非线性数学物理中的对称和达布变换
|
摘要
对称方法及达布变换是求解非线性数学物理方程精确解的强有力工具。首先我们学习和研究了对称群和约化的理论和方法,其中包括:经典李群方法、非经典李群方法、CK直接约化法及改进的CK方法。以这些方法为基础,研究了一个非等谱的KP方程。其次,微分方程的李对称群理论的重要应用之一是其群不变解的构造和分类,这就涉及到优化系统的问题。在这方面,我们研究了一般的含参数的(2+1)维的非线性Klein-Gordon方程。最后,学习和研究了达布变换的相关知识,构造了修正KP方程的一种二元达布变换。
论文安排如下:
第一章绪论:简要介绍了求解非线性数学物理方程精确解的几种方法。
第二章对称约化:介绍了对称和相似约化的知识,分别利用CK直接约化法、经典李群方法和改进的CK方法研究了一个非等谱的KP方程的对称及其相似约化,从中可以看出这三种方法的相似和不同之处。
第三章对称优化:利用对称优化方法构造群不变解及其分类,研究了一个一般的(2+1)维的非线性Klein-Gordon方程,根据其李代数维数的不同将其分成了三种类型,分别构造了其群不变解及相应的一维优化系统。
第四章二元达布变换:介绍了达布变换的相关知识,构造了修正KP方程的一种二元达布变换,并且给出了它的n次迭代形式。
The symmetry method and Darboux transformation are the powerful tools to solve the equations in nonlinear mathematical physics. Firstly, we studyed the theories and methods of the symmetry group and reductions, including classical Lie group method, nonclassical Lie group method, CK direct method and the modified CK method. Based on these methods, we investigated a nonispectral KP equation. Secondly, one of the main applications of Lie theory of symmetry groups for differential equations is the construction and classification of group invariant solutions, which is in relation to the optimal system. In this respect, we considered a general (2+1) dimensional nonlinear Klein-Gordon equation. Lastly, after studying the related knowledge of Darboux transformation, we constructed a binary Darboux transformation for the modified KP equation.
The thesis is arranged as follows:
Chapter 1 Introduction. Several methods for obtaining exact solutions of the nonlinear mathematical physics equations are introduced in this chapter.
Chapter 2 Symmetry and Reductions. This chapter introduces some relevant knowledge about the symmetry and Reductions. We make use of CK direct method, the classical Lie group method and the modified CK method to investigate the symmetry of a nonispectral KP equation, respectively. From the results, one can see the similarities and differences among the above three methods.
Chapter 3 Symmetry Optimization. One can make use of symmetry optimization to construct group invariant solutions and its classification. In this chapter, we investigate a general (2+1) dimensional nonlinear Klein-Gordon equation. In accordance with the dimensions of its Lie algebra, it is divided into three types. For every type, we construct its group invariant solutions and the corresponding one dimentional optimal system, respectively.
Chapter 4 Binary Darboux Transformation. This chapter firstly introduces the relevant knowledge about Darboux transformation. Then we construct a binary Darboux transformation for the modified KP equation and give out its N-step iteration.
论文安排如下:
第一章绪论:简要介绍了求解非线性数学物理方程精确解的几种方法。
第二章对称约化:介绍了对称和相似约化的知识,分别利用CK直接约化法、经典李群方法和改进的CK方法研究了一个非等谱的KP方程的对称及其相似约化,从中可以看出这三种方法的相似和不同之处。
第三章对称优化:利用对称优化方法构造群不变解及其分类,研究了一个一般的(2+1)维的非线性Klein-Gordon方程,根据其李代数维数的不同将其分成了三种类型,分别构造了其群不变解及相应的一维优化系统。
第四章二元达布变换:介绍了达布变换的相关知识,构造了修正KP方程的一种二元达布变换,并且给出了它的n次迭代形式。
The symmetry method and Darboux transformation are the powerful tools to solve the equations in nonlinear mathematical physics. Firstly, we studyed the theories and methods of the symmetry group and reductions, including classical Lie group method, nonclassical Lie group method, CK direct method and the modified CK method. Based on these methods, we investigated a nonispectral KP equation. Secondly, one of the main applications of Lie theory of symmetry groups for differential equations is the construction and classification of group invariant solutions, which is in relation to the optimal system. In this respect, we considered a general (2+1) dimensional nonlinear Klein-Gordon equation. Lastly, after studying the related knowledge of Darboux transformation, we constructed a binary Darboux transformation for the modified KP equation.
