非线性微分方程求解和群分析
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摘要
非线性微分方程一直以来都是备受关注的研究对象.本文基于数学机械化思想,以计算机符号计算软件为工具,围绕非线性微分方程这一课题开展了三个方面的研究:非线性发展方程的精确解;非线性分数阶微分方程的近似解;非线性微分方程的群分析.
本文由以下五章组成:
第一章介绍本文所涉及的学科:数学机械化、孤立子理论、分数微积分和分数阶微分方程、微分方程群理论分析的历史发展及现状,同时介绍了国内外学者在这些领域所取得的成果.最后,介绍了本文的主要工作.
第二章介绍本文的理论基础:AC=BD理论和微分伪带余除法,并且阐述在AC=BD这一理论框架下的微分方程精确解的构造问题.
第三章是在第二章理论的指导下,基于将非线性发展方程精确求解代数化、算法化、机械化的思想,提出两种求解非线性发展方程精确解的有效算法:第一种是广义的有理形式展开法,该方法是对已有的有理形式展开法的扩展,本章以高维耦合Burgers方程为例,验证了该方法的有效性;第二种是基于变系数Korteweg-de Vries(KdV)方程的子方程法,该方法用变系数KdV方程取代常微分方程作为辅助方程,本章应用该方法获得3+1维potential-YTSF方程的精确解.
第四章是关于非线性分数阶微分方程的近似求解.首先介绍了本章涉及到的分数微积分的定义及其性质,然后介绍了非线性分数阶微分方程近似求解的方法及其应用.本章应用变分迭代法、Adomian分解法和同伦摄动法获得非线性分数阶Sharma-Tasso-Olever(STO)方程的有理近似解,并且通过对具体的数值、绝对误差和图形的分析,阐述这三种方法的自身特征、有效性和可靠性.本章将用于求解整数阶微分方程精确解和数值解的同伦分析法应用到非线性分数阶Benjamin-Bona-Mahony-Burgers(BBM-Burgers)方程中,获得其近似解.
第五章主要研究非线性微分方程的群分类和守恒律分类.首先介绍标准型和符号计算软件Maple中的软件包K_(IF)S_(IMP),此软件包可以化超定的微分方程组为标准型.本章成功地利用软件包R_(IF)S_(IMP)获得非线性变系数电报方程,f(x)u_(tt)=(H(u)u_x)_x+h(x)K(u)u_x的群分类,并且给出在等价变换群下该方程的守恒律分类.
The nonlinear differential equation remains a major concern all the time. In the dissertation, under the guidance of mathematics mechanization and by means of symbolic-numeric computation software, the following problems around the nonlinear differential equation are discussed as follows: the exact solutions of nonlinear evolution equations; the approximation solutions of nonlinear fractional differential equations; the group analysis of nonlinear differential equations.
The paper consists of the following chapters.
Chapter 1 is devoted to reviewing the history and development of the mathematics mechanization, soliton theory, fractional calculous, fractional differential equation, and group analysis of differential equations. Some works and achievements on these subjects involved in this dissertation are presented at home and abroad. Finally, our main works are listed.
Chapter 2 introduces the basic theories of AC = BD and pseudo-differential division with remainder, as well as the construction of exact solutions of differential equations under the instruction of the theory of AC = BD.
Based on the theories in Chapter 2 and the idea of algebraic method、algorithm reality and mechanization for solving nonlinear evolution equations, Chapter 3 presents two kinds methods for obtaining the exact solutions of nonlinear evolution equations: the one is the generalized rational expansion method, which extended the known rational expansion method, the chapter takes the Burgers equation for example to illuminate the effectiveness of the method; the other is the variable coefficient Korteweg-de Vries (KdV) equation-based sub-equation method, which takes the variable coefficient KdV equation substituting ordinary differential equation as the sub-equation, and the exact solutions of the 3+1-dimensional potential-YTSF equation are obtained by the method.
Chapter 4 is on the approximation solutions of nonlinear fractional differential equations.The first section is to introduce some corresponding basic definiens and properties on the fractional calculous. The rest of the chapter is to present solving methods for the approximation solutions of the nonlinear fractional differential equations and their application.The chapter applies the variational iteration method、the Adomian decomposition method and the homotopy perturbation method to obtain the rational approximation solutions of the nonlinear fractional Sharma-Tasso-Olever (STO) equation, and demonstratesthe significant features, efficiency and reliability of three methods by investigating some numerical results, absolute error and figures. The homotopy analysis which traditionally developed for differential equations of integer order are directly extended to derive approximation solutions of the nonlinear fractional Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation in the chapter.
Chapter 5 is to study the group classification and the classification of conservation law for the nonlinear differential equations. Here, standard form and the package R_(IF)S_(IMP) in the symbolic computation software Maple are firstly introduced.. The package is used to reduce over-determined differential equations to standard form. The chapter applies successfullythe package R_(IF)S_(IMP) to obtain the group classification of the nonlinear variable coefficient telegraph equations f(x)u_(tt) = (H(u)u_x)_x + h(x)K(u)u_x, and gives the classification of conservation law with respect to the group of equivalence transformations.
本文由以下五章组成:
第一章介绍本文所涉及的学科:数学机械化、孤立子理论、分数微积分和分数阶微分方程、微分方程群理论分析的历史发展及现状,同时介绍了国内外学者在这些领域所取得的成果.最后,介绍了本文的主要工作.
第二章介绍本文的理论基础:AC=BD理论和微分伪带余除法,并且阐述在AC=BD这一理论框架下的微分方程精确解的构造问题.
第三章是在第二章理论的指导下,基于将非线性发展方程精确求解代数化、算法化、机械化的思想,提出两种求解非线性发展方程精确解的有效算法:第一种是广义的有理形式展开法,该方法是对已有的有理形式展开法的扩展,本章以高维耦合Burgers方程为例,验证了该方法的有效性;第二种是基于变系数Korteweg-de Vries(KdV)方程的子方程法,该方法用变系数KdV方程取代常微分方程作为辅助方程,本章应用该方法获得3+1维potential-YTSF方程的精确解.
第四章是关于非线性分数阶微分方程的近似求解.首先介绍了本章涉及到的分数微积分的定义及其性质,然后介绍了非线性分数阶微分方程近似求解的方法及其应用.本章应用变分迭代法、Adomian分解法和同伦摄动法获得非线性分数阶Sharma-Tasso-Olever(STO)方程的有理近似解,并且通过对具体的数值、绝对误差和图形的分析,阐述这三种方法的自身特征、有效性和可靠性.本章将用于求解整数阶微分方程精确解和数值解的同伦分析法应用到非线性分数阶Benjamin-Bona-Mahony-Burgers(BBM-Burgers)方程中,获得其近似解.
第五章主要研究非线性微分方程的群分类和守恒律分类.首先介绍标准型和符号计算软件Maple中的软件包K_(IF)S_(IMP),此软件包可以化超定的微分方程组为标准型.本章成功地利用软件包R_(IF)S_(IMP)获得非线性变系数电报方程,f(x)u_(tt)=(H(u)u_x)_x+h(x)K(u)u_x的群分类,并且给出在等价变换群下该方程的守恒律分类.
The nonlinear differential equation remains a major concern all the time. In the dissertation, under the guidance of mathematics mechanization and by means of symbolic-numeric computation software, the following problems around the nonlinear differential equation are discussed as follows: the exact solutions of nonlinear evolution equations; the approximation solutions of nonlinear fractional differential equations; the group analysis of nonlinear differential equations.
The paper consists of the following chapters.
Chapter 1 is devoted to reviewing the history and development of the mathematics mechanization, soliton theory, fractional calculous, fractional differential equation, and group analysis of differential equations. Some works and achievements on these subjects involved in this dissertation are presented at home and abroad. Finally, our main works are listed.
Chapter 2 introduces the basic theories of AC = BD and pseudo-differential division with remainder, as well as the construction of exact solutions of differential equations under the instruction of the theory of AC = BD.
Based on the theories in Chapter 2 and the idea of algebraic method、algorithm reality and mechanization for solving nonlinear evolution equations, Chapter 3 presents two kinds methods for obtaining the exact solutions of nonlinear evolution equations: the one is the generalized rational expansion method, which extended the known rational expansion method, the chapter takes the Burgers equation for example to illuminate the effectiveness of the method; the other is the variable coefficient Korteweg-de Vries (KdV) equation-based sub-equation method, which takes the variable coefficient KdV equation substituting ordinary differential equation as the sub-equation, and the exact solutions of the 3+1-dimensional potential-YTSF equation are obtained by the method.
Chapter 4 is on the approximation solutions of nonlinear fractional differential equations.The first section is to introduce some corresponding basic definiens and properties on the fractional calculous. The rest of the chapter is to present solving methods for the approximation solutions of the nonlinear fractional differential equations and their application.The chapter applies the variational iteration method、the Adomian decomposition method and the homotopy perturbation method to obtain the rational approximation solutions of the nonlinear fractional Sharma-Tasso-Olever (STO) equation, and demonstratesthe significant features, efficiency and reliability of three methods by investigating some numerical results, absolute error and figures. The homotopy analysis which traditionally developed for differential equations of integer order are directly extended to derive approximation solutions of the nonlinear fractional Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation in the chapter.
Chapter 5 is to study the group classification and the classification of conservation law for the nonlinear differential equations. Here, standard form and the package R_(IF)S_(IMP) in the symbolic computation software Maple are firstly introduced.. The package is used to reduce over-determined differential equations to standard form. The chapter applies successfullythe package R_(IF)S_(IMP) to obtain the group classification of the nonlinear variable coefficient telegraph equations f(x)u_(tt) = (H(u)u_x)_x + h(x)K(u)u_x, and gives the classification of conservation law with respect to the group of equivalence transformations.
引文
[1] 吴文俊.吴文俊论数学机械化[M].山东:山东教育出版社,1996.
[2] 高小山.数学机械化进展综述[J].数学进展,2001,30(5):385-404.
[3] 石赫.机械化数学引论[M].湖南:湖南教育出版社,1998.
[4] 吴文俊.几何定理机器证明的基本原理[M].北京:科学出版社,1984.
[5] WU W T. On the decision problem and the mechanization of theorem-proving in elementary geometry [J]. Scientia Sinica, 1978, 21: 159-172.
[6] WU W T. On zeros of algebraic equations: an application of Ritt principle [J]. Kexue Tongbao, 1986, 31: 1-5.
[7] RITT J F. Differential Algebra [M]. New York: American mathematical Society, 1950.
[8] WU W T. On the foundation of algebraic differential geometry [J]. Sys. Sci. & Math. Scis., 1989, 2(4): 289-312.
[9] GAO X S, ZHANG J Z, CHOU S C. Geometry Expert (in Chinese) [M]. Taiwan: Nine Chapters Pub, 1998.
[10] CHOU S C, GAO X S, ZHANG J Z. Machine Proofs in Geometry [M]. Singapore: World Scientific, 1994.
[11] CHOU S C. Automated reasoning in differential geometry and mechanics using the characteristic set method, Ⅰ. An improved version of Ritt-Wu's decomposition algorithm [J]. J. Auto. Reasoning, 1993, 10: 161-172.
[12] GAO X S, CHOU S C, ZHANG J Z. Automated production of traditional proofs for constructive geometry theorems [C]. Proceedings of ISSAC-93, 48.
[13] GAO X S. An Introduction to Wu's Method of Mechanical Geometry Theorem Proving [C]. IFIP Transaction on, Automated Reasoning, North-Holland, 1993, 13.
[14] WANG D M, GAO X S. Geometry theorems proved mechanically using Wu's method [J]. MM-Reseach Preprints, 1987(2).
[15] WANG D M, HU S. A mechanical proving system for constructible theorems in elementry geometry [J]. J. Sys. Sci. & Math. Scis., 1987, 7: 163-172.
[16] LIN D D, LIU Z J. Some results on theorem proving in finite geometry [C]. Proc. of IWMM, Inter Academic Pub, 1992.
[17] WU J Z, LIU Z J. Proceedings of 1st Asian Symposium on Computer Mathematics [C]. Beijing of China, 1995.
[18] WU J Z, LIU Z J. On first-order theorem proving using generalized odd-superpositions Ⅱ [J]. Science in China (Series E), 1996, 39: 50-61.
[19] CHOU S C, GAO X S. Ritt-Wu's decomposition algorithm and geometry theorem proving [M]. CADE'10, Stikel M E, Ed. Berlin: Springer-Verlag, 1990, 207-220.
[20] KAPUR D, WAN H K. Refutational proofs of geometry theorems via characteristic sets [C]. Proc. of ISSAC-90, Tokyo, Japan, 1990, 277-284.
[21] KO H. Geometry theorem proving by decomposition of quasi-algebraic sets: an application of the Ritt-Wu principle [J]. Artif. Intell., 1988, 37: 95-122.
[22] LI H B, CHENG M T. Proving theorems in elementary geometry with Clifford algebraic method [J]. Chinese Math. Progress, 1997, 26(4): 357-371.
[23] LI H B. Automated reasoning with differential forms [C]. Proc. of ASCM'92, 1992.
[24] LI H B, CHENG M T. Clifford algebraic reduction method for mechanical theorem proving in differential geometry [J]. J. Auto. Reasoning, 1998, 21: 1-21.
[25] WANG D M. Clifford algebraic calculus for geometric reasoning with applications to computer vision [M]. In: Automated deduction in geom., Wang, ed., LNAI 1360,115-140, Berlin: Springer-Verlag, 1997.
[26] SUN X D, WANG K, WU K. Solutions of Yang-Baxter equation with spectral parameters for a six-vertex model [J]. Acta Phys. Sinica (Chinese), 1995, 44: 1-8.
[27] SUN X D, WANG K, WU K. Classification of six-vertex-type solutions of the colored Yang-Baxter equation [J]. J. Math. Phys., 1995, 36(10): 6043-6063.
[28] WANG M L, LI Z B. Proc. Of the 1994 Beijing Symposium on nonlinear evolution equations and infinite dimensional dynamics systems [C]. Zhongshan University Press, 1995, 181.
[29] LI Z B, WANG M L. Travelling wave solutions to the two-dimensional KdV-Burgers equation [J]. J. Phys. A: Math. Gen., 1993, 26: 6027-6031.
[30] 李志斌,张善卿.非线性波方程准确孤立波解的符号计算[J].数学物理学报,1997,17:81-89.
[31] LI Z B, SHI H. Exact solutions for Belousov-Zhabotinski reaction-diffusion system [J]. Appl. Math. J. Chinese Univ. Ser. B, 1996, 11: 1-6.
[32] LI Z B. Proceeding of asian symposition on computer mathematics [C]. Lanzhou University, 1998, 153.
[33] LI Z B, LIU Y P. RATH: A Maple package for finding travelling solitary wave solutions to nonlinear evolution equations [J]. Compu. Phys. Commun., 2002, 148: 256-266.
[34] 范恩贵.可积系统与计算机代数[M].北京:科技出版社,2004.
[35] 闰振亚.非线性波与可积系统[D].大连:大连理工大学应用数学系,2002.
[36] 郑学东.一类非线性偏微分方程组的机械化求解与P-性质的机械化检验[D].大连:大连理工大学应用数学系,2002.
[37]朝鲁.微分方程(组)对称向量的吴-微分特征列算法及其应用[J].数学物理学报,1999,19(3):326-332.
[38]朝鲁.微分多项式系统的约化算法理论[J].数学进展,2003,32(2):208-220.
[39]陈勇.孤立子理论中的若干问题的研究及机械化实现[D].大连:大连理工大学应用数学系,2003.
[40]谢福鼎.Wu-Ritt消元法在偏微分代数方程中的应用[D].大连:大连理工大学应用数学系,2003.
