482．
[163] Conte R. Painlevé analysis of nonlinear PDE and related topics: a computer algebra program, in: E. Tournier(Ed.), Computer algebra and differential equations, Academic Press, New York, 1989, 219.
[164] Pickering A. Testing nonlinear evolution equations for complete integrability, Ph.D. Thesis, University of Leeds, 1992.
[165] Hearn A C. Reduce: User's Manual vet.3.5, The RAND Corporation, Santa Monica, RAND publication CP78, 1993.
[166] 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，(2002)，51：950．
[167] 徐桂琼，李志斌．扩展的混合指数方法及其应用．物理学报，(2002)，51：1424．
[168] 郑云，张鸿庆．一个非线性方程的显式行波解，物理学报．(2000)，49：389，
[169] Porubov A V. Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid layer, Phys. Left. A, (1996), 221: 391.
[170] Porubov A V, Parker D F. Some general periodic solutions to coupled nonlinear Schroedinger equations, Wave Motion, (1999), 29: 97.
[171] 李画眉．(3+1)．维Nizhnik-Novikov-Vesslov方程的孤子解与周期解，物理学报， (2002)，51：467．
[172] 张金良，王跃明．王明亮，方宗德．两类非线性方程的精确解，物理学报，(2003)，52：1574．
[173] Feng X. Exploratory approach to explicit solution of nonlinear evolution equation, Int. J. Theor. Phys., (2000), 39: 207.
[174] Xu G Q, Li Z B. Travelling wave solutions to a special type of nonlinear evolution equations, Commun. Theor. Phys., (2003), 39: 39.
[175] Xu G Q, Li Z B, Liu Y P. Exact solutions to a large class of nonlinear evolution equation, Chinese Journal of Physics(in Taiwan), (2003), 41: 232.
[176] 陈勇．孤立子理论的若干问题的研究及机械化实现，博士学位论文，大连理工大学，2003．
[177] Hirota R. Soliton structure of the Drinfel'd SokolovWilson equation, J. Math. Phys., (1986), 27: 1499.
[178] Fan E G , Hon B Y. Double periodic solutions with Jacobi elliptic functions for two generalized Hirota-Satsuma coupled KdV systems, Phys. Left. A, (2002), 292: 335.
[179] Tam H W, Hu X B, Wang D L. Two integrable coupled nonlinear systems, J. Phys. Soc. Jap., (1999), 68: 369.
[180] Painlevé. Sur les équations différentielles d'ordre superieur dont l'integerale n'admet qu'un nombre fini determinations, C.R.Acad.Sc. Paris, 1888, 116: 88; 116: 173; Surles équations différentielles du premier ordre, C.R.Acad.Sc.Paris, (1888), 107: 221; 320; 724.
[181] Steeb W H, Kloke M, Spieker B M, Kunick A. Integrability of dynamical systems and the singular point analysis, Found. Phys., (1985), 15: 637.
[182] Zakharov V E. On stochastization of one-dimensional chains of nonlinear oscillations, Soy. Phys. Tetp, (1974), 38: 108.
[183] Wadati M. The exact solution of the modified Korteweg-de Vries equation, J.Phys. Soc. Japan, (1972), 32: 1681.
[184] Weiss J. On classes of integrable systems and the Painlevé property, J. Math. Phys., (1984), 25: 13.
[185] Clarkson P A, Mcleod J B, Olver P J and Ramani A. Integrability of Klein-Gordon equations, SIAM J. Math. Anal., (1986), 17: 798.
[186] Gibbon J D, Radmore P, Tabor M and Wood D. The Painlevé property and Hirota's methodd, Stud. Appl. Math.,(1985), 72: 39.
[187] Steeb W H, Grauel A, Kloke M and Spieker B M. Nonlinear diffusion equations, integrability and the Painlevé property, Phys. Scripta, (1985), 31: 5.
[188] Steeb W H, Kloke M, Spiekcr B M and Grensing D. Relativistic field equations and the Painlevé test, Z. Phys. C, (1985), 28: 241.
[189] Xu G Q, Li Z B. A maple package for the Painlevé test of nonlinear partial differential equations, Chin. Phys. Left., (2003), 20: 975.
[190] Xu G Q, Li Z B. Symbolic computation of the Painlevé test for nonlinear partial differential equations using Maple, Comput. Phys. Commun., 2004 (accepted).
[191] Lou S Y. Painlevé test for the integrable dispersive long wave equations in two space dimensiona,
Phys. Lett. A, (1993), 176: 96.
[192] Tang X Y, Hu H C. Characteristic manifold and Painlevé integrability: fifth order Schwarzian KdV type equation, Chin. Phys. Lett., (2002), 19: 1225.
[193] Xie F D, Chen Y. An algorithmic method in Painlevé analysis of PDE, Comput. Phys. Commun., (2003), 154: 197.
[194] Lou S Y, Chen C L and Tang X Y. (2+1)-dimensional (M+N)-component AKNS system: Painlevé integrability, infinitely many symmetries and simlarity reductions, J. Math. Phys., (2002), 43: 4078.