The thesis is arranged as follows:
Chapter 1 Introduction. Several methods for obtaining exact solutions of the nonlinear mathematical physics equations are introduced in this chapter.
Chapter 2 Symmetry and Reductions. This chapter introduces some relevant knowledge about the symmetry and Reductions. We make use of CK direct method, the classical Lie group method and the modified CK method to investigate the symmetry of a nonispectral KP equation, respectively. From the results, one can see the similarities and differences among the above three methods.
Chapter 3 Symmetry Optimization. One can make use of symmetry optimization to construct group invariant solutions and its classification. In this chapter, we investigate a general (2+1) dimensional nonlinear Klein-Gordon equation. In accordance with the dimensions of its Lie algebra, it is divided into three types. For every type, we construct its group invariant solutions and the corresponding one dimentional optimal system, respectively.
Chapter 4 Binary Darboux Transformation. This chapter firstly introduces the relevant knowledge about Darboux transformation. Then we construct a binary Darboux transformation for the modified KP equation and give out its N-step iteration.
引文
[1]Garder C S,Greene J M,Kruskal R M,Miura R M.The Korteweg-de Vries equation and generations VI.Method for exact solutions.Comm.Pure.Appl.1974,27:97-133.
[2]Gardner C S,Green J M,Kruskal M D,Miura R M.Method for solving the Korteweg-de Vries equation.Phys.Rev.Lett.1976,19:1059-1095.
[3]Gu C H,Hu H S,Zhou Z X.Darboux Transformation in soliton theory and its Geometric Applications.Shanghai Science and Technical Publishers,1999.谷超豪,胡和生,周子翔.孤子理论中的达布变换及其几何应用.上海科学技术出版社,2001.
[4]Rogers C,Schief W K.B(a|¨)cklund and Darboux transformation geometry and modern applications in soliton theory.Cambridge University Press,Cambridge Texts in Applied Mathematics,2002.
[5]Wadati M,Sanuki H,Konno K.Simple derivation of B(a|¨)cklund transformation from Riccati form of inverse method.Prog.Theor.Phys.1975,53:419-436.
[6]Matveev V B,Salle M A.Darboux Transformation and Solitons.Springer,Berlin,1991.
[7]Miura M R.B(a|¨)cklund Transformation.Brelin,Springer-Verlag,1978.
[8]Hirota R.Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons.Phys.Rev.Lett.1971,27:1192-1194.
[9]Hirota R.The direct method in soliton theory.Edited and tranlated by Nagai A,Nimmo J,Gilson C.Cambridge Tracts in Mathematics.No.155,2004.
[10]Matsukdaira J,Satsuma J,Strampp W.Soliton equations expressed by trilinear forms and their solutions.Phys.Lett.A.1990,147:467-471.
[11]Bluman G W,Kumei S.Symmetries and Differential Euqation.Appl.Math.Sci.81,Springer,Berlin,1989.
[12]Olver P J.Application of Lie Groups to Differential Equation.2nd ed.Graduate Texts Math,107,Springer,New York,1993.
[13]Clarkson P A,Kruskal M D.New similarity solutions of the Boussinesq equations.J.Math.Phys.1989,30:2201-2213.
[14]Clarkson P A.Nonclassical symmetry reductions of the Boussinesq equation.Chaos,Soliton and Fractals,1995,5:2261-2301.
[15]Clarkson P A,Mansfield E L.Symmetry reductions and exact solutions of shallow water wave equations. Acta Appl. Math. 1995, 39: 245-276.
[16] Lou S Y. A note on the new similarity reductions of the Boussinesq equation. Phys. Lett. A, 1990, 151: 133-135.
[17] Lou S Y. Similarity solutions of the Kadomtsev-Petviashvili equation. J. Phys. A: Math. Gen., 1990, 23: 649-654;
Lou S Y. Symmetries and similarity reductions of the Dym equayion. Physica Scripta, 1996, 54: 428-435;
Lou S Y, Ruang H Y, Chen D F, Chen W Z. Similarity reductions of the KP equation by a direct method. J. Phys. A, 1991, 24: 1455-1467;
Lou S Y, Tang X Y, Lin J. Similarity and conditional similarity reductions of a (2+1)- dimensional KdV equation via a direct method. J. Math. Phys., 2000, 41: 8286-8303.