[41]李彪.孤立子理论中若干精确求解方法的研究及应用[D].大连:大连理工大学应用数学系,2005.
[42]FAN E G, ZHANG H Q. Backlund transformation and exact solutions for Whitham-Broer-Kaup equations in shallow water [J]. Appl. Math. Mech., 1998, 19: 713-716.
[43]范恩贵,张鸿庆.非线性波动方程的孤波解[J].物理学报,1997,46:1254-1258.
[44]YAN Z Y, ZHANG H Q. New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water [J]. Phys. Lett. A., 2001, 285: 355-260.
[45]YAN Z Y, ZHANG H Q. On a new algorithm of constructing solitary wave solutions for systems of nonlinear evolution equations in mathematical physics [J]. Appl. Math. Mech., 2000, 21: 342-347.
[46]YAN Z Y, ZANG H Q. Multiple soliton-like and periodic-like solutions to the generalization of integrable (2+1)-dimensional dispersive long-wave equations [J]. J. Phys. Soc. Jpn., 2002, 71: 437-442.
[47]闰振亚,张鸿庆.New explicit travelling wave solutions for a class of nonlinear evolutionequations[J].物理学报,1999,48:1-5.
[48]KALTOFEN E. Challenges of Symbolic Computation: My Favorite Open Problems [J]. J. Symb. Comput., 2000, 29: 891-919.
[49]BUCHBERGER B, COLLINS G E, LOOS R. Computer Algebra-Symbolic and Algebraic Computation [M]. Beijing: World Publishing Corporation, 1988.
[50]RISH R H. The problem of integration in finite terms [J]. Trans. Ams. Math. Soc, 1969, 139: 167-189.
[51]BERLEKAMP E R. Factoring polynomials over large finite fields [J]. Math. Comput., 1970, 24: 713-735.
[52]BROWN W S. On the partition function of a finite set [J]. J. ACM., 1971, 18: 478-504.
[53]GOSPER R W. Decision Procedures for Indefinite Hypergeometric Summation [J]. Proc Nat. Acad. Sci. USA, 1978, 75: 40-42.
[54]LENSTRA A K. Factoring multivariate polynomials over algebraic number fields [J]. SIAM J. Comput., 1987, 16: 591-598.
[55]LENSTRA A K, LENSTRA H W JR, LOVASZ L. Factoring polynomials with rational coefficients [J]. Math. Ann., 1982, 261: 515-534.
[56] KALTOFEN E, TRAGER B. Computing with polynomials given by black boxes for their evaluations: greatest common divisors, factorization, separation of numerators and denominators [J]. J. Symb. Comput., 1990, 9: 301-320.
[57] SHOUP V. A new polynomial factorization algorithm and its implementation [J]. J. Symb. Comput., 1995, 20: 363-397.
[58] RUSSELL J S. Report on waves [C]. Fourteen meeting of the British association for the advancement of science, John Murray, London, 1844, 311-390.
[59] BOUSSINESQ J. Theorie des ondes et de remous qui se propagent [J]. J. Math. Pure Appl., 1972, 17: 55-108.
[60] KORTEWEG D J, DEVRIES G. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves [J]. Phil. Mag., 1895, 39: 422-433.
[61] FERMI E, PASTA J, ULAM S. Studies of nonlinear problems [C]. Los Alamos Scient. Lab., Rep. LA-1940, 1955.
[62] PERRING J K, SKYRME T H R. A model unified field equation [J]. Nucl. Phys., 1962, 31: 550-555.
[63] ZABUSKY N J, KRUSKAL M D. Interaction of solitons in a collisionless plasma and the recurrence of initial states [J]. Phys. Rev. Lett., 1965, 15: 240-243.
[64] ABLOWITZ M J, SEGUR H. Solitons and the inverse scattering transformation [M]. Philadelphia: SIAM, 1981.
[65] ABLOWITZ M J, CLARKSON P A. Solitons, nonlinear evolution equations and inverse scattering [M]. Cambridge: Cambridge University Press, 1991.
[66] NEWELL A C. Soliton in mathematics and physics [M]. Philadelphia: SIAM, 1985.
[67] FADDEEV L D, TAKHTAJAN L A. Hamiltonian method in the theory of solitons [M]. Berlin: Springer-Verlag, 1987.
[68] MATVEEV V B, SALLE M A. Darboux transformation and Solitons [M]. Berlin: Springer-Verlag, 1991.
[69] 郭柏灵,庞小峰.孤立子[M].北京:科学出版社,1987.
[70] 谷超豪等.孤立子理论与应用[M].杭州:浙江科技出版社,1990.
[71] 谷超豪等.孤立子理论中的Darboux变换及其几何应用[M].上海:上海科技出版社,1999.
[72] 李翊神.孤子与可积系统[M].上海:上海科技出版社,1999.
[73] HIROTA R. The direct method in soliton theory [M]. New York: Cambridge University Press, 2004.
[74] MIURA M R. B(?)cklund transformation [M]. Berlin: Springer-Verlag, 1978.
[75] WAHLQUIST H D, ESTABROOK F B. Backlund transformations for solitons of the Korteweg-de Vries equation [J]. Phys. Rev. Lett., 1973, 31: 1386-1390.
[76] WEISS J, TABOR M, CARNVALE G. The Painleve property for partial differential equations [J]. J. Math. Phys., 1983, 24: 522-526.
[77] WEISS J. The Painlev(?) property for partial differential equations, Ⅱ. B(?)cklund transformation, Lax pairs, and the Schwarzian derivative [J]. J. Math. Phys., 1983, 24: 1405-1413.
[78] DARBOUX G. Compts Rendus Hebdomadaires des Seances de l'Academie des Sciences [J]. Pairs, 1882, 94: 1456-1459.
[79] WADATI M et al. Simple derivation of B(?)cklund transformation from Riccati form of inverse method [J]. Prog. Theor. Phys., 1975, 53: 1652-1656.
[80] GU C H. A unified explicit form of B(?)cklund transformations for generalized hierarchies of KdV equations [J]. Lett. Math. Phys., 1986, 11; 325-335.
[81] GU C H, ZHOU Z X. On the Darboux matrices of B(?)cklund transformations for AKNS systems [J]. Lett. Math. Phys., 1987, 13: 179-187.
[82] HU X B, CLARKSON P A. B(?)cklund transformations and nonlinear superposition formulae of a differential-difference KdV equation [J]. J. Phys. A: Math. Gen., 1998, 31: 1405-1414.
[83] HU X B, ZHU Z N. A B(?)cklund transformation and nonlinear superposition formula for the Belov-Chaltikian lattice [J]. J. Phys. A: Math. Gen., 1998, 31: 4755-4761.
[84] MATVEEV V B, SKLYANIN E K. On B(?)cklund transformations for many-body systems [J]. J. Phys. A: Math. Gen., 1998, 31: 2241-2251.
[85] SKLYANIN E K. Canonicity of B(?)cklund transformation:r-matrix approach. I. L. D. Faddeev's Seminar on Mathematical Physics [J]. Amer. Math. Soc. Transt. Ser., 2000, 201: 277-282.
[86] HONE A N W et al. B(?)cklund transformations for many-body systems related to KdV [J]. J. Phys. A: Math. Gen., 1999, 32: L299-L306.
[87] CHOUDHURY A G, CHOWHURY A R. Canonical and B(?)cklund transformations for discrete integrable systems and classical r-matrix [J]. Phys. Lett. A, 2001, 280: 37-44.
[88] ZENG Y B et al. Canonical explicit Backlund transformations with spectrality for constrained flows of soliton hierarchies [J]. Phys. A, 2002, 303: 321-338.
[89] LOU S Y et al. Vortices, circumfluence, symmetry groups and Darboux transformations of the Euler equations [J]. arXiv: nlin.PS/0509039.
[90] GARDNER C S et al. Method for solving the Korteweg-deVries equation [J]. Phys. Rev. Lett., 1967, 19: 1095-1097.
[91] LAX P D. Integrals of nonlinear equations of evolution and solitary waves [J]. Commun. Pure Appl. Math., 1968, 21: 467-490.
[92] WADATI M. The modified Korteweg-de Vries equation [J]. J. Phys. Soc. Jpn., 1973, 32: 1289-1296.
[93] ABLOWITZ M J et al. Method for solving the sine-Gordon equation [J]. Phys. Rev. Lett., 1973, 30: 1262-1264.
[94] WAHLQUIST H D, Estabrook F B. Prolongation structures of nonlinear evolution equations [J]. J. Math. Phys., 1975, 16: 1-7.
[95] GUO H Y et al. On the prolongation structure of Ernst equation [J]. Commun. Theor. Phys., 1982, 1: 661-664.
[96] GUO H Y, WU K, WANG S K. Prolongation structure, B(?)cklund transformation and principal homogeneous Hilbert problem in general relativity [J]. Commun. Theor. Phys., 1983, 2: 883-898.
[97] GUO H Y, WU K, WANG S K. Inverse scattering transform and regular Riemann-Hilbert problem [J]. Commun. Theor. Phys., 1983, 2: 1169-1173.
[98] WU K, GUO H Y, WANG S K. Prolongation structures of nonlinear systems in higher dimensions [J]. Commun. Theor. Phys., 1983, 2: 1425-1437.
[99] CASE K M, KAC M. A discrete version of the inverse scattering problem [J]. J. Math. Phys., 1973, 14: 594-603.
[100] FLASCHKA H. On the Toda lattice. Ⅱ. Inverse-scattering solution [J]. Prog. Theor. Phys., 1974, 51: 703-716.
[101] ABLOWITZ M J, Ladik J F. Nonlinear differential-difference equations [J]. J. Math. Phys., 1975, 16: 598-603.
[102] 曹策问.孤立子与反散射[M].郑州:郑州大学出版社,1983.
[103] 李翊神,庄大尉.两类非线性演化方程的等价[J].北京大学学报,1983,2:107-118.
[104] 顾新身.相应的特征值问题有二重特征根的复KdV方程的一个求解实例[J].中国科学技术大学学报数学专辑,1983,5:82-89.
[105] ZENG Y B, MA W X, LIN R L. Integration of the soliton hierarchy with self-consistent sources [J]. J. Math. Phys., 2000, 41: 5453-5489.
[106] NING T K, CHEN D Y, ZHANG D J. Soliton-like solutions for a nonisospectral KdV hierarchy [J]. Chaos, Solitons & Fractals, 2004, 21: 395-401.
[107] ABLOWITZ M J et al. A connection between nonlinear evolution equations and ordinary differential equations of P-type [J]. J. Math. Phys., 1980, 21: 715-721.
[108] JIMBO M et al. Painleve test for the self-dual Yang-Mills equation [J]. Phys. Lett. A, 1982, 92: 59-60.
[109] FORDY A P, PICKERING A. Analysing negative resonances in the Painleve test [J]. Phys. Lett. A, 1991, 160: 347-354
[110] MUSETTE M, CONTE R. The two-singular-manifold method: Ⅰ. Modified Korteweg-de Vries and sine-Gordon equations [J]. J. Phys. A: Math. Gen., 1994, 27: 3895-3913.
[111] CONTE R, MUSETTE M, PICKERING A. The two-singular manifold method: Ⅱ. Classical Boussinesq system [J]. J. Phys. A: Math. Gen., 1995, 28: 179-187.
[112] 曾云波.与A_l相联系的半Toda方程的Lax对与B(?)cklund变换[J].数学学报,1992.35:454-459.
[113] 曾云波.递推算子与plinlev(?)性质[J].数学年刊,1991,12A:78-88.
[114] XU G Q, LI Z B. Symbolic computation of the Painleve test for nonlinear partial differential equations using Maple [J]. Comput. Phys. Commun., 2004, 161: 65-75.
[115]XU G Q, LI Z B. PDEPtest: a package for the Painleve test of nonlinear partial differential equations [J]. Appl. Math. Comput., 2005, 169: 1364-1379.
[116]LOU S Y. KdV extensions with Painlev(?) property [J]. J. Math. Phys., 1998, 39: 2112-2121.
[117]CHEN L L, LOU S Y. Painlev(?) analysis of a (2+1)-dimensional Burgers equation [J]. Commun. Theor. Phys., 1998, 29: 313-316.
[118]楼森岳.推广的Painlev(?)展开及KdV方程的非标准截断解[J].物理学报,1998,47:1937-1949.
[119] WHITHAM G B. Nonlinear dispersive waves [J]. SIAM J. Appl. Math., 1966,14: 956-958.
[120] KRUSKAL M D, ZABUSKY N J. Progress on the Fermi-Pasta-Ulam nonlinear string problem [J]. Princeton Plasma Physics Lab. Ann. Rep., MATT-Q-21, Princeton, N. J. 1963, 301-308.
[121] KRUSKAL M D. Korteweg-de Vries equation and generalizations: Ⅴ. Uniqueness and nonexistence of polynomial conservation laws [J]. J. Math. Phys., 1970, 11: 952-960.
[122]屠规彰,秦孟兆.非线性演化方程的不变群与守恒律-对称函数方法[J].中国科学A辑,1980.24:13-26.
[123]屠规彰.Boussinesq方程的B(?)cklund变换与守恒律[J].应用数学学报,1981,4:63-68.
[124]曾云波.Darboux transformations of families of AKNS equatiOIlS with additional terms[J].数学学报,1995,15:337-345.
[125]屠规彰,秦盂兆.Relationship between symmetries and conservation laws of nonlinear evolution equations[J].科学通报,1979,29:913-917.
[126]HU X B. Rational solutions of integrable equations via nonlinear superposition formulae [J]. J. Phys. A: Math. Gen., 1997, 30: 8225-8240.
[127]HU X B, Zhu Z N. Some new results on the Blaszak-Marciniak lattice: Backlund transformation and nonlinear superposition formula [J]. J. Math. Phys., 1998, 39: 4766-4772.
[128]HU X B, WU Y T. A new integrable differential-difference system and its explicit solutions [J]. J. Phys. A: Math. Gen., 1999, 32: 1515-1521.
[129]HU X B, TAM H W. Some new results on the Blaszak-Marciniak, 3-field and 4-field lattices [J]. Rep. Math. Phys., 2000, 46: 99-105.
[130]TAM H W et al. The Hirota-Satsuma coupled KdV equation and a coupled Ito system revisited [J]. J. Phys. Soc. Jpn., 2000, 69: 45-52.
[131]HU X B et al. Lax pairs and B(?)cklund transformations for a coupled Ramani equation and its related system [J]. Appl. Math. Lett., 2000, 13: 45-48.
[132]BOITI M et al. On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions [J]. Inv. Probl., 1986, 2: 271-280.
[133]RADHA R, LAKSHMANAN M. Singularity analysis and localized coherent structures in (2+1)-dimensional generalized Korteweg-de Vries equations [J]. J. Math. Phys., 1994, 35: 4746-4756.
[134] LOU S Y. Generalized dromion solutions of the (2+1)-dimensional KdV equation [J]. J. Phys. A: Math. Gen., 1995, 28: 7227-7232.
[135] ZHANG J F. Abundant dromion-like structures to the (2+1)-dimensional KdV equation [J]. Chin. Phys., 2000, 9: 1-4.
[136] LOU S Y, RUAN H Y. Revisitation of the localized excitations of the (2+1)-dimensional KdV equation [J]. J. Phys. A: Math. Gen., 2001, 34: 305-316.
[137] ROSENAU P, HYMAN M. Compactons: Solitons with finite wavelength [J]. Phys. Rev. Lett., 1993, 70: 564-567.
[138] ROSENAU P. Nonlinear dispersion and compact structures [J]. Phys. Rev. Lett., 1994, 73: 1737-1741.