[195] Zhang S L, Wu B, Lou S Y. Painlevé analysis and special solutions of of generalized Broer-Kaup equations, Phys. Lett. A, (2002), 300: 40.
[196] Lou S Y, Wu Q X. Painlevé integrability of two sets of nonlinear evolution equations with nonlinear dispersions, Phys. Lett. A, (1999), 262: 344.
[197] Mohammad A A, Can M. Painlevé analysis and symmetries of the Hirota-Satsuma equation, Nonlinear Math. Phys., (1996), 3: 152.
[198] Conte R. Invariant Painlevé analysis of partial differential equations, Phys. Lett. A, (1989), 140: 383.
[199] Liu Q P. Painlevé analysis and exact solutions of a modified Boussinesq equation, CUMT-MATH-9504, 1995.
[200] Alagesan T, Uthayakumar A, Porsezian K. Painlevé analysis and Bcklund transformation for a three-dimensional Kadomtsev-Petviashvili equation, Chaos, Solitons and Fractals,(1997), 8: 893.
[201] Yan Z Y. New families of nontravelling wave solutions to a new (3+1)-dimensional potential YTSF-equation, Phys. Lett. A, (2003), 318: 78.
[202] Yu S J, Toda K, Sasa N, Fukuyama T. N soliton solutions to the Bogoyavlenskii-Schiff equation and a quest for the soliton solution in (3+1)-dimensions, J. Phys. A: Math. Gen.,(1998), 31: 3337.
[203] Zhang J F, Zhu Y J, Lin J. Similarity reductions for the KZ equation, Commun. Theor. Phys., (1995), 24: 69.
[204] Isojima W, Willox R and Satsuma J. On various solutions of the coupled KP equation, J. Phys. A: Math. Gen., (2002), 35: 6893.
[205] Geng X G. Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations, J. Phys. A: Math. Gen., (2003), 36: 2289.
[206] Conte R, Musette M. Link between solitary waves and projective Riccati equations, J. Phys. A: Math. Gen., (1992), 25: 5609.
[207] Radha R, Lakshmanan M. Singularity structure analysis and bilinear form of a (2+1) dimensional non-linear Schrdinger (NLS) equation, Inverse Problems, (1994), 10: L29.
[208] Newell A C, Tabor M and Zeng Y B. A unified approach to Painlevé expansion, Physica D, (1987), 29: 1.
[209] Kalkanli A K, Sakovich S Y, Yurdusen I. Integrability of KerstenKrasil'shchik coupled KdVmKdV equations: singularity analysis and Lax pair, J. Math. Phys., (2003), 44: 1703.
[210] Hon Y C, Fan E G. Solitary wave and doubly periodic wave solutions for the Kersten-Krasil'shchik coupled KdV-mKdV system, Chaos, Solitons and Fractals, (2003), 19: 1141.
[211] 范恩贵．张鸿庆．获得非线性演化方程Bcklund变换的一种新的途径．应用数学与力学．(1998)，19：603．
[212] 范恩贵，张鸿庆．Whitham-Broer-Kaup浅水波方程的Bcklund变换和精确解，应用数学与力学，
(1998), 19: 667.
[213] Xu G Q, Zhang S Q, Li Z B. Simplified Hirota Method and its Applications, Journal of Shanghai University, (2003), 7: 143.
[214] Picketing A. A new truncation in Painleve analysis, J.Phys. A: Math.Gen., (1993), 26: 4395.
[215] Conte R. Universal invariance properties of Painlevé analysis and Bcklund transformation in nonlinear partial differential equations, Phys. Lett. A, (1988), 134: 100.
[216] 楼森岳．推广的Painlevé展开及KdV方程的非标准截断解，(1998)，47：1937．
[217] Lou S Y. Extended Painlevé expansion, nonstandard truncation and special reductions of nonlinear evolution equations, Z. Naturforsch A, (1998), 53: 251.
[218] Pickering A. The singular manifold method revisited, J. Math. Phys., (1996), 37: 1894.
[219] Flaschka H, Newell A C and Tabor M. Integrability In "What is Integrability", 73, Zakaharov V E, ed. Springer Verlag, New York, 1991.
[220] Hereman W, Nuseir A. Symbolic methods to construct exact solutions of nonlinear partial differential equations, Math. Comp. Sim., (1997), 43: 13.
[221] Dye J M, Parker A. On bidirectional fifth-order nonlinear evolution equations, Lax pairs, and directionally dependent solitary waves, J. Math. Phys., (2001), 42: 2567.
[222] Dye J M, Parker A. A bidirectional KaupKupershmidt equation and directionally dependent solitons, J. Math. Phys., (2002), 43: 4921.
[223] 李志斌，潘素起．广义五阶KdV方程的孤子解与孤波解，物理学报，(2001)，50：402．
[224] Hu X B. Soliton solutions and symmetries of the 2+1 dimensional KaupKupershmidt equation, Phys. Lett. A. (1999). 262: 409.
[225] 范恩贵．齐次平衡法、Weiss-Tabor-Carnevale法与Clarkson-Kruskal约化法之间的联系，(2000)，49：1409．