[18] Pucci E. J. Potential symmetries and solutions by reduction of partial differential equations. Phys. A: Math. Gen., 1993, 26: 681-690.
[19] Saccomandi G. Potential symmetries and direct reduction methods of order two. J. Phys. A. 1997, 30: 2211-2217.
[20] Lou S Y, Ni G J. The relations among a special type of solutions in some (D+1) dimensional nonlinear equations. J. Math. Phys. 1989, 30: 1614-1630.
[21] Hereman W, Banerjee P P, Korpel A et al. Exact solitary wave solutions of non-linear evolution and wave equations using a direct algebraic method. J. Phys. A: Math. Gen. 1986, 19: 607-628.
[22] Dolye P W. Separation of variables for scalar evolution equations in one space dimension. J. Phys. A. 1996, 29: 7581-7595.
[23] Dolye P W, Vassiliou P J. Separation of variables for the 1-dimensional non-linear diffusion equation. Int. J. Nonlear Mech. 1998, 33: 315-326.
[24] Alexeyev N N. A multidimensional superposition principle: classical solitons III. Phys. Lett. A. 2005, 335: 197-206.
[25] Chen C L, Tang X Y, Lou S Y. Solutions of (2+l)-dimensional dispersive long wave equation. Phys. Rev. E. 2002, 66: 036605.
[26] Conte R. Invariant Painleve analysis of partial differential equations. Phys. Lett.A. 1989, 140: 383-390.
[27] Lou S Y. Extended Painleve expansion, nonstand truncation and special reductions of nonlinear evolution equations. Z Naturforsch a. 1998, 53: 251-258.
[28] Picking A. A new truncation in Painleve analysis. J. Phys. A: Math. Gen. 1993, 26: 4395- 4406;Picking A. The singular manifold method revisted. J. Math. Phys. 1996, 37: 1894-1927.
[29] Weiss J, Tabor M, Carnevale G. The Painleve property for partial differential equations. J. Math. Phys. 1983, 24: 522-526.
[30] Chen Y, Li B. New exact travelling wave solutions for generalized Zakharov-Kuzentsov equations using general projective Riccati equation method. Commun. Theor. Phys. 2004, 41: 1-6.
[31] Chen Y, Li Y S. The constraint of the Kadomtsev-Petviashvili equation and its special solutions. Phys. Lett. A. 1991, 157: 22-26.
[32] Fan E G. Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A. 2000, 277: 212-218.
[33] Huang G X, Lou S Y, Dai X X. Exact and explicit solitary wave solutions to a model equation for water waves. Phys. Lett. A. 1989, 139: 373-374.
[34] Lan H B, Wang K L. Exact solutions for two nonlinear equations. J. Phys. A: Math. Gen. 1990, 23: 4097-4105.
[35] Zakharov V E, Shabat A B. A scheme for integrating nonlinear equations of mathematical physics by the method of the inverse scattering transform. Funct. Anal. Appl. 1974, 8(3): 43-53;A scheme for integrating nonlinear equations of mathematical physics by the method of the inverse scattering transform. II. Funct. Anal. Appl. 1979, 13(3): 13-23.
[36] Zenchuk A I. A unified dressing method for C- and S-integrable hierarchies; the particular example of a (3 + l)-dimensional n-wave equation. J. Phys. A: Math. Gen. 2004, 37:6557-6571.
[37] Gardner C S et al. Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 1967, 19: 1059-1097.
[38] Lax P D. Integrals of nonlinear equations of evolution and soliton waves. Commun, Pure Appl. Math. 1968, 21: 467-490.
[39] Zabusky N J, Kruskal M D. Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 1965, 15: 240-243.
[40] Ablowitz M J et al. Method for solving the sine-Gordon equation. Phys. Rev. Lett. 1973, 30: 1262-1264.
[41] Gu C H. On the Backlund transformation for the generalized hierarchies of compound MKdV-SG equation. Lett. Math. Phys. 1986, 11: 31-41.
[42] Gu C H. On the Darboux form of Backlund transformations. In Integrable System, Nankai Lectures on Math. Phys. World Scientic, Singapore, 1989, 162-168.
[43] Gu C H. Darboux transformations for a class of integarable systems in n variables. Analyse, Varietes et Physique, Proc. of Colloque International enl'honneur d'Yvonne Choquet- Bruhat, Kluwer, 1992.