[139] OLVER P J, ROSENAU P. Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support [J]. Phys. Rev. E, 1996, 53: 1900-1906.
[140] DUSUEL S et al. Predictability in systems with many characteristic times: The case of turbulence [J]. Phys. Rev. E, 1998, 57: 2320-2326.
[141] ISMAIL M S, TAHA T A. A numerical study of compactons [J]. Math. Computer Simulation, 1998, 47: 519-530.
[142] WAZWAZ A M. Exact special solutions with solitary patterns for the nonlinear dispersive K(m,n) equations [J]. Chaos, Solitons & Fractals, 2002, 13: 161-170.
[143] LOU S Y, WANG Q X. Painleve' integrability of two sets of nonlinear evolution equations with nonlinear dispersions [J]. Phys. Lett. A, 1999, 262: 344-349.
[144] LIU Q P, HU X B, ZHANG M X. Supersymmetric modified Korteweg-de Vries equation: bilinear approach [J]. Nonlinearity, 2005, 18: 1597-1603.
[145] LIU Q P, HU X B. Bilinearization of N = 1 supersymmetric Korteweg-de Vries equation revisited [J]. J. Phys. A: Math. Gen., 2005, 38: 6371-6378.
[146] MIURA R M. Korteweg-de Vries equation and generalizations. Ⅰ. A remarkable explicit nonlinear transformation [J]. J. Math. Phys., 1968, 9: 1202-1204.
[147] ABLOWITZ M J et al. A note on Miura's transformation [J]. J. Math. Phys., 1979, 20: 991-1003.
[148] WANG M L. Solitary wave solutions for variant Boussinesq equations [J]. Phys. Lett. A, 1995, 199: 169-172.
[149] LOU S Y, RUAN H Y, HUANG G X. Exact solitary waves in a converting fluid [J]. J. Phys. A: Math. Gen., 1991, 24: L587-L590.
[150] CONTE R, MUSETTE M. Link between solitary waves and projective Riccati equations [J]. J. Phys. A: Math. Gen., 1992, 25: 5609-5623.
[151] LIU S K, FU Z T, LIU S D, ZHAO Q. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations [J]. Phys. Lett. A, 2001. 289: 69-74.
[152] FU Z T, LIU S K, LIU S D, ZHAO Q. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations [J]. Phys. Lett. A, 2001, 290: 72-76.
[153] FAN E G. Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a coupled mKdV equation [J]. Phys. Lett. A, 2001, 282: 18-22.
[154] YAN C T. A simple transformation for nonlinear waves [J]. Phys. Lett. A, 1996, 224: 77-84.
[155] YAN Z Y, ZHANG H Q. New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics [J]. Phys. Lett. A, 1999, 252: 291-296.
[156] TIAN B, GAO Y T. Variable-coefficient balancing-act method and variable-coefficient KdV equation from fluid dynamics and plasma physics [J]. Eur. Phys. J. B, 2001, 22: 351-360.
[157] TIAN B, GAO Y T. Extension of generalized tanh method and soliton-like solutions for a set of nonlinear evolution equations [J]. Chaos, Solitons & Fractals, 1997, 8: 1651-1653.
[158] GAO Y T, TIAN B. New families of exact solutions to the integrable dispersive long wave equations in (2+1)-dimensional spaces [J]. J. Phys. A: Math. Gen., 1996, 29: 2895-2903.
[159] 昌卓生.计算微分方程对称与精确解的机械化算法及实现[D].大连:大连理工大学应用数学系,2004.
[160] BARNETT M P et al. Symbolic calculation in chemistry: Selected examples [J]. Int. J. Quantum Chem., 2004, 100: 80-104.
[161] 王琪.非线性微分方程求解和混沌同步[D].大连:大连理工大学应用数学系,2006.
[162] Lü Z S. A Burgers equation-based constructive method for solving nonlinear evolution equations [J]. Phys. Lett. A, 2006, 353: 158-160.
[163] 张鸿庆.弹性力学方程组一般解的统一理论[J].大连理工大学学报,1978,18:23-47.
[164] 张鸿庆,王震宇.胡海昌解的完备性和逼近性[J].科学通报,1986,30:342-344.
[165] 张鸿庆,吴方向.一类偏微分方程组的一般解及其在壳体理论中的应用[J].力学学报,1992,24:700-707.
[166] 张鸿庆,冯红.构造弹性力学位移函数的机械化算法[J].应用数学和力学,1995,16:315-322.
[167] 张鸿庆,杨光.变系数偏微分方程组一般解的构造[J].应用数学和力学,1991,12:135-139.
[168] ZHANG H Q, CHEN Y F. Proceeding of the 3rd ACM [C]. Lanzhou University Press, 1998, 147.
[169] ZHANG H Q. C-D integrable system and computer aided solver for differential equations [C]. Proceeding of the 5rd ACM, World Scientific Press, 2001, 221-226.
[170] 梅建琴.微分方程组精确解及其解的规模的机械化算法[D].大连:大连理工大学应用数学系,2006.
[171] 张鸿庆.数学机械化中的AC=BD模式[J].系统科学与数学,2008,28:1030-1039.
[172] 张鸿庆,丁琦.一类非线性偏微分方程组的解析解[J].应用数学和力学,2008,29:8621-8721.
[173] LACROIX S F. Traite du Calcul Differentiel et du Calcul Integral (Second edition) [M]. Paris: Courcier, 1812.
[174] OLDHAM K B, SPANIER J. The Fractional Calculus [M]. New York: Academic Press, 1974.
[175] PODLUBNY I. Fractional Differential Equations [M]. San Diego: Academic Press, 1999.
[176] CARPINTERI A, MAINARDIF. Fractals and Fractional Calculus in Continuum Mechanics [M]. Berlin: Springer-Verlag, 1997.
[177] HILFER E (Ed.). Applications of Fractional Calculus in Physics [M]. World Sci. Publishing, New York, 2000.
[178] ZASLAVSKY G M. Chaos and Fractional Dynamics, vol. 511 of Lect [M]. Notes in Phys., Oxford University Press, Oxford, 2005.
[179] KILBAS A A, SRIVASTAVA H M, TRUJILLO J J. Theory and Applications of Fractional Differential Equations [M]. New York: Elsevier Science Inc., 2006.
[180] SABATIER J, AGRAWAL O P, TENREIRO MACHADO J A. Advances in Fractional Calculus [M]. Berlin: Springer-Verlag, 2007.
[181] WYSS W. The fractional diffusion equation [J]. J. Math. Phys., 1986, 27: 2782-2785.
[182] SCHNEIDER W R, WYSS W. Fractional diffusion and wave equations [J]. J. Math. Phys., 1989, 30: 134-144.
[183] HE J H. Approximate analytical solution for seepage flow with fractional derivatives in porousmedia [J]. Comput. Methods Appl. Mech. Engrg., 1998, 167: 57-68.
[184] DRAGANESCU G E. Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives [J]. J. Math. Phys., 2006, 47: 082902.
[185] ODIBAT Z M, MOMANI S. Application of variational iteration method to nonlinear differential equations of fractional order [J]. Internat. J. Nonlinear Sci. Numer. Simul., 2006, 7: 27-34.
[186] GORENFLO R, LUCHKO Y, MAINARDI F. Wright function as scale-invariant solutions of the diffusion-wave equation [J]. J. Comp. Appl. Math., 2000, 118: 175-191.
[187] SHAWAGFEH N T. Analytical approximate solutions for nonlinear fractional differential equations [J]. Appl. Math. Comput., 2002, 131: 517-529.
[188] GHORBANI A, SABERI-NADJAFI J. He's Homotopy perturbation method for calculating Adomian polynomials [J]. Int. J. NonlinearSci. Numer. Simul., 2007, 8: 229-232.
[189] LIU F, ANH V V, TURNER I, ZHUANG P. Time fractional advection-dispersion equation [J]. J. Appl. Math. Computing, 2003, 13: 233-245.
[190] MOMANI S, SHAWAGFEH N. Decomposition method for solving the fractional Riccati differential equation [J]. Appl. Math. Comput., 2006, 182: 1083-1092.
[191] MOMANI S, ODIBAT Z. Homotopy perturbation method for nonlinear partial differential equations off ractional order [J]. Phys. Lett. A, 2007, 365: 345-350.
[192] ODIBAT Z. Compact and noncompact structures for nonlinear fractional evolution equations [J]. Phys. Lett. A, 2008, 372: 1219-1227.
[193] RIEWE F. Nonconservative Lagrangian and Hamiltonian mechanics [J]. Phys. Rev. E. 1996, 53: 1890-1899.
[194] LIE S. Uber die integration durch bestimmte integrate von einer klasse linear partieller differentialgleichung [J]. Arch for Math., 1881, 6: 328-368; Gesamm elts Abhandlungen, Vol. Ⅲ, Teubner B. G., Leipzig, 1922, 492-523.
[195] LIE S. Classification und Integration von gewohnlichen Differential gleichungen zwischenxy, die eine Gruppe von Transformationen gestatten: Die nachstehende Arbeit erschien zum ersten Male im Frühling 1883 im norwegischen [J]. Archiv. Math. Ann., 1908, 32: 213-281.
[196] BATEMAN H. The conformal transformations of a spsce of four dimensions and their applications to geometrical optics [J]. Proc. London Math. Soc, 1909, 7: 70-92.
[197] CUNNINGHAM E. The principle of relativity in electrodynamics and an extension thereof [J]. Proc. London Math. Soc, 1909, 8: 77-97.
[198] NOTHER E. Invariante varlationsprobleme [J]. Nachr Koning. Gesell. Wissen. Gottingen, Meth. Phys., 1918, Kl: 235-257.
[199] BIRKHOF. Hydrodynamics-A study in logic, fact and simitude [M]. Princeton: Princeton University press, 1950.
[200] OVSIANNIKOV L V. Group properties of differential equations [M]. Moscow: Novosibirsk, 1962.
[201] OVSIANNIKOV L V. Group analysis of differential equations [M]. New York: Academic Press, 1982.
[202] WARNER F W. Foundations of differential manifolds and Lie groups [M]. Scott, Foresman & Glenview, 1971.
[203] BOOTHY W M. An introduction to differential manifolds and Reimamnian geometry [M]. New York: Academic press, 1975.
[204] THRIRING W E. A course in mathematical physics, Vol.1 [M]. New York: Springer Vorlag, 1978.
[205] MILLER W. Symmetry groups and their applications [M]. New York: Academic press, 1972.
[206] POULRYAGIN L S. Topological groups [M]. New York: Ind ed. Gordon and Breach, 1966.
[207] AMES W F. Some exact solutions for wave propagation in viscoelastic and electrical transmission [J]. Int. J. Nonlinear Mech., 1982, 17: 223-230.
[208] KUMEI S, BLUMAN G. When nonlinear differential equations are equivalent to linear differential equations [J]. SIAMJ. Appl. Math., 1982, 42: 1157-1173.
[209] QU C Z. Allowed transformations and symmetry classes of variable coefficient Burgers equations [J]. IMA J. Appl. Math., 1995, 54(3): 203-225.
[210] ZHAANOV R Z, FUSHCHYCH W L. Conditional symmetry and new classical solutions of the Yang-Mills equations [J]. J. Phys. A: Math. Gen., 1995, 28: 6253-6264.
[211] OLVER P J. Application of Lie groups to differential equations [M]. GTM, No. 107, New York: Springer-Verlag, 1986.
[212] BLUMAN G W, KUMEI S. Symmetries and Differential Equations [M]. Applied Mathematical Sciences, No. 81, New York: Springer-Verlag, 1989.
[213] BAUMAN G W. Symmetry analysis of differential equations with Mathematica [M]. New York: Springer-Verlag, 1998.
[214] CANTWELL B J. Introduction to symmetry analysis [M]. Cambridge: Cambridge University Press, 2002.
[215] 田畴.李群及其在微分方程中的应用[M].北京:科学出版社,2001.
[216] BLUMAN G W, COLE J D. The general similarity solution of the heat equation [J]. J. Math. Mech., 1969, 18: 1025-1042.
[217] BLUMAN G W, COLE J D. Similarity method for differential equation [J]. Appl. Math. Sci. No. 13, New York: Springer-Verlag, 1974.
[218] LEVI D, WINTERNITZ P. Non-classical symmetry reduction: example of the Boussinesq equation [J]. J. Phys. A: Math. Gen., 1989, 22: 2915-2924.
[219] OLVER P J. Evolution equations possessing infinitely many symmetries [J]. J. Math. Phys., 1977, 18: 1212-1215.
[220] OLVER P J. On the Hamiltonian structure of evolution equations [J]. Math. Proc. Camb. Phill Soc., 1980, 88: 71-88.
[221] FUCHAATEINER B, FOKAS A S. Symplectic structures, their Backlund transformations and hereditary symmetries [J]. Phys. D, 1981, 4: 47-66.
[222] BLUMAN G W, KUMEI S. On invariance properties of the wave equation [J]. J. Math. Phys., 1987, 28: 307-318.
[223] GANDARIAS M. New symmetries for a model of fast diffusion [J]. Phys. Lett. A, 2001, 286: 153-160.
[224] OLVER P J, ROSENAU P. The construction of special solutions to partial differential equations [J]. Phys. Lett. A, 1986, 114: 107-278.
[225] PUCCI E et al. On the weak symmetry groups of partial differential equations [J]. J. Math. Anay. Appl., 1992, 163: 588-598.
[226] 李翊神,朱国城.一个谱可变演化方程的对称[J].科学通报,1986,19:1449-1453.
[227] 李翊神,朱国城.可积方程新的对称,李代数及谱可变演化方程[J].中国科学A辑,1987,30:1243-1250.
[228] 李翊神,朱国城.“C可积”非线性方程的代数性质-(Ⅰ)Burgers方程和Calogero方程[J].中国科学A辑,1990,33:513-520.
[229] 田畴.Burgers方程的新的强对称,对称和李代数[J].中国科学A辑,1987,31:141-151.
[230] ZHU G C, CHEN H H. Symmetries and integrability of the cylindrical Korteweg-de Vries equation [J]. J. Math. Phys., 1986, 27: 100-103.
[231] LOU S Y. Twelve sets of symmetries of the Caudrey-Dodd-Gibbon-Sawada-Kotera equation [J]. Phys. Lett. A, 1993, 175: 23-26.
[232] SUN T, WANG W. Symmetries and their Lie algebra properties for the higher-order Burgers equations [J]. J. Phys. A: Math. Gen., 1989, 22: 3743-3751.
[233] 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]. J. Phys. A: Math. Gen., 1986, 19: 3713-3725.
[234] MA W X. K symmetries and tau symmetries of evolution equations and their Lie algebras [J]. J. Phys. A: Math. Gen., 1990, 23: 2707-2716.
[235] LOU S Y. Symmetries of the Kadomtsev-Petviashvili equation [J]. J. Phys. A: Math. Gen., 1993, 26: 4387-4394.
[236] LOU S Y, MA H C. Finite symmetry transformation groups and exact solutions of Lax integrable systems [J]. Chaos, Solitons & Fractals, 2006, 30: 804-821.
[237] ZHANG S L, QU C Z. Approximate generalized conditional symmetries for the perturbed nonlinear Diffusion-Convection equations [J]. Chinese Physics Letters, 2006, 23: 527-530.
[238] CLARKSON P A, KRUSKAL M D. New similarity reductions of the Boussinesq equation [J]. J. Math. Phys., 1989, 30: 2201-2213.
[239] LOU S Y. A note on the new similarity reductions of the Boussinesq equation [J]. Phys. Lett. A, 1990, 151: 133-135.