[44] Gu C H, Hu H S. A unified explicit form of Backlund transformations for the AKNS system. Lett. Math. Phys. 1986, 11: 325-335.
[45] Gu C H, Zhou Z X. On the Darboux matrices of Backlund transformations for the AKNS system. Lett. Math. Phys. 1987, 13: 179-187.
[46] Gu C H, Zhou Z X. On the Darboux transformations for soliton equations in high dimen- tional space-time. Lett. Math. Phys. 1994, 32: 1-10.
[47] Hu H S. Darboux transformations of Su-chain. Proc. Symposium in honor of Prof. Su Buchin, World Scientific, 1993, 108-113.
[48] Hu H S. Darboux transformations between △α = sinhα and △α = sinα, and the applications to pseuso-spherical congruences in R~(2,1). Lett. Math. Phys. 1999, 48: 187-195.
[49] Zhou Z X. On the Darboux transformations for 1+2 dimentional equations. Lett. math. Phys. 1988, 16: 9-17.
[50] Zhou Z X. Binary Darboux transformations for Manakov triad. Phys. Lett. A. 1994, 195: 339-345.
[51] Zhou Z X. Binary Darboux transformations for Manakov triad (II). Phys. Lett. A. 1996, 211: 191-198.
[52] Liu Q P, M. Manas. Darboux transformation for the Manin-Radul supersymmetric KdV equation. Phys. Lett. B. 1997, 394: 337-342.
[53] Levi D, Ragnisco O. The inhomogeneous Toda lattice: its hierarchy and Darboux-Backlund transformations. J. Phys. A: Math. Gen. 1991, 24: 1729-1739.
[54] Matveev V B, Salle M A. Darboux transformations and Solitons. Springer, Berlin, 1991.
[55] Rogers C, Schief W K. Backlund and Darboux transformations geometry and modern applications
[56] Matveev V B, Sklyanin E K. On Backlund transformations for many-body systems. J. Phys. A. 1998, 31: 2241-2251.
[57] Sklyanin E K. Canonicity of Backlund transformation: 7-matrix approach. Amer. Math. Soc. Transt. Ser. 2000, 2: 201.
[58] Choudhury A G, Chowhury A R. Canonical and Backlund transformations for discrete integrable systems and classical γ-matrix.Phys.Lett.A.2001,280:37-44.
[59]Zeng Y B,et al.Canonical and B(a|¨)cklund transformations with spectrality for constrained flows of soliton hierarchies.Physica A,2002,303:321-338.
[60]Hu X B.Rational solutions of integrable equations via nonlinear superposition formular.J.Phys.A:Math.Gen.1997,30:8225-8240.
[61]Hu X B,Zhu Z N.Some new results on the Blaszak-Marciniak lattice:B(a|¨)cklund transformation and nonlinear superposition formular.J.Math.Phys.1998,39:4766-4772.
[62]Hu X B,Wu Y T.A new integrable differential-difference system and its explicit solutions.J.Phys.A:Math.Gen.1999,32:1515-1521.
[63]Hu X B,Tam H W.Some new results on the Blaszak-Marcinizk,3-field and 4-field lattices.Rep.Math.Phys.2000,46:99-105.
[64]Tam H W et al.The Hirota-Satsuma coupled KdV equation and a coupled Ito system revisited.J.Phys.Soc.Jpn.2000,69:45-52.
[65]Hu X B et al.Lax pairs and B(a|¨)cklund transformation for a coupled Ramani equation and its related system.Appl.Math.Lett.2000,13:45-48.
[66]Lou S Y.Generalized dromion solutions of the(2+1)-dimensional KdV equation.J.Phys.A:Math.Gen.1995,28:7227-7232.
[67]Liu Q P,Hu X B,Zhang M X.Supersymmetric modified Korteweg-de Vries equation:bininear approach.Nonlinearity,2005,18:1597-1603.
[68]Liu Q P,Hu X B.Blilinearization of N=1 supersymmetric Korteweg-de Vries equation revisited.J.Phys.A:Math.Gen.2005,38:6371-6378.
[69]曾云波。与A_l相联系的半Toda方程的Lax对与B(a|¨)cklund变换.数学学报,1992,35:454-459.
[70]曾云波.递推算子与Painleve性质.数学年刊,1991,12A:78-88.
[71]郑学东.大连理工大学博士学位论文.大连:大连理工大学,2002.
[72]陈勇.大连理工大学博士学位论文.大连:大连理工大学,2003.