[240] CLARKSON P A. New similarity solutions for the modified Boussinesq equation [J]. J. Phys. A: Math. Gen., 1989, 22: 2355-2367.
[241] EUROPEAN H S. Nonclassical symmetry reductions of the Boussinesq equation [J]. Chaos, Solitons & Fractals, 1995, 5: 2261-2301.
[242] CLARKSON P A, EUROPEAN H S. Symmetry reductions of a generalized, cylindrical nonlinear Schr(?)dinger equation [J]. J. Phys. A: Math. Gen., 1993, 26: 133-150.
[243] CLARKSON P A, WINTERNITZ P. Nonclassical symmetry reductions for the Kadomtsev-Petviashvili equation [J]. Phys. D, 1991, 49: 257-272.
[244] CLARKSON P A, LUDLOW D K. Symmetry reductions, exact solutions, and Painleve analysis for a generalised Boussinesq equation [J]. J. Math. Anal. Appl., 1994, 186: 132-155.
[245] LOU S Y. Generalized Boussinesq equation and KdV equation-Painlev(?) properties, B(?)cklund transformations and Lax pairs [J]. Sci. China A, 1991, 34: 1098-1108.
[246] LOU S Y, RUAN H Y. Nonclassical analysis and Painlev(?) property for the Kupershmidt equations [J]. J. Phys. A: Math. Gen., 1993, 26: 4679-4688.
[247] QU C Z. Nonclassical symmetry reductions for the integrable super KdV equations [J]. Commun. Theor. Phys., 1995, 24: 177-184.
[248] ZHANG J Z, LIN J. Similarity reductions for the Khokhlov-Zabolotskayaequation [J]. Commun. Theor. Phys., 1995, 24: 69-72.
[249] 王烈衍.K(m,n)方程的对称约化[J].物理学报,2000,49:181-185.
[250] NUCCI M C, CLARKSON P A. The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitz-Hugh-Nagumo equation [J]. Phys. Lett. A, 1992, 164: 49-56.
[251] OLVER P J. Direct reduction and differential constraints [J]. Proc. R. Soc. Lond. A, 1994, 444: 509-523.
[252] PUCCI E. Similarity reductions of partial differential equations [J]. J. Phys. A: Math. Gen., 1992, 25: 2631-2640.
[253] ARRIGO P J et al. Nonclassical symmetry solutions and the methods of Bluman-Cole and Clarkson-Kruskal [J]. J. Math. Phys., 1993, 34: 4692-4703.
[254] 范恩贵,齐次平衡法、 Weiss-Tabor-Carnevale 法及 Clarkson-Kruskal 约化法之间的联系[J]. 物理学报, 2000, 49(8): 1409-1412.
[255] 张玉峰.孤立子方程求解与可积系统[D].大连:大连理工大学应用数学系,2002.
[256] LOU S Y et al. Similarity and conditional similarity reductions of a (2+1)-dimensional KdV equation via a direct method [J]. J. Math. Phys., 2000, 41: 8286-9303.
[257] LOU S Y, TANG X Y. Conditional similarity reduction approach: Jimbo-Miwa equation [J]. Chin. Phys., 2001, 10: 897-901.
[258] TANG X Y et al. Conditional similarity solutions of the Boussinesq equation [J]. Commun. Theor. Phys., 2001, 35: 399-404.
[259] TANG X Y, LIN J. Conditional similarity reductions of Jimbo-Miwa equation via the classical Lie group approach [J]. Commun. Theor. Phys., 2003, 39: 6-8.
[260] TANG X Y, QIAN X M, LIN J, LOU S Y. Conditional similarity reductions of the (2+1)-dimensional KdV Equation via the extended Lie group approach [J]. J. Phys. Soc. Jpn., 2004, 73: 1464-1475.
[261] ZHAANOV R Z, FUSHCHYCH W I. Conditional symmetry and new classical solutions of the Yang-Mills equations [J]. J. Phys. A: Math. Gen., 1995, 28: 6253-6264.
[262] 朝鲁,张鸿庆,唐立民.一个计算微分方程(组)对称群的Mathematica程序包及其应用[J].计算物理,1997,14(3):375-379.
[263] HEREMAN W. Review of symbolic software for Lie symmetry analysis [J]. Math. Comput. Modelling, 1997, 25: 115-132.
[264] NICOLETA B, JITSE N. On a new procedure for finding nonclassical symmetries [J]. Journal of Symbolic Computation, 2004, 38: 1523-1533.
[265] LIE S. Gesammelte Abhandlungen, vol 5 [M]. Leipzig: Teubner, 1924, pp 767-73.
[266] OLVER P J. Application of Lie groups to differential equations [M]. New York: Springer-Verlag, 1986.
[267] FUSHCHYCH W I, SHTELEN W M, SEROV N I. Symmetry Analysis and Exact Solutions of Nonlinear Equations of Mathematical Physics [M]. Dordrecht: Kluwer, (English transl.) 1993.
[268] OLVER P J, HEREDERO R H. Classification of invariant wave equations [J]. J. Math. Phys., 1996, 37: 6419-38.
[269] GONZ'ALEZ-L'OPEZ A, KAMRAN N, OLVER P J. Quasi-exactly solvable Lie algebras of first order differential operators in two complex variables [J]. J. Phys. A: Math. Gen., 1991, 24: 3995-4008.
[270] GONZ'ALEZ-L'OPEZ A, KAMRAN N, OLVER P J. Quasi-exact solvability [J]. Commun. Math. Phys., 1994, 159: 503-537.
[271] AKHATOV I Sh, GAZIZOV R K, IBRAGIMOV N Kh. Group classification of equation ofnonlinear filtration [J]. Dokl. AN SSSR, 1987, 293: 1033-1035.
[272] AKHATOV I Sh, GAZIZOV R K, IBRAGIMOV N Kh. Nonlocal symmetries: A heuristic approach [J]. Itogi Nauki i Tekhniki, Current problems in mathematics. Newest results, 1989, 34, 3-83 (Russian, translated in J. Soviet Math., 1991, 55, 1401-1450).
[273] TORRISI M, TRACINA R, VALENTI A. A group analysis approach for a nonlinear differ-ential system arising in diffusion phenomena [J]. J. Math. Phys., 1996, 37: 4758-67.
[274] TORRISI M, TRACINA R. Equivalence transformations and symmetries fora heat conduc-tion model [J]. Int. J. of Non-Linear Mechanics, 1998, 33: 473-87.
[275] IBRAGIMOV N H, TORRISI M, VALENTI A. Preliminary group classification of equations v_(tt) =f(x,vx)v_(xx)+g(x,v_x) [J]. J. Math. Phys., 1991, 32: 2988-2995.
[276] IBRAGIMOV N H, TORRISI M. A simple method for group analysis and its applications to a model of detonation [J]. J. Math. Phys., 1992, 33: 3931-3937.
[277] QU C Z. Allowed transformations and symmetry class of variable-coeffcient Burgers equa-tions [J]. IMA J. Appl. Math., 1995, 54 (3): 203-225.
[278] QU C Z. Preliminary group classification of equation u_t = f(x,u_x)u_(xx) + g(x,u_x) [J]. Acta Mathematica Scientia, 1997, 17(3): 255-261.
[279] KINGSTON J G, SOPHOCLEOUS C On form-preserving point transformations of partial differential equations [J]. J. Phys. A: Math. Gen., 1998. 31: 1597-1619.
[280] ZHDANOV R Z, LAHNO V I. Group classification of heat conductivity equations with a nonlinear source [J]. J. Phys. A.: Math. Gen., 1999, 32: 7405-7418.
[281] BASARAB-HORWATH P, LAHNO V, ZHDANOV R. The structure of Lie algebras and the classification problem for partial differential equations [J]. Acta Applicandae Mathematicae, 2001, 69: 43-94.
[282] BARONE A, ESPOSITO F, MAGEE C G, SCOTT A C, Theory and applications of the sine- Gordon equation [J]. Riv. Nuovo Cimento, 1971, 1: 227-267.
[283] KUMEI S. Invariance transformations, invariance group transformations and invariance groups of the sine-Gordon equations [J]. J. Math. Phys., 1975, 16: 2461-2468.
[284] PUCCI E, SALVATORI M C. Group properties of a class of semilinear hyperbolic equations [J]. Int. J. Non-Linear Mech., 1986, 21: 147-155.
[285] AMES W F, ADAMS E, LOHNER R J. Group properties of u_(tt) = (f(u)u_x)_x [J]. Internat. J. Non-Linear Mech., 1981, 16: 439-447.
[286] ORON A, ROSENAU P. Some symmetries of the nonlinear heat and wave equations [J]. Phys. Lett. A, 1986, 118: 172-176.
[287] SUHUBI E S, BAKKALOGLU A. Group properties and similarity solutions for a quasi-linear wave equation in the plane [J]. Int. J. Non-Linear Mech., 1991, 26: 567-584.
[288] CHIKWENDU S C. Non-linear wave propagation solutions by Fourier transform perturba-tion [J]. Int. J. Non-Linear Mech., 1981, 16: 117-128.
[289] PUCCI E. Group analysis of the equation u_(tt) + λu_(xx) = g(u, u_x) [J]. Riv. Mat. Univ. Parma, 1987, 4: 71-87.
[290] TORRISI M, VALENTI A. Group properties and invariant solutions for infinitesimal trans-formations of a nonlinear wave equation [J]. Int. J. Non-Linear Mech., 1985, 20: 135-144.
[291] DONATO A. Similarity analysis and nonlinear wave propagation [J]. Int. J. Non-Linear Mech., 1987, 22: 307-314.
[292] ARRIGO J. Group properties of u_(xx) - u~mu_(yy) = f(u) [J]. Int. J. Non-Linear Mech., 1991, 26: 619-629.
[293] KINGSTON J G, SOPHOCLEOUS C. Symmetries and form-preserving transformations of one- dimensional wave equations with dissipation [J]. Int. J. Non-Linear Mech., 2001, 36: 987-997.
[294] GANDARIAS M L, TORRISI M, VALENTI A. Symmetry classification and optimal systems of a non-linear wave equation [J]. Int. J. Non-Linear Mech., 2004, 39: 389-398.
[295] BLUMAN G, TEMUERCHAOLU. Comparing symmetries and conservation laws of nonlin-ear telegraph equations [J]. J. Math. Phys., 2005, 46: 073513.
[296] BLUMAN G, TEMUERCHAOLU. Conservation laws for nonlinear telegraph equations [J]. J. Math. Anal. Appl., 2005, 310: 459-476.
[297] BLUMAN G, TEMUERCHAOLU, SAHADEVAN R. Local and nonlocal symmetries for nonlinear telegraph equation [J]. J. Math. Phys., 2005, 46: 023505.
[298] LAHNO V, ZHDANOV R, MAGDA O. Group classificaion and exact solutions of nonlinear wave equations [J]. Acta. Appl. Math., 2006, 91: 253-313.
[299] HUANG D J, IVANOVA N M. Group analysis and exact solutions of a class of variable coeffcient nonlinear telegraph equations [J]. J. Math. Phys., 2007, 48: 073507.
[300]黄定江.非线性波、几何可积性与群分类[D].大连:大连理工大学应用数学系,2007.
[301]智红燕.非线性偏微分方程求解和对称约化[D].大连:大连理工大学应用数学系,2007.
[302] MA W X. Complexiton solution to the KdV equation [J]. Phys. Lett. A, 2002, 301: 35-44.
[303] MA W X, Maruon K. Complexiton solutions of the Toda lattice equation [J]. Phys. A, 2004,343: 219-237.
[304] MA W X. Complexiton solutions to integrable equations [J]. Nonlinear Analysis, 2005, 63:e2461-e2471.
[305] MA W X, YOU Y. Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions [J]. Trans. Am. Math. Soc, 2005, 357: 1753-1778.
[306] LOU S Y, HU H C, TANG X Y. Interactions among periodic waves and solitary waves of the (n+1)-dimensional sine-Gordon field [J]. Phys. Rev. E, 2005, 71: 036604.
[307] TAMIZHMANI K M et al. Similarity reductions and Painleve property of the coupled higher-dimensional Burgers' equation [J]. Int. J. Nonlinear Meth., 1991, 26: 427-438.
[308] HONG W, JUNG Y D. Auto-B(?)cklund transformation and analytic solutions for general variable-coeffcient KdV equation [J]. Phys. Lett. A, 1999, 257: 149-152.
[309] WANG M L, WANG Y M. A new B(?)cklund transformation and multi-soliton solutions to the KdV equation with general variable coeffcients [J]. Phys. Lett. A, 2001, 287: 211-216.
[310] FAN E G. Auto-B(?0cklund transformation and similarity reductions for general variable co-effcient KdV equations [J]. Phys. Lett. A, 2002, 294: 26-30.
[311] ZHAO X Q, TANG D B, WANG L M. New soliton-like solutions for KdV equation with variable coeffcient [J]. Phys. Lett. A, 2005, 346: 288-291.
[312] 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]. J. Phys. A, 1998, 31: 3337-3347.
[313] OCHMANN M, MAKAROV S. Representation of the absorption of nonlinears by fractional derivatives [J]. J. Amer. Acoust. Soc. 1993, 94: 3392-3399.
[314] INOKUTI M, SEKINE H, MURA T. General use of the Lagrange multiplier in nonlinear mathematical physics, in: Varivational Method in the Mechanics of Solids [M]. New York: Pergamon Press, 1978, 156-162.
[315] HE J H. Variational iteration methoda kind of nonlinear analytical technique: Some examples [J]. Internat. J. Nonlinear Mech., 1999, 34: 699-708.
[316] HE J H. Variational iteration method for autonomous ordinary differential systems [J]. Appl. Math. Comput., 2000, 114: 115-123.
[317] HE J H, WU X H. Variational iteration method: New development and applications [J]. Comput. Math. Appl., 2007, 54: 881-894.
[318] HE J H. Variational iteration method-some recent results and new interpretations [J]. J. Comput. Appl. Math., 2007, 207: 3-17.
[319] TATARI M, DEHGHAN M. On the convergence of He's variational iteration method [J]. J. Comput. Appl. Math., 2007, 207: 121-128.
[320] ADOMIAN G. Solving Frontier Problems of Physics: The Decomposition Method [M]. Boston: Kluwer Academic Publishers, 1994.
[321] ADOMIAN G. A review of the decomposition method in applied mathematics [J]. J. Math. Anal. Appl., 1988, 135: 501-544.
[322] CHERRULT Y. Convergence of Adomian's method [J]. Kybernetes, 1989, 18: 31-38.
[323] ABBAOUI K, CHERRUAULT Y. Convergence of Adomian's method applied to differential equations [J]. Comput. Math. Appl., 1994, 28: 103-109.
[324] ABBAOUI K, CHERRUAULT Y. New ideas for proving convergence of decomposition methods [J]. Comput. Math. Appl., 1995, 29: 103-108.
[325] HIMOUN N, ABBAOUI K, CHERRUAULT Y. New results of convergence of Adomian's method [J]. Kybernetes, 1999, 28: 423-429.
[326] HE J H. Homotopy perturbation techique [J]. Comput. Methods Appl. Mech. Eng., 1999, 178: 257-262.
[327] HE J H. A coupling method of a homotopy technique and a perturbation technique for non-linear problems [J]. Internat. J. Non-Linear Mech., 2000, 35: 37-43.
[328] HE J H. Perturbation Methods: Basic and Beyond [M]. Amsterdam: Elsevier, 2006.
[329] LIAO S J. The proposed homotopy analysis technique for the solution of nonlinear problems [D]. Shanghai: Jiao Tong University, 1992.
[330] LIAO S J. Comparison between the homotopy analysis method and homotopy perturbation method [J]. Appl. Math. Comput., 2005, 169: 1186-1194.