[73]谢福鼎.大连理工大学博士学位论文.大连:大连理工大学,2003.
[74]李彪.大连理工大学博士学位论文.大连:大连理工大学,2005.
[75]Xu G Q,Li Z B.PDEPtest:a package for the Painleve test of nonlinear partial differential equations.Appl.Math.Comput.2005,169:1364-1379.
[76]Lou S Y.KdV extensions with Painleve property.J.Math.Phys.1998,39:2112-2121.
[77]Chen L L,Lou S Y.Painleve analysis of a(2+1)-dimentional Burgers equation.Commun. Theor.Phys.1998,29:313-316.
[78]楼森岳.推广的Painleve展开及KdV方程的非标准截断解.物理学报,1998,47:1937-1949.
[79]Lie S.(U|¨)ber die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichungen.Arch.Math.1881,6:328-368.
[80]Ovsiannikov L.V.Group Analysis of Differential Equations,New York,Academic(1982).
[81]Lou S Y,Ma H C.Non-Lie symmetry groups of(2+1)-dimensional nonlinear systems obtained from a simple direct method.J.Phys.A:Math.Gen.2005,38:129-137.
[82]Lou S Y,Ma H C.Finite symmetry transformation groups and exact solutions of Lax integrable systems.Chaos,Solitons and Fractals,2006,30:804-821.
[83]Lou S Y,Tang X Y.Conditional similarity reduction approach:Jimbo-Miwa equation.Chin.Phys.2001,10:897-901.
[84]Tang X Y,Lin J,Lou S Y.Conditional similarity solution of the Boussinesq equation.2001,35:399-404.
[85]李翊神,程艺.新KdV方程族的对称与守恒量.中国科学.A,1988,31:1-7.
[86]Zhdanov R Z.Separation of variables in the nonlinear wave equation.J.Phys.A:Math.Gen.,1994,27:291-297;Zhdanov R Z,Revenko I V,Fushchych W I.Orthogonal and non-orthogonal separation of variables in the wave equation u_tt-u_xx+V(x)u=0.J.Phys.A:Math.Gen.1993,26:5959-5562;Zhdanov R Z.On the new approach to variable separation in the time-dependent Schrodinger equation with two space dimensions.J.Math.Phys.1995,36:5506-5521.
[87]Blaszak M,Ma W X.New Liouville integrable noncanonical Hamiltonian systems from the AKNS spectral problem.J.Math.Phys.2002,43:3107-3123.
[88]Cao C W.Nonlinearization of the Lax system for AKNS hierarchy.Sci.China A,1990,33:528-536.
[89]Cao C W,Geng X G,Wu Y T.From the special(2+1) Toda lattice to the Kadomtsev-Petviashvili equation.J.Phys.A:Math.Gen.1999,32:8059-8078.
[90]Ma W X,Fuchssteiner B,Oevel W.A 3×3 matrix spectral problem for AKNS hierarchy and its binary nonlinearization.Physica A,1996,233:331-354.
[91]Konopelchenko B G,Sidorenko V,Strampp W.Phys.Lett.A,1991,175:17.
[92]Lou S Y,Hu X B.Infinitely many Lax pairs and symmetry constraints of the KP equation.J.Math.Phys.1997,38:6401-6427.
[93]Zeng Y B, Ma W X. Separation of variables for soliton equations via their binary constrained flows. J. Math. Phys. 1999, 40: 6526-6557. Zeng Y B. A family of separable integrable Hamiltonian systems and their classical dynamical r-matrix Poisson structures. Inverse Problem, 1996, 12: 797-809.
[94] Lou S Y, Chen L L. Formally variable separation approach for nonintegrable models. J. Math. Phys. 1999, 40: 6491-6500.
[95] Chou K S, Qu C Z. Symmetry groups and separation of variables of a class of nonlinear diffusion-convection equations. J. Phys. A: Math. Gen. 1999, 32: 6271-6286.
[96] Estevez P G, Qu C Z, Zhang S L. Separation of variables of a generalized porous medium equation with nonlinear source. J. Math. Anal. Appl. 2002, 275: 44-59.
[97] Estevez P G, Qu C Z. Separation of variables in a nonlinear wave equation with a variable wave speed. Theor. Math. Phys. 2002, 133: 1490-1497.