[331] LIAO S J, SU J, CHWANG A T. Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body [J]. Int. J. Heat. Mass. Transfer., 2006, 49: 2437-2445.
[332] LIAO S J, MAGYARI E. Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones [J]. Z. Angew. Math. Phys., 2006, 57: 777-792.
[333] LIAO S J. Series solutions of unsteady boundary-layer flows over a stretching flat plate [J]. Stud. Appl. Math., 2006, 117: 239-264.
[334] ZHU S P. A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield [J]. Anziam J., 2006, 47: 477-494.
[335] ABBASBANDY S. The application of homotopy analysis method to nonlinear equations arising in heat transfer [J]. Phys. Lett. A, 2006, 360: 109-113.
[336] ABBASBANDY S. The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation [J]. Phys. Lett. A, 2007, 361: 478-483.
[337] WANG Z, ZOU L, ZHANG H Q. Applying homotopy analysis method for solving differential-difference equation [J]. Phys. Lett. A, 2007, 369: 77-84.
[338] RIQUIER C. Les systemes d'(?)quations aux d(?)riv(?)es partielles [M]. Paris: Gauthier-Villars, 1910.
[339] JANET M. Sur les systemes d'(?)quations aux d(?)riv(?)es partielles [J]. J. de Marh., 1920; 3: 65-151.
[340] REID G J. Algorithms for reducing a system of PDEs to standard form determing the dimension of its solution space and calculationg its Taylor series solution [J]. Eur. J. Appl. Math., 1991, 2: 293-318.
[341] REID G J. Finding abstract Lie symmetry algebras of differential equations without integrating determining equations. Euro [J]. J. Appl. Math., 1991, 2: 319-340.
[342] REID G J, WITTKOPF A D, BOULTON A. Reduction of systems of nonlinear partial differential equations to simplified involutivc forms [J]. Eur. J. Appl. Math., 1996, 7: 604-635.
[343] WITTKOPF A D, REID G J. The Reduced Involutive Form package [CP]. First distributed as part of Maple 7, 2001.
[344] ZHOU W, JEFFREY D J, REID G J, SCHMITKE C, MCPHEE J. Implicit Reduced Involutive Forms and their application to engineering multibody aystems [J]. Lecture notes in Computer Science, 2005, 3519: 31-43.
[345] LISLE I G, REID G J. Symmetry Classification Using Noncommutative Invariant Differential [J]. Found. Comput. Math, 2006, 6: 353-386.
[346] AMES W F. Nonlinear Partial Differential Equations in Engineering [M]. New York: Academic, 1972.
[347] NIKITIN A G, POPOVYCH R O. Group classification of nonlinear Schroinger equations [J]. Ukr. Math. J., 2001, 53: 1053-1060.
[348] POPOVUCH R O, IVANOVA N M. New results on group classification of nonlinear diffusion-convection equations [J]. J. Phys. A, 2004, 37: 7547-7565.
[349] IVANOVA N M, SOPHOCLEOUS C. On the group classification of variable coefficient nonlinear diffusion-convection equations [J]. J. Comput. Appl. Math., 2006, 197: 322-344.
[350] CORLESS R M, GONNET G H, HARE DEG, JEFFREY D J, KNUTH D E. On The Lambert W Function [J]. Advances in Computational Mathematics, 1996, 5: 329-359.
[351] ANCO S C, BLUMAN G. Direct construction of conservation laws from field equations [J]. Phys. Rev. Lett., 1997, 78: 2869-2873.
[352] ANCO S C, BLUMAN G. Intergrating factors and first integrals of ordinary diferential equations [J]. European J. Appl. Math., 1998, 9: 245-259.
[353] ANCO S C, BLUMAN G. Direct construction method for conservation laws of partial differential equations, Part Ⅰ: Examples of conservation law classifications [J]. European, J. Appl. Math., 2002, 13: 545-566.
[354] ANCO S C, BLUMAN G. Direct construction method for conservation laws of partial differential equations, Part Ⅱ: General treatment [J]. European J. Appl. Math., 2002,13: 567-585.
[355] BLUMAN G W, ANCO S C. Symmetry and integration methods for differential equations [M]. New York: Springer-Verlag, 2002.
[356] CHEVIAKOV A F. GeM software package for computation of symmetries and conservation laws of differential equations [J]. Comput. Phys. Comm., 2007, 176: 48-61.
[357] BLUMAN G W, TEMUERCHAO. Conservation laws for nonliear telegraph equations [J]. J. Math. Anal. Appl., 2005, 310: 459-476.
[358] BLUMAN G W, TEMUERCHAO. New conservation laws are obtained directly from symmetry action on a known conservation law [J]. J. Math. Anal. Appl., 2006, 322: 233-250.
[359]XIE F D,ZHANG Y,LU Z S.Symbolic computation in non-linear evolution equation:application to(3+1)-dimensional Kadomtsev-Petviashvili equation[J].Chaos,& Solitons Eract.,2005,24:257.
[2] 高小山.数学机械化进展综述[J].数学进展,2001,30(5):385-404.
[3] 石赫.机械化数学引论[M].湖南:湖南教育出版社,1998.
[4] 吴文俊.几何定理机器证明的基本原理[M].北京:科学出版社,1984.
[5] WU W T. On the decision problem and the mechanization of theorem-proving in elementary geometry [J]. Scientia Sinica, 1978, 21: 159-172.
[6] WU W T. On zeros of algebraic equations: an application of Ritt principle [J]. Kexue Tongbao, 1986, 31: 1-5.
[7] RITT J F. Differential Algebra [M]. New York: American mathematical Society, 1950.
[8] WU W T. On the foundation of algebraic differential geometry [J]. Sys. Sci. & Math. Scis., 1989, 2(4): 289-312.
[9] GAO X S, ZHANG J Z, CHOU S C. Geometry Expert (in Chinese) [M]. Taiwan: Nine Chapters Pub, 1998.
[10] CHOU S C, GAO X S, ZHANG J Z. Machine Proofs in Geometry [M]. Singapore: World Scientific, 1994.
[11] CHOU S C. Automated reasoning in differential geometry and mechanics using the characteristic set method, Ⅰ. An improved version of Ritt-Wu's decomposition algorithm [J]. J. Auto. Reasoning, 1993, 10: 161-172.
[12] GAO X S, CHOU S C, ZHANG J Z. Automated production of traditional proofs for constructive geometry theorems [C]. Proceedings of ISSAC-93, 48.
[13] GAO X S. An Introduction to Wu's Method of Mechanical Geometry Theorem Proving [C]. IFIP Transaction on, Automated Reasoning, North-Holland, 1993, 13.
[14] WANG D M, GAO X S. Geometry theorems proved mechanically using Wu's method [J]. MM-Reseach Preprints, 1987(2).
[15] WANG D M, HU S. A mechanical proving system for constructible theorems in elementry geometry [J]. J. Sys. Sci. & Math. Scis., 1987, 7: 163-172.
[16] LIN D D, LIU Z J. Some results on theorem proving in finite geometry [C]. Proc. of IWMM, Inter Academic Pub, 1992.
[17] WU J Z, LIU Z J. Proceedings of 1st Asian Symposium on Computer Mathematics [C]. Beijing of China, 1995.
[18] WU J Z, LIU Z J. On first-order theorem proving using generalized odd-superpositions Ⅱ [J]. Science in China (Series E), 1996, 39: 50-61.
[19] CHOU S C, GAO X S. Ritt-Wu's decomposition algorithm and geometry theorem proving [M]. CADE'10, Stikel M E, Ed. Berlin: Springer-Verlag, 1990, 207-220.
[20] KAPUR D, WAN H K. Refutational proofs of geometry theorems via characteristic sets [C]. Proc. of ISSAC-90, Tokyo, Japan, 1990, 277-284.
[21] KO H. Geometry theorem proving by decomposition of quasi-algebraic sets: an application of the Ritt-Wu principle [J]. Artif. Intell., 1988, 37: 95-122.
[22] LI H B, CHENG M T. Proving theorems in elementary geometry with Clifford algebraic method [J]. Chinese Math. Progress, 1997, 26(4): 357-371.
[23] LI H B. Automated reasoning with differential forms [C]. Proc. of ASCM'92, 1992.
[24] LI H B, CHENG M T. Clifford algebraic reduction method for mechanical theorem proving in differential geometry [J]. J. Auto. Reasoning, 1998, 21: 1-21.
[25] WANG D M. Clifford algebraic calculus for geometric reasoning with applications to computer vision [M]. In: Automated deduction in geom., Wang, ed., LNAI 1360,115-140, Berlin: Springer-Verlag, 1997.
[26] SUN X D, WANG K, WU K. Solutions of Yang-Baxter equation with spectral parameters for a six-vertex model [J]. Acta Phys. Sinica (Chinese), 1995, 44: 1-8.
[27] SUN X D, WANG K, WU K. Classification of six-vertex-type solutions of the colored Yang-Baxter equation [J]. J. Math. Phys., 1995, 36(10): 6043-6063.
[28] WANG M L, LI Z B. Proc. Of the 1994 Beijing Symposium on nonlinear evolution equations and infinite dimensional dynamics systems [C]. Zhongshan University Press, 1995, 181.
[29] LI Z B, WANG M L. Travelling wave solutions to the two-dimensional KdV-Burgers equation [J]. J. Phys. A: Math. Gen., 1993, 26: 6027-6031.
[30] 李志斌,张善卿.非线性波方程准确孤立波解的符号计算[J].数学物理学报,1997,17:81-89.
[31] LI Z B, SHI H. Exact solutions for Belousov-Zhabotinski reaction-diffusion system [J]. Appl. Math. J. Chinese Univ. Ser. B, 1996, 11: 1-6.
[32] LI Z B. Proceeding of asian symposition on computer mathematics [C]. Lanzhou University, 1998, 153.
[33] LI Z B, LIU Y P. RATH: A Maple package for finding travelling solitary wave solutions to nonlinear evolution equations [J]. Compu. Phys. Commun., 2002, 148: 256-266.
[34] 范恩贵.可积系统与计算机代数[M].北京:科技出版社,2004.
[35] 闰振亚.非线性波与可积系统[D].大连:大连理工大学应用数学系,2002.
[36] 郑学东.一类非线性偏微分方程组的机械化求解与P-性质的机械化检验[D].大连:大连理工大学应用数学系,2002.
[37]朝鲁.微分方程(组)对称向量的吴-微分特征列算法及其应用[J].数学物理学报,1999,19(3):326-332.
[38]朝鲁.微分多项式系统的约化算法理论[J].数学进展,2003,32(2):208-220.
[39]陈勇.孤立子理论中的若干问题的研究及机械化实现[D].大连:大连理工大学应用数学系,2003.
[40]谢福鼎.Wu-Ritt消元法在偏微分代数方程中的应用[D].大连:大连理工大学应用数学系,2003.
[41]李彪.孤立子理论中若干精确求解方法的研究及应用[D].大连:大连理工大学应用数学系,2005.
[42]FAN E G, ZHANG H Q. Backlund transformation and exact solutions for Whitham-Broer-Kaup equations in shallow water [J]. Appl. Math. Mech., 1998, 19: 713-716.
[43]范恩贵,张鸿庆.非线性波动方程的孤波解[J].物理学报,1997,46:1254-1258.
[44]YAN Z Y, ZHANG H Q. New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water [J]. Phys. Lett. A., 2001, 285: 355-260.
[45]YAN Z Y, ZHANG H Q. On a new algorithm of constructing solitary wave solutions for systems of nonlinear evolution equations in mathematical physics [J]. Appl. Math. Mech., 2000, 21: 342-347.
[46]YAN Z Y, ZANG H Q. Multiple soliton-like and periodic-like solutions to the generalization of integrable (2+1)-dimensional dispersive long-wave equations [J]. J. Phys. Soc. Jpn., 2002, 71: 437-442.
[47]闰振亚,张鸿庆.New explicit travelling wave solutions for a class of nonlinear evolutionequations[J].物理学报,1999,48:1-5.
[48]KALTOFEN E. Challenges of Symbolic Computation: My Favorite Open Problems [J]. J. Symb. Comput., 2000, 29: 891-919.
[49]BUCHBERGER B, COLLINS G E, LOOS R. Computer Algebra-Symbolic and Algebraic Computation [M]. Beijing: World Publishing Corporation, 1988.
[50]RISH R H. The problem of integration in finite terms [J]. Trans. Ams. Math. Soc, 1969, 139: 167-189.
[51]BERLEKAMP E R. Factoring polynomials over large finite fields [J]. Math. Comput., 1970, 24: 713-735.
[52]BROWN W S. On the partition function of a finite set [J]. J. ACM., 1971, 18: 478-504.
[53]GOSPER R W. Decision Procedures for Indefinite Hypergeometric Summation [J]. Proc Nat. Acad. Sci. USA, 1978, 75: 40-42.
[54]LENSTRA A K. Factoring multivariate polynomials over algebraic number fields [J]. SIAM J. Comput., 1987, 16: 591-598.
[55]LENSTRA A K, LENSTRA H W JR, LOVASZ L. Factoring polynomials with rational coefficients [J]. Math. Ann., 1982, 261: 515-534.
[56] KALTOFEN E, TRAGER B. Computing with polynomials given by black boxes for their evaluations: greatest common divisors, factorization, separation of numerators and denominators [J]. J. Symb. Comput., 1990, 9: 301-320.
[57] SHOUP V. A new polynomial factorization algorithm and its implementation [J]. J. Symb. Comput., 1995, 20: 363-397.
[58] RUSSELL J S. Report on waves [C]. Fourteen meeting of the British association for the advancement of science, John Murray, London, 1844, 311-390.
[59] BOUSSINESQ J. Theorie des ondes et de remous qui se propagent [J]. J. Math. Pure Appl., 1972, 17: 55-108.
[60] KORTEWEG D J, DEVRIES G. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves [J]. Phil. Mag., 1895, 39: 422-433.
[61] FERMI E, PASTA J, ULAM S. Studies of nonlinear problems [C]. Los Alamos Scient. Lab., Rep. LA-1940, 1955.
[62] PERRING J K, SKYRME T H R. A model unified field equation [J]. Nucl. Phys., 1962, 31: 550-555.
[63] ZABUSKY N J, KRUSKAL M D. Interaction of solitons in a collisionless plasma and the recurrence of initial states [J]. Phys. Rev. Lett., 1965, 15: 240-243.
[64] ABLOWITZ M J, SEGUR H. Solitons and the inverse scattering transformation [M]. Philadelphia: SIAM, 1981.
[65] ABLOWITZ M J, CLARKSON P A. Solitons, nonlinear evolution equations and inverse scattering [M]. Cambridge: Cambridge University Press, 1991.
[66] NEWELL A C. Soliton in mathematics and physics [M]. Philadelphia: SIAM, 1985.
[67] FADDEEV L D, TAKHTAJAN L A. Hamiltonian method in the theory of solitons [M]. Berlin: Springer-Verlag, 1987.
[68] MATVEEV V B, SALLE M A. Darboux transformation and Solitons [M]. Berlin: Springer-Verlag, 1991.
[69] 郭柏灵,庞小峰.孤立子[M].北京:科学出版社,1987.
[70] 谷超豪等.孤立子理论与应用[M].杭州:浙江科技出版社,1990.
[71] 谷超豪等.孤立子理论中的Darboux变换及其几何应用[M].上海:上海科技出版社,1999.
[72] 李翊神.孤子与可积系统[M].上海:上海科技出版社,1999.
[73] HIROTA R. The direct method in soliton theory [M]. New York: Cambridge University Press, 2004.
[74] MIURA M R. B(?)cklund transformation [M]. Berlin: Springer-Verlag, 1978.
[75] WAHLQUIST H D, ESTABROOK F B. Backlund transformations for solitons of the Korteweg-de Vries equation [J]. Phys. Rev. Lett., 1973, 31: 1386-1390.
[76] WEISS J, TABOR M, CARNVALE G. The Painleve property for partial differential equations [J]. J. Math. Phys., 1983, 24: 522-526.
[77] WEISS J. The Painlev(?) property for partial differential equations, Ⅱ. B(?)cklund transformation, Lax pairs, and the Schwarzian derivative [J]. J. Math. Phys., 1983, 24: 1405-1413.
[78] DARBOUX G. Compts Rendus Hebdomadaires des Seances de l'Academie des Sciences [J]. Pairs, 1882, 94: 1456-1459.
[79] WADATI M et al. Simple derivation of B(?)cklund transformation from Riccati form of inverse method [J]. Prog. Theor. Phys., 1975, 53: 1652-1656.
[80] GU C H. A unified explicit form of B(?)cklund transformations for generalized hierarchies of KdV equations [J]. Lett. Math. Phys., 1986, 11; 325-335.
[81] GU C H, ZHOU Z X. On the Darboux matrices of B(?)cklund transformations for AKNS systems [J]. Lett. Math. Phys., 1987, 13: 179-187.
[82] HU X B, CLARKSON P A. B(?)cklund transformations and nonlinear superposition formulae of a differential-difference KdV equation [J]. J. Phys. A: Math. Gen., 1998, 31: 1405-1414.
[83] HU X B, ZHU Z N. A B(?)cklund transformation and nonlinear superposition formula for the Belov-Chaltikian lattice [J]. J. Phys. A: Math. Gen., 1998, 31: 4755-4761.
[84] MATVEEV V B, SKLYANIN E K. On B(?)cklund transformations for many-body systems [J]. J. Phys. A: Math. Gen., 1998, 31: 2241-2251.
[85] SKLYANIN E K. Canonicity of B(?)cklund transformation:r-matrix approach. I. L. D. Faddeev's Seminar on Mathematical Physics [J]. Amer. Math. Soc. Transt. Ser., 2000, 201: 277-282.
[86] HONE A N W et al. B(?)cklund transformations for many-body systems related to KdV [J]. J. Phys. A: Math. Gen., 1999, 32: L299-L306.
[87] CHOUDHURY A G, CHOWHURY A R. Canonical and B(?)cklund transformations for discrete integrable systems and classical r-matrix [J]. Phys. Lett. A, 2001, 280: 37-44.
[88] ZENG Y B et al. Canonical explicit Backlund transformations with spectrality for constrained flows of soliton hierarchies [J]. Phys. A, 2002, 303: 321-338.
[89] LOU S Y et al. Vortices, circumfluence, symmetry groups and Darboux transformations of the Euler equations [J]. arXiv: nlin.PS/0509039.
[90] GARDNER C S et al. Method for solving the Korteweg-deVries equation [J]. Phys. Rev. Lett., 1967, 19: 1095-1097.
[91] LAX P D. Integrals of nonlinear equations of evolution and solitary waves [J]. Commun. Pure Appl. Math., 1968, 21: 467-490.
[92] WADATI M. The modified Korteweg-de Vries equation [J]. J. Phys. Soc. Jpn., 1973, 32: 1289-1296.
[93] ABLOWITZ M J et al. Method for solving the sine-Gordon equation [J]. Phys. Rev. Lett., 1973, 30: 1262-1264.
[94] WAHLQUIST H D, Estabrook F B. Prolongation structures of nonlinear evolution equations [J]. J. Math. Phys., 1975, 16: 1-7.
[95] GUO H Y et al. On the prolongation structure of Ernst equation [J]. Commun. Theor. Phys., 1982, 1: 661-664.
[96] GUO H Y, WU K, WANG S K. Prolongation structure, B(?)cklund transformation and principal homogeneous Hilbert problem in general relativity [J]. Commun. Theor. Phys., 1983, 2: 883-898.
[97] GUO H Y, WU K, WANG S K. Inverse scattering transform and regular Riemann-Hilbert problem [J]. Commun. Theor. Phys., 1983, 2: 1169-1173.
[98] WU K, GUO H Y, WANG S K. Prolongation structures of nonlinear systems in higher dimensions [J]. Commun. Theor. Phys., 1983, 2: 1425-1437.
[99] CASE K M, KAC M. A discrete version of the inverse scattering problem [J]. J. Math. Phys., 1973, 14: 594-603.
[100] FLASCHKA H. On the Toda lattice. Ⅱ. Inverse-scattering solution [J]. Prog. Theor. Phys., 1974, 51: 703-716.
[101] ABLOWITZ M J, Ladik J F. Nonlinear differential-difference equations [J]. J. Math. Phys., 1975, 16: 598-603.
[102] 曹策问.孤立子与反散射[M].郑州:郑州大学出版社,1983.
[103] 李翊神,庄大尉.两类非线性演化方程的等价[J].北京大学学报,1983,2:107-118.
[104] 顾新身.相应的特征值问题有二重特征根的复KdV方程的一个求解实例[J].中国科学技术大学学报数学专辑,1983,5:82-89.
[105] ZENG Y B, MA W X, LIN R L. Integration of the soliton hierarchy with self-consistent sources [J]. J. Math. Phys., 2000, 41: 5453-5489.
[106] NING T K, CHEN D Y, ZHANG D J. Soliton-like solutions for a nonisospectral KdV hierarchy [J]. Chaos, Solitons & Fractals, 2004, 21: 395-401.
[107] ABLOWITZ M J et al. A connection between nonlinear evolution equations and ordinary differential equations of P-type [J]. J. Math. Phys., 1980, 21: 715-721.
[108] JIMBO M et al. Painleve test for the self-dual Yang-Mills equation [J]. Phys. Lett. A, 1982, 92: 59-60.
[109] FORDY A P, PICKERING A. Analysing negative resonances in the Painleve test [J]. Phys. Lett. A, 1991, 160: 347-354
[110] MUSETTE M, CONTE R. The two-singular-manifold method: Ⅰ. Modified Korteweg-de Vries and sine-Gordon equations [J]. J. Phys. A: Math. Gen., 1994, 27: 3895-3913.
[111] CONTE R, MUSETTE M, PICKERING A. The two-singular manifold method: Ⅱ. Classical Boussinesq system [J]. J. Phys. A: Math. Gen., 1995, 28: 179-187.
[112] 曾云波.与A_l相联系的半Toda方程的Lax对与B(?)cklund变换[J].数学学报,1992.35:454-459.
[113] 曾云波.递推算子与plinlev(?)性质[J].数学年刊,1991,12A:78-88.
[114] XU G Q, LI Z B. Symbolic computation of the Painleve test for nonlinear partial differential equations using Maple [J]. Comput. Phys. Commun., 2004, 161: 65-75.
[115]XU G Q, LI Z B. PDEPtest: a package for the Painleve test of nonlinear partial differential equations [J]. Appl. Math. Comput., 2005, 169: 1364-1379.
[116]LOU S Y. KdV extensions with Painlev(?) property [J]. J. Math. Phys., 1998, 39: 2112-2121.
[117]CHEN L L, LOU S Y. Painlev(?) analysis of a (2+1)-dimensional Burgers equation [J]. Commun. Theor. Phys., 1998, 29: 313-316.
[118]楼森岳.推广的Painlev(?)展开及KdV方程的非标准截断解[J].物理学报,1998,47:1937-1949.
[119] WHITHAM G B. Nonlinear dispersive waves [J]. SIAM J. Appl. Math., 1966,14: 956-958.
[120] KRUSKAL M D, ZABUSKY N J. Progress on the Fermi-Pasta-Ulam nonlinear string problem [J]. Princeton Plasma Physics Lab. Ann. Rep., MATT-Q-21, Princeton, N. J. 1963, 301-308.
[121] KRUSKAL M D. Korteweg-de Vries equation and generalizations: Ⅴ. Uniqueness and nonexistence of polynomial conservation laws [J]. J. Math. Phys., 1970, 11: 952-960.
[122]屠规彰,秦孟兆.非线性演化方程的不变群与守恒律-对称函数方法[J].中国科学A辑,1980.24:13-26.
[123]屠规彰.Boussinesq方程的B(?)cklund变换与守恒律[J].应用数学学报,1981,4:63-68.
[124]曾云波.Darboux transformations of families of AKNS equatiOIlS with additional terms[J].数学学报,1995,15:337-345.
[125]屠规彰,秦盂兆.Relationship between symmetries and conservation laws of nonlinear evolution equations[J].科学通报,1979,29:913-917.
[126]HU X B. Rational solutions of integrable equations via nonlinear superposition formulae [J]. J. Phys. A: Math. Gen., 1997, 30: 8225-8240.
[127]HU X B, Zhu Z N. Some new results on the Blaszak-Marciniak lattice: Backlund transformation and nonlinear superposition formula [J]. J. Math. Phys., 1998, 39: 4766-4772.
[128]HU X B, WU Y T. A new integrable differential-difference system and its explicit solutions [J]. J. Phys. A: Math. Gen., 1999, 32: 1515-1521.
[129]HU X B, TAM H W. Some new results on the Blaszak-Marciniak, 3-field and 4-field lattices [J]. Rep. Math. Phys., 2000, 46: 99-105.
[130]TAM H W et al. The Hirota-Satsuma coupled KdV equation and a coupled Ito system revisited [J]. J. Phys. Soc. Jpn., 2000, 69: 45-52.
[131]HU X B et al. Lax pairs and B(?)cklund transformations for a coupled Ramani equation and its related system [J]. Appl. Math. Lett., 2000, 13: 45-48.
[132]BOITI M et al. On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions [J]. Inv. Probl., 1986, 2: 271-280.
[133]RADHA R, LAKSHMANAN M. Singularity analysis and localized coherent structures in (2+1)-dimensional generalized Korteweg-de Vries equations [J]. J. Math. Phys., 1994, 35: 4746-4756.
[134] LOU S Y. Generalized dromion solutions of the (2+1)-dimensional KdV equation [J]. J. Phys. A: Math. Gen., 1995, 28: 7227-7232.
[135] ZHANG J F. Abundant dromion-like structures to the (2+1)-dimensional KdV equation [J]. Chin. Phys., 2000, 9: 1-4.
[136] LOU S Y, RUAN H Y. Revisitation of the localized excitations of the (2+1)-dimensional KdV equation [J]. J. Phys. A: Math. Gen., 2001, 34: 305-316.
[137] ROSENAU P, HYMAN M. Compactons: Solitons with finite wavelength [J]. Phys. Rev. Lett., 1993, 70: 564-567.
[138] ROSENAU P. Nonlinear dispersion and compact structures [J]. Phys. Rev. Lett., 1994, 73: 1737-1741.
[139] OLVER P J, ROSENAU P. Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support [J]. Phys. Rev. E, 1996, 53: 1900-1906.
[140] DUSUEL S et al. Predictability in systems with many characteristic times: The case of turbulence [J]. Phys. Rev. E, 1998, 57: 2320-2326.
[141] ISMAIL M S, TAHA T A. A numerical study of compactons [J]. Math. Computer Simulation, 1998, 47: 519-530.
[142] WAZWAZ A M. Exact special solutions with solitary patterns for the nonlinear dispersive K(m,n) equations [J]. Chaos, Solitons & Fractals, 2002, 13: 161-170.
[143] LOU S Y, WANG Q X. Painleve' integrability of two sets of nonlinear evolution equations with nonlinear dispersions [J]. Phys. Lett. A, 1999, 262: 344-349.
[144] LIU Q P, HU X B, ZHANG M X. Supersymmetric modified Korteweg-de Vries equation: bilinear approach [J]. Nonlinearity, 2005, 18: 1597-1603.
[145] LIU Q P, HU X B. Bilinearization of N = 1 supersymmetric Korteweg-de Vries equation revisited [J]. J. Phys. A: Math. Gen., 2005, 38: 6371-6378.
[146] MIURA R M. Korteweg-de Vries equation and generalizations. Ⅰ. A remarkable explicit nonlinear transformation [J]. J. Math. Phys., 1968, 9: 1202-1204.
[147] ABLOWITZ M J et al. A note on Miura's transformation [J]. J. Math. Phys., 1979, 20: 991-1003.
[148] WANG M L. Solitary wave solutions for variant Boussinesq equations [J]. Phys. Lett. A, 1995, 199: 169-172.
[149] LOU S Y, RUAN H Y, HUANG G X. Exact solitary waves in a converting fluid [J]. J. Phys. A: Math. Gen., 1991, 24: L587-L590.
[150] CONTE R, MUSETTE M. Link between solitary waves and projective Riccati equations [J]. J. Phys. A: Math. Gen., 1992, 25: 5609-5623.
[151] LIU S K, FU Z T, LIU S D, ZHAO Q. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations [J]. Phys. Lett. A, 2001. 289: 69-74.
[152] FU Z T, LIU S K, LIU S D, ZHAO Q. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations [J]. Phys. Lett. A, 2001, 290: 72-76.
[153] FAN E G. Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a coupled mKdV equation [J]. Phys. Lett. A, 2001, 282: 18-22.
[154] YAN C T. A simple transformation for nonlinear waves [J]. Phys. Lett. A, 1996, 224: 77-84.
[155] YAN Z Y, ZHANG H Q. New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics [J]. Phys. Lett. A, 1999, 252: 291-296.
[156] TIAN B, GAO Y T. Variable-coefficient balancing-act method and variable-coefficient KdV equation from fluid dynamics and plasma physics [J]. Eur. Phys. J. B, 2001, 22: 351-360.
[157] TIAN B, GAO Y T. Extension of generalized tanh method and soliton-like solutions for a set of nonlinear evolution equations [J]. Chaos, Solitons & Fractals, 1997, 8: 1651-1653.
[158] GAO Y T, TIAN B. New families of exact solutions to the integrable dispersive long wave equations in (2+1)-dimensional spaces [J]. J. Phys. A: Math. Gen., 1996, 29: 2895-2903.
[159] 昌卓生.计算微分方程对称与精确解的机械化算法及实现[D].大连:大连理工大学应用数学系,2004.
[160] BARNETT M P et al. Symbolic calculation in chemistry: Selected examples [J]. Int. J. Quantum Chem., 2004, 100: 80-104.
[161] 王琪.非线性微分方程求解和混沌同步[D].大连:大连理工大学应用数学系,2006.
[162] Lü Z S. A Burgers equation-based constructive method for solving nonlinear evolution equations [J]. Phys. Lett. A, 2006, 353: 158-160.
[163] 张鸿庆.弹性力学方程组一般解的统一理论[J].大连理工大学学报,1978,18:23-47.
[164] 张鸿庆,王震宇.胡海昌解的完备性和逼近性[J].科学通报,1986,30:342-344.
[165] 张鸿庆,吴方向.一类偏微分方程组的一般解及其在壳体理论中的应用[J].力学学报,1992,24:700-707.
[166] 张鸿庆,冯红.构造弹性力学位移函数的机械化算法[J].应用数学和力学,1995,16:315-322.
[167] 张鸿庆,杨光.变系数偏微分方程组一般解的构造[J].应用数学和力学,1991,12:135-139.
[168] ZHANG H Q, CHEN Y F. Proceeding of the 3rd ACM [C]. Lanzhou University Press, 1998, 147.
[169] ZHANG H Q. C-D integrable system and computer aided solver for differential equations [C]. Proceeding of the 5rd ACM, World Scientific Press, 2001, 221-226.
[170] 梅建琴.微分方程组精确解及其解的规模的机械化算法[D].大连:大连理工大学应用数学系,2006.
[171] 张鸿庆.数学机械化中的AC=BD模式[J].系统科学与数学,2008,28:1030-1039.
[172] 张鸿庆,丁琦.一类非线性偏微分方程组的解析解[J].应用数学和力学,2008,29:8621-8721.
[173] LACROIX S F. Traite du Calcul Differentiel et du Calcul Integral (Second edition) [M]. Paris: Courcier, 1812.
[174] OLDHAM K B, SPANIER J. The Fractional Calculus [M]. New York: Academic Press, 1974.
[175] PODLUBNY I. Fractional Differential Equations [M]. San Diego: Academic Press, 1999.
[176] CARPINTERI A, MAINARDIF. Fractals and Fractional Calculus in Continuum Mechanics [M]. Berlin: Springer-Verlag, 1997.
[177] HILFER E (Ed.). Applications of Fractional Calculus in Physics [M]. World Sci. Publishing, New York, 2000.
[178] ZASLAVSKY G M. Chaos and Fractional Dynamics, vol. 511 of Lect [M]. Notes in Phys., Oxford University Press, Oxford, 2005.
[179] KILBAS A A, SRIVASTAVA H M, TRUJILLO J J. Theory and Applications of Fractional Differential Equations [M]. New York: Elsevier Science Inc., 2006.
[180] SABATIER J, AGRAWAL O P, TENREIRO MACHADO J A. Advances in Fractional Calculus [M]. Berlin: Springer-Verlag, 2007.
[181] WYSS W. The fractional diffusion equation [J]. J. Math. Phys., 1986, 27: 2782-2785.
[182] SCHNEIDER W R, WYSS W. Fractional diffusion and wave equations [J]. J. Math. Phys., 1989, 30: 134-144.
[183] HE J H. Approximate analytical solution for seepage flow with fractional derivatives in porousmedia [J]. Comput. Methods Appl. Mech. Engrg., 1998, 167: 57-68.
[184] DRAGANESCU G E. Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives [J]. J. Math. Phys., 2006, 47: 082902.
[185] ODIBAT Z M, MOMANI S. Application of variational iteration method to nonlinear differential equations of fractional order [J]. Internat. J. Nonlinear Sci. Numer. Simul., 2006, 7: 27-34.
[186] GORENFLO R, LUCHKO Y, MAINARDI F. Wright function as scale-invariant solutions of the diffusion-wave equation [J]. J. Comp. Appl. Math., 2000, 118: 175-191.
[187] SHAWAGFEH N T. Analytical approximate solutions for nonlinear fractional differential equations [J]. Appl. Math. Comput., 2002, 131: 517-529.
[188] GHORBANI A, SABERI-NADJAFI J. He's Homotopy perturbation method for calculating Adomian polynomials [J]. Int. J. NonlinearSci. Numer. Simul., 2007, 8: 229-232.
[189] LIU F, ANH V V, TURNER I, ZHUANG P. Time fractional advection-dispersion equation [J]. J. Appl. Math. Computing, 2003, 13: 233-245.
[190] MOMANI S, SHAWAGFEH N. Decomposition method for solving the fractional Riccati differential equation [J]. Appl. Math. Comput., 2006, 182: 1083-1092.
[191] MOMANI S, ODIBAT Z. Homotopy perturbation method for nonlinear partial differential equations off ractional order [J]. Phys. Lett. A, 2007, 365: 345-350.
[192] ODIBAT Z. Compact and noncompact structures for nonlinear fractional evolution equations [J]. Phys. Lett. A, 2008, 372: 1219-1227.
[193] RIEWE F. Nonconservative Lagrangian and Hamiltonian mechanics [J]. Phys. Rev. E. 1996, 53: 1890-1899.
[194] LIE S. Uber die integration durch bestimmte integrate von einer klasse linear partieller differentialgleichung [J]. Arch for Math., 1881, 6: 328-368; Gesamm elts Abhandlungen, Vol. Ⅲ, Teubner B. G., Leipzig, 1922, 492-523.
[195] LIE S. Classification und Integration von gewohnlichen Differential gleichungen zwischenxy, die eine Gruppe von Transformationen gestatten: Die nachstehende Arbeit erschien zum ersten Male im Frühling 1883 im norwegischen [J]. Archiv. Math. Ann., 1908, 32: 213-281.
[196] BATEMAN H. The conformal transformations of a spsce of four dimensions and their applications to geometrical optics [J]. Proc. London Math. Soc, 1909, 7: 70-92.
[197] CUNNINGHAM E. The principle of relativity in electrodynamics and an extension thereof [J]. Proc. London Math. Soc, 1909, 8: 77-97.
[198] NOTHER E. Invariante varlationsprobleme [J]. Nachr Koning. Gesell. Wissen. Gottingen, Meth. Phys., 1918, Kl: 235-257.
[199] BIRKHOF. Hydrodynamics-A study in logic, fact and simitude [M]. Princeton: Princeton University press, 1950.
[200] OVSIANNIKOV L V. Group properties of differential equations [M]. Moscow: Novosibirsk, 1962.
[201] OVSIANNIKOV L V. Group analysis of differential equations [M]. New York: Academic Press, 1982.
[202] WARNER F W. Foundations of differential manifolds and Lie groups [M]. Scott, Foresman & Glenview, 1971.
[203] BOOTHY W M. An introduction to differential manifolds and Reimamnian geometry [M]. New York: Academic press, 1975.
[204] THRIRING W E. A course in mathematical physics, Vol.1 [M]. New York: Springer Vorlag, 1978.
[205] MILLER W. Symmetry groups and their applications [M]. New York: Academic press, 1972.
[206] POULRYAGIN L S. Topological groups [M]. New York: Ind ed. Gordon and Breach, 1966.
[207] AMES W F. Some exact solutions for wave propagation in viscoelastic and electrical transmission [J]. Int. J. Nonlinear Mech., 1982, 17: 223-230.
[208] KUMEI S, BLUMAN G. When nonlinear differential equations are equivalent to linear differential equations [J]. SIAMJ. Appl. Math., 1982, 42: 1157-1173.
[209] QU C Z. Allowed transformations and symmetry classes of variable coefficient Burgers equations [J]. IMA J. Appl. Math., 1995, 54(3): 203-225.
[210] ZHAANOV R Z, FUSHCHYCH W L. Conditional symmetry and new classical solutions of the Yang-Mills equations [J]. J. Phys. A: Math. Gen., 1995, 28: 6253-6264.
[211] OLVER P J. Application of Lie groups to differential equations [M]. GTM, No. 107, New York: Springer-Verlag, 1986.
[212] BLUMAN G W, KUMEI S. Symmetries and Differential Equations [M]. Applied Mathematical Sciences, No. 81, New York: Springer-Verlag, 1989.
[213] BAUMAN G W. Symmetry analysis of differential equations with Mathematica [M]. New York: Springer-Verlag, 1998.
[214] CANTWELL B J. Introduction to symmetry analysis [M]. Cambridge: Cambridge University Press, 2002.
[215] 田畴.李群及其在微分方程中的应用[M].北京:科学出版社,2001.
[216] BLUMAN G W, COLE J D. The general similarity solution of the heat equation [J]. J. Math. Mech., 1969, 18: 1025-1042.
[217] BLUMAN G W, COLE J D. Similarity method for differential equation [J]. Appl. Math. Sci. No. 13, New York: Springer-Verlag, 1974.
[218] LEVI D, WINTERNITZ P. Non-classical symmetry reduction: example of the Boussinesq equation [J]. J. Phys. A: Math. Gen., 1989, 22: 2915-2924.
[219] OLVER P J. Evolution equations possessing infinitely many symmetries [J]. J. Math. Phys., 1977, 18: 1212-1215.
[220] OLVER P J. On the Hamiltonian structure of evolution equations [J]. Math. Proc. Camb. Phill Soc., 1980, 88: 71-88.
[221] FUCHAATEINER B, FOKAS A S. Symplectic structures, their Backlund transformations and hereditary symmetries [J]. Phys. D, 1981, 4: 47-66.
[222] BLUMAN G W, KUMEI S. On invariance properties of the wave equation [J]. J. Math. Phys., 1987, 28: 307-318.
[223] GANDARIAS M. New symmetries for a model of fast diffusion [J]. Phys. Lett. A, 2001, 286: 153-160.
[224] OLVER P J, ROSENAU P. The construction of special solutions to partial differential equations [J]. Phys. Lett. A, 1986, 114: 107-278.
[225] PUCCI E et al. On the weak symmetry groups of partial differential equations [J]. J. Math. Anay. Appl., 1992, 163: 588-598.
[226] 李翊神,朱国城.一个谱可变演化方程的对称[J].科学通报,1986,19:1449-1453.
[227] 李翊神,朱国城.可积方程新的对称,李代数及谱可变演化方程[J].中国科学A辑,1987,30:1243-1250.
[228] 李翊神,朱国城.“C可积”非线性方程的代数性质-(Ⅰ)Burgers方程和Calogero方程[J].中国科学A辑,1990,33:513-520.
[229] 田畴.Burgers方程的新的强对称,对称和李代数[J].中国科学A辑,1987,31:141-151.
[230] ZHU G C, CHEN H H. Symmetries and integrability of the cylindrical Korteweg-de Vries equation [J]. J. Math. Phys., 1986, 27: 100-103.
[231] LOU S Y. Twelve sets of symmetries of the Caudrey-Dodd-Gibbon-Sawada-Kotera equation [J]. Phys. Lett. A, 1993, 175: 23-26.
[232] SUN T, WANG W. Symmetries and their Lie algebra properties for the higher-order Burgers equations [J]. J. Phys. A: Math. Gen., 1989, 22: 3743-3751.
[233] 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]. J. Phys. A: Math. Gen., 1986, 19: 3713-3725.
[234] MA W X. K symmetries and tau symmetries of evolution equations and their Lie algebras [J]. J. Phys. A: Math. Gen., 1990, 23: 2707-2716.
[235] LOU S Y. Symmetries of the Kadomtsev-Petviashvili equation [J]. J. Phys. A: Math. Gen., 1993, 26: 4387-4394.
[236] LOU S Y, MA H C. Finite symmetry transformation groups and exact solutions of Lax integrable systems [J]. Chaos, Solitons & Fractals, 2006, 30: 804-821.
[237] ZHANG S L, QU C Z. Approximate generalized conditional symmetries for the perturbed nonlinear Diffusion-Convection equations [J]. Chinese Physics Letters, 2006, 23: 527-530.
[238] CLARKSON P A, KRUSKAL M D. New similarity reductions of the Boussinesq equation [J]. J. Math. Phys., 1989, 30: 2201-2213.
[239] LOU S Y. A note on the new similarity reductions of the Boussinesq equation [J]. Phys. Lett. A, 1990, 151: 133-135.
[240] CLARKSON P A. New similarity solutions for the modified Boussinesq equation [J]. J. Phys. A: Math. Gen., 1989, 22: 2355-2367.
[241] EUROPEAN H S. Nonclassical symmetry reductions of the Boussinesq equation [J]. Chaos, Solitons & Fractals, 1995, 5: 2261-2301.
[242] CLARKSON P A, EUROPEAN H S. Symmetry reductions of a generalized, cylindrical nonlinear Schr(?)dinger equation [J]. J. Phys. A: Math. Gen., 1993, 26: 133-150.
[243] CLARKSON P A, WINTERNITZ P. Nonclassical symmetry reductions for the Kadomtsev-Petviashvili equation [J]. Phys. D, 1991, 49: 257-272.
[244] CLARKSON P A, LUDLOW D K. Symmetry reductions, exact solutions, and Painleve analysis for a generalised Boussinesq equation [J]. J. Math. Anal. Appl., 1994, 186: 132-155.
[245] LOU S Y. Generalized Boussinesq equation and KdV equation-Painlev(?) properties, B(?)cklund transformations and Lax pairs [J]. Sci. China A, 1991, 34: 1098-1108.
[246] LOU S Y, RUAN H Y. Nonclassical analysis and Painlev(?) property for the Kupershmidt equations [J]. J. Phys. A: Math. Gen., 1993, 26: 4679-4688.
[247] QU C Z. Nonclassical symmetry reductions for the integrable super KdV equations [J]. Commun. Theor. Phys., 1995, 24: 177-184.
[248] ZHANG J Z, LIN J. Similarity reductions for the Khokhlov-Zabolotskayaequation [J]. Commun. Theor. Phys., 1995, 24: 69-72.
[249] 王烈衍.K(m,n)方程的对称约化[J].物理学报,2000,49:181-185.
[250] NUCCI M C, CLARKSON P A. The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitz-Hugh-Nagumo equation [J]. Phys. Lett. A, 1992, 164: 49-56.
[251] OLVER P J. Direct reduction and differential constraints [J]. Proc. R. Soc. Lond. A, 1994, 444: 509-523.
[252] PUCCI E. Similarity reductions of partial differential equations [J]. J. Phys. A: Math. Gen., 1992, 25: 2631-2640.
[253] ARRIGO P J et al. Nonclassical symmetry solutions and the methods of Bluman-Cole and Clarkson-Kruskal [J]. J. Math. Phys., 1993, 34: 4692-4703.
[254] 范恩贵,齐次平衡法、 Weiss-Tabor-Carnevale 法及 Clarkson-Kruskal 约化法之间的联系[J]. 物理学报, 2000, 49(8): 1409-1412.
[255] 张玉峰.孤立子方程求解与可积系统[D].大连:大连理工大学应用数学系,2002.
[256] LOU S Y et al. Similarity and conditional similarity reductions of a (2+1)-dimensional KdV equation via a direct method [J]. J. Math. Phys., 2000, 41: 8286-9303.
[257] LOU S Y, TANG X Y. Conditional similarity reduction approach: Jimbo-Miwa equation [J]. Chin. Phys., 2001, 10: 897-901.
[258] TANG X Y et al. Conditional similarity solutions of the Boussinesq equation [J]. Commun. Theor. Phys., 2001, 35: 399-404.
[259] TANG X Y, LIN J. Conditional similarity reductions of Jimbo-Miwa equation via the classical Lie group approach [J]. Commun. Theor. Phys., 2003, 39: 6-8.
[260] TANG X Y, QIAN X M, LIN J, LOU S Y. Conditional similarity reductions of the (2+1)-dimensional KdV Equation via the extended Lie group approach [J]. J. Phys. Soc. Jpn., 2004, 73: 1464-1475.
[261] ZHAANOV R Z, FUSHCHYCH W I. Conditional symmetry and new classical solutions of the Yang-Mills equations [J]. J. Phys. A: Math. Gen., 1995, 28: 6253-6264.
[262] 朝鲁,张鸿庆,唐立民.一个计算微分方程(组)对称群的Mathematica程序包及其应用[J].计算物理,1997,14(3):375-379.
[263] HEREMAN W. Review of symbolic software for Lie symmetry analysis [J]. Math. Comput. Modelling, 1997, 25: 115-132.
[264] NICOLETA B, JITSE N. On a new procedure for finding nonclassical symmetries [J]. Journal of Symbolic Computation, 2004, 38: 1523-1533.
[265] LIE S. Gesammelte Abhandlungen, vol 5 [M]. Leipzig: Teubner, 1924, pp 767-73.
[266] OLVER P J. Application of Lie groups to differential equations [M]. New York: Springer-Verlag, 1986.
[267] FUSHCHYCH W I, SHTELEN W M, SEROV N I. Symmetry Analysis and Exact Solutions of Nonlinear Equations of Mathematical Physics [M]. Dordrecht: Kluwer, (English transl.) 1993.
[268] OLVER P J, HEREDERO R H. Classification of invariant wave equations [J]. J. Math. Phys., 1996, 37: 6419-38.
[269] GONZ'ALEZ-L'OPEZ A, KAMRAN N, OLVER P J. Quasi-exactly solvable Lie algebras of first order differential operators in two complex variables [J]. J. Phys. A: Math. Gen., 1991, 24: 3995-4008.
[270] GONZ'ALEZ-L'OPEZ A, KAMRAN N, OLVER P J. Quasi-exact solvability [J]. Commun. Math. Phys., 1994, 159: 503-537.
[271] AKHATOV I Sh, GAZIZOV R K, IBRAGIMOV N Kh. Group classification of equation ofnonlinear filtration [J]. Dokl. AN SSSR, 1987, 293: 1033-1035.
[272] AKHATOV I Sh, GAZIZOV R K, IBRAGIMOV N Kh. Nonlocal symmetries: A heuristic approach [J]. Itogi Nauki i Tekhniki, Current problems in mathematics. Newest results, 1989, 34, 3-83 (Russian, translated in J. Soviet Math., 1991, 55, 1401-1450).
[273] TORRISI M, TRACINA R, VALENTI A. A group analysis approach for a nonlinear differ-ential system arising in diffusion phenomena [J]. J. Math. Phys., 1996, 37: 4758-67.
[274] TORRISI M, TRACINA R. Equivalence transformations and symmetries fora heat conduc-tion model [J]. Int. J. of Non-Linear Mechanics, 1998, 33: 473-87.
[275] IBRAGIMOV N H, TORRISI M, VALENTI A. Preliminary group classification of equations v_(tt) =f(x,vx)v_(xx)+g(x,v_x) [J]. J. Math. Phys., 1991, 32: 2988-2995.
[276] IBRAGIMOV N H, TORRISI M. A simple method for group analysis and its applications to a model of detonation [J]. J. Math. Phys., 1992, 33: 3931-3937.
[277] QU C Z. Allowed transformations and symmetry class of variable-coeffcient Burgers equa-tions [J]. IMA J. Appl. Math., 1995, 54 (3): 203-225.
[278] QU C Z. Preliminary group classification of equation u_t = f(x,u_x)u_(xx) + g(x,u_x) [J]. Acta Mathematica Scientia, 1997, 17(3): 255-261.
[279] KINGSTON J G, SOPHOCLEOUS C On form-preserving point transformations of partial differential equations [J]. J. Phys. A: Math. Gen., 1998. 31: 1597-1619.
[280] ZHDANOV R Z, LAHNO V I. Group classification of heat conductivity equations with a nonlinear source [J]. J. Phys. A.: Math. Gen., 1999, 32: 7405-7418.
[281] BASARAB-HORWATH P, LAHNO V, ZHDANOV R. The structure of Lie algebras and the classification problem for partial differential equations [J]. Acta Applicandae Mathematicae, 2001, 69: 43-94.
[282] BARONE A, ESPOSITO F, MAGEE C G, SCOTT A C, Theory and applications of the sine- Gordon equation [J]. Riv. Nuovo Cimento, 1971, 1: 227-267.
[283] KUMEI S. Invariance transformations, invariance group transformations and invariance groups of the sine-Gordon equations [J]. J. Math. Phys., 1975, 16: 2461-2468.
[284] PUCCI E, SALVATORI M C. Group properties of a class of semilinear hyperbolic equations [J]. Int. J. Non-Linear Mech., 1986, 21: 147-155.
[285] AMES W F, ADAMS E, LOHNER R J. Group properties of u_(tt) = (f(u)u_x)_x [J]. Internat. J. Non-Linear Mech., 1981, 16: 439-447.
[286] ORON A, ROSENAU P. Some symmetries of the nonlinear heat and wave equations [J]. Phys. Lett. A, 1986, 118: 172-176.
[287] SUHUBI E S, BAKKALOGLU A. Group properties and similarity solutions for a quasi-linear wave equation in the plane [J]. Int. J. Non-Linear Mech., 1991, 26: 567-584.
[288] CHIKWENDU S C. Non-linear wave propagation solutions by Fourier transform perturba-tion [J]. Int. J. Non-Linear Mech., 1981, 16: 117-128.
[289] PUCCI E. Group analysis of the equation u_(tt) + λu_(xx) = g(u, u_x) [J]. Riv. Mat. Univ. Parma, 1987, 4: 71-87.
[290] TORRISI M, VALENTI A. Group properties and invariant solutions for infinitesimal trans-formations of a nonlinear wave equation [J]. Int. J. Non-Linear Mech., 1985, 20: 135-144.
[291] DONATO A. Similarity analysis and nonlinear wave propagation [J]. Int. J. Non-Linear Mech., 1987, 22: 307-314.
[292] ARRIGO J. Group properties of u_(xx) - u~mu_(yy) = f(u) [J]. Int. J. Non-Linear Mech., 1991, 26: 619-629.
[293] KINGSTON J G, SOPHOCLEOUS C. Symmetries and form-preserving transformations of one- dimensional wave equations with dissipation [J]. Int. J. Non-Linear Mech., 2001, 36: 987-997.
[294] GANDARIAS M L, TORRISI M, VALENTI A. Symmetry classification and optimal systems of a non-linear wave equation [J]. Int. J. Non-Linear Mech., 2004, 39: 389-398.
[295] BLUMAN G, TEMUERCHAOLU. Comparing symmetries and conservation laws of nonlin-ear telegraph equations [J]. J. Math. Phys., 2005, 46: 073513.
[296] BLUMAN G, TEMUERCHAOLU. Conservation laws for nonlinear telegraph equations [J]. J. Math. Anal. Appl., 2005, 310: 459-476.
[297] BLUMAN G, TEMUERCHAOLU, SAHADEVAN R. Local and nonlocal symmetries for nonlinear telegraph equation [J]. J. Math. Phys., 2005, 46: 023505.
[298] LAHNO V, ZHDANOV R, MAGDA O. Group classificaion and exact solutions of nonlinear wave equations [J]. Acta. Appl. Math., 2006, 91: 253-313.
[299] HUANG D J, IVANOVA N M. Group analysis and exact solutions of a class of variable coeffcient nonlinear telegraph equations [J]. J. Math. Phys., 2007, 48: 073507.
[300]黄定江.非线性波、几何可积性与群分类[D].大连:大连理工大学应用数学系,2007.
[301]智红燕.非线性偏微分方程求解和对称约化[D].大连:大连理工大学应用数学系,2007.
[302] MA W X. Complexiton solution to the KdV equation [J]. Phys. Lett. A, 2002, 301: 35-44.
[303] MA W X, Maruon K. Complexiton solutions of the Toda lattice equation [J]. Phys. A, 2004,343: 219-237.
[304] MA W X. Complexiton solutions to integrable equations [J]. Nonlinear Analysis, 2005, 63:e2461-e2471.
[305] MA W X, YOU Y. Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions [J]. Trans. Am. Math. Soc, 2005, 357: 1753-1778.
[306] LOU S Y, HU H C, TANG X Y. Interactions among periodic waves and solitary waves of the (n+1)-dimensional sine-Gordon field [J]. Phys. Rev. E, 2005, 71: 036604.
[307] TAMIZHMANI K M et al. Similarity reductions and Painleve property of the coupled higher-dimensional Burgers' equation [J]. Int. J. Nonlinear Meth., 1991, 26: 427-438.
[308] HONG W, JUNG Y D. Auto-B(?)cklund transformation and analytic solutions for general variable-coeffcient KdV equation [J]. Phys. Lett. A, 1999, 257: 149-152.
[309] WANG M L, WANG Y M. A new B(?)cklund transformation and multi-soliton solutions to the KdV equation with general variable coeffcients [J]. Phys. Lett. A, 2001, 287: 211-216.
[310] FAN E G. Auto-B(?0cklund transformation and similarity reductions for general variable co-effcient KdV equations [J]. Phys. Lett. A, 2002, 294: 26-30.
[311] ZHAO X Q, TANG D B, WANG L M. New soliton-like solutions for KdV equation with variable coeffcient [J]. Phys. Lett. A, 2005, 346: 288-291.
[312] 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]. J. Phys. A, 1998, 31: 3337-3347.
[313] OCHMANN M, MAKAROV S. Representation of the absorption of nonlinears by fractional derivatives [J]. J. Amer. Acoust. Soc. 1993, 94: 3392-3399.
[314] INOKUTI M, SEKINE H, MURA T. General use of the Lagrange multiplier in nonlinear mathematical physics, in: Varivational Method in the Mechanics of Solids [M]. New York: Pergamon Press, 1978, 156-162.
[315] HE J H. Variational iteration methoda kind of nonlinear analytical technique: Some examples [J]. Internat. J. Nonlinear Mech., 1999, 34: 699-708.
[316] HE J H. Variational iteration method for autonomous ordinary differential systems [J]. Appl. Math. Comput., 2000, 114: 115-123.
[317] HE J H, WU X H. Variational iteration method: New development and applications [J]. Comput. Math. Appl., 2007, 54: 881-894.
[318] HE J H. Variational iteration method-some recent results and new interpretations [J]. J. Comput. Appl. Math., 2007, 207: 3-17.
[319] TATARI M, DEHGHAN M. On the convergence of He's variational iteration method [J]. J. Comput. Appl. Math., 2007, 207: 121-128.
[320] ADOMIAN G. Solving Frontier Problems of Physics: The Decomposition Method [M]. Boston: Kluwer Academic Publishers, 1994.
[321] ADOMIAN G. A review of the decomposition method in applied mathematics [J]. J. Math. Anal. Appl., 1988, 135: 501-544.
[322] CHERRULT Y. Convergence of Adomian's method [J]. Kybernetes, 1989, 18: 31-38.
[323] ABBAOUI K, CHERRUAULT Y. Convergence of Adomian's method applied to differential equations [J]. Comput. Math. Appl., 1994, 28: 103-109.
[324] ABBAOUI K, CHERRUAULT Y. New ideas for proving convergence of decomposition methods [J]. Comput. Math. Appl., 1995, 29: 103-108.
[325] HIMOUN N, ABBAOUI K, CHERRUAULT Y. New results of convergence of Adomian's method [J]. Kybernetes, 1999, 28: 423-429.
[326] HE J H. Homotopy perturbation techique [J]. Comput. Methods Appl. Mech. Eng., 1999, 178: 257-262.
[327] HE J H. A coupling method of a homotopy technique and a perturbation technique for non-linear problems [J]. Internat. J. Non-Linear Mech., 2000, 35: 37-43.
[328] HE J H. Perturbation Methods: Basic and Beyond [M]. Amsterdam: Elsevier, 2006.
[329] LIAO S J. The proposed homotopy analysis technique for the solution of nonlinear problems [D]. Shanghai: Jiao Tong University, 1992.
[330] LIAO S J. Comparison between the homotopy analysis method and homotopy perturbation method [J]. Appl. Math. Comput., 2005, 169: 1186-1194.
[331] LIAO S J, SU J, CHWANG A T. Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body [J]. Int. J. Heat. Mass. Transfer., 2006, 49: 2437-2445.
[332] LIAO S J, MAGYARI E. Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones [J]. Z. Angew. Math. Phys., 2006, 57: 777-792.
[333] LIAO S J. Series solutions of unsteady boundary-layer flows over a stretching flat plate [J]. Stud. Appl. Math., 2006, 117: 239-264.
[334] ZHU S P. A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield [J]. Anziam J., 2006, 47: 477-494.
[335] ABBASBANDY S. The application of homotopy analysis method to nonlinear equations arising in heat transfer [J]. Phys. Lett. A, 2006, 360: 109-113.
[336] ABBASBANDY S. The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation [J]. Phys. Lett. A, 2007, 361: 478-483.
[337] WANG Z, ZOU L, ZHANG H Q. Applying homotopy analysis method for solving differential-difference equation [J]. Phys. Lett. A, 2007, 369: 77-84.
[338] RIQUIER C. Les systemes d'(?)quations aux d(?)riv(?)es partielles [M]. Paris: Gauthier-Villars, 1910.
[339] JANET M. Sur les systemes d'(?)quations aux d(?)riv(?)es partielles [J]. J. de Marh., 1920; 3: 65-151.
[340] REID G J. Algorithms for reducing a system of PDEs to standard form determing the dimension of its solution space and calculationg its Taylor series solution [J]. Eur. J. Appl. Math., 1991, 2: 293-318.
[341] REID G J. Finding abstract Lie symmetry algebras of differential equations without integrating determining equations. Euro [J]. J. Appl. Math., 1991, 2: 319-340.
[342] REID G J, WITTKOPF A D, BOULTON A. Reduction of systems of nonlinear partial differential equations to simplified involutivc forms [J]. Eur. J. Appl. Math., 1996, 7: 604-635.
[343] WITTKOPF A D, REID G J. The Reduced Involutive Form package [CP]. First distributed as part of Maple 7, 2001.
[344] ZHOU W, JEFFREY D J, REID G J, SCHMITKE C, MCPHEE J. Implicit Reduced Involutive Forms and their application to engineering multibody aystems [J]. Lecture notes in Computer Science, 2005, 3519: 31-43.
[345] LISLE I G, REID G J. Symmetry Classification Using Noncommutative Invariant Differential [J]. Found. Comput. Math, 2006, 6: 353-386.
[346] AMES W F. Nonlinear Partial Differential Equations in Engineering [M]. New York: Academic, 1972.
[347] NIKITIN A G, POPOVYCH R O. Group classification of nonlinear Schroinger equations [J]. Ukr. Math. J., 2001, 53: 1053-1060.
[348] POPOVUCH R O, IVANOVA N M. New results on group classification of nonlinear diffusion-convection equations [J]. J. Phys. A, 2004, 37: 7547-7565.
[349] IVANOVA N M, SOPHOCLEOUS C. On the group classification of variable coefficient nonlinear diffusion-convection equations [J]. J. Comput. Appl. Math., 2006, 197: 322-344.
[350] CORLESS R M, GONNET G H, HARE DEG, JEFFREY D J, KNUTH D E. On The Lambert W Function [J]. Advances in Computational Mathematics, 1996, 5: 329-359.
[351] ANCO S C, BLUMAN G. Direct construction of conservation laws from field equations [J]. Phys. Rev. Lett., 1997, 78: 2869-2873.
[352] ANCO S C, BLUMAN G. Intergrating factors and first integrals of ordinary diferential equations [J]. European J. Appl. Math., 1998, 9: 245-259.
[353] ANCO S C, BLUMAN G. Direct construction method for conservation laws of partial differential equations, Part Ⅰ: Examples of conservation law classifications [J]. European, J. Appl. Math., 2002, 13: 545-566.
[354] ANCO S C, BLUMAN G. Direct construction method for conservation laws of partial differential equations, Part Ⅱ: General treatment [J]. European J. Appl. Math., 2002,13: 567-585.
[355] BLUMAN G W, ANCO S C. Symmetry and integration methods for differential equations [M]. New York: Springer-Verlag, 2002.
[356] CHEVIAKOV A F. GeM software package for computation of symmetries and conservation laws of differential equations [J]. Comput. Phys. Comm., 2007, 176: 48-61.
[357] BLUMAN G W, TEMUERCHAO. Conservation laws for nonliear telegraph equations [J]. J. Math. Anal. Appl., 2005, 310: 459-476.
[358] BLUMAN G W, TEMUERCHAO. New conservation laws are obtained directly from symmetry action on a known conservation law [J]. J. Math. Anal. Appl., 2006, 322: 233-250.
[359]XIE F D,ZHANG Y,LU Z S.Symbolic computation in non-linear evolution equation:application to(3+1)-dimensional Kadomtsev-Petviashvili equation[J].Chaos,& Solitons Eract.,2005,24:257.