[98]Qu C Z, Zhang S L, Liu R C. Separation of variables and exact solutions to diffusion equations with nonlinear source. Physica D. 2000, 144: 97-123.
[99]Qu C Z, He W L, Dou J H. Separation of variables and exact solutions of generalized nonlinear Klein-Gordon equations. Prog. Theor. Phys. 2001, 105: 379-398.
[100] Zhang S L, Lou S Y, Qu C Z. Variable separation and exact solutions to generalized nonlinear diffusion equations. Chin. Phys. Lett. 2002, 12: 1471-1744.
[101] Zhang S L, Lou S Y. Variable separation and derivative-dependent functional separable solutions to generalized KdV equations. Commun. Theor. Phys. 2003, 40: 401-406.Zhang S L, Lou S Y. Derivative-dependent functional separation solutions for the Kdv- type equations. Physica A. 2004, 335: 430-444. Zhang S L, Lou S Y, Qu C Z. New variable separation approach: application to nonlinear diffusion equations. J. Phys. A: Math. Gen. 2003, 36: 12223-12242.
[102] Lou S Y, Rogers C, Schief W K. Virasoro structure and localized excitations of the LKR system. J. Math. Phys. 2003, 44: 5869-5887. Lou S Y, Rogers C, Schief W K. On a nonlinear discretized LKR system: reductions and underlying Virasoro symmetry algebras. Stud. Appl. Math. 2004, 113: 353-380.
[103] Lou S Y, Tang X Y. Equations of arbitrary order invariant under the Kadomtsev- Petviashvili symmetry group. J. Math. Phys. 2004, 45: 1020-1030.
[104] Darboux G. Comptes Rendus Acad. 1982, 94: 1456-1459.
[105] Nimmo J J C. Darboux Transformations from Reductions of the KP Hierarchy (Singapore: World Scientific). 1995.
[106] Nimmo J J C. Inverse Problem. 1992, 81: 219-243.
[107] Deng S F. Darboux transformations for the isospectral and nonisospectral mKP equation. Physica A. 2007, 382: 487-493.
[2]Gardner C S,Green J M,Kruskal M D,Miura R M.Method for solving the Korteweg-de Vries equation.Phys.Rev.Lett.1976,19:1059-1095.
[3]Gu C H,Hu H S,Zhou Z X.Darboux Transformation in soliton theory and its Geometric Applications.Shanghai Science and Technical Publishers,1999.谷超豪,胡和生,周子翔.孤子理论中的达布变换及其几何应用.上海科学技术出版社,2001.
[4]Rogers C,Schief W K.B(a|¨)cklund and Darboux transformation geometry and modern applications in soliton theory.Cambridge University Press,Cambridge Texts in Applied Mathematics,2002.
[5]Wadati M,Sanuki H,Konno K.Simple derivation of B(a|¨)cklund transformation from Riccati form of inverse method.Prog.Theor.Phys.1975,53:419-436.
[6]Matveev V B,Salle M A.Darboux Transformation and Solitons.Springer,Berlin,1991.
[7]Miura M R.B(a|¨)cklund Transformation.Brelin,Springer-Verlag,1978.
[8]Hirota R.Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons.Phys.Rev.Lett.1971,27:1192-1194.
[9]Hirota R.The direct method in soliton theory.Edited and tranlated by Nagai A,Nimmo J,Gilson C.Cambridge Tracts in Mathematics.No.155,2004.
[10]Matsukdaira J,Satsuma J,Strampp W.Soliton equations expressed by trilinear forms and their solutions.Phys.Lett.A.1990,147:467-471.
[11]Bluman G W,Kumei S.Symmetries and Differential Euqation.Appl.Math.Sci.81,Springer,Berlin,1989.
[12]Olver P J.Application of Lie Groups to Differential Equation.2nd ed.Graduate Texts Math,107,Springer,New York,1993.
[13]Clarkson P A,Kruskal M D.New similarity solutions of the Boussinesq equations.J.Math.Phys.1989,30:2201-2213.
[14]Clarkson P A.Nonclassical symmetry reductions of the Boussinesq equation.Chaos,Soliton and Fractals,1995,5:2261-2301.
[15]Clarkson P A,Mansfield E L.Symmetry reductions and exact solutions of shallow water wave equations. Acta Appl. Math. 1995, 39: 245-276.
[16] Lou S Y. A note on the new similarity reductions of the Boussinesq equation. Phys. Lett. A, 1990, 151: 133-135.
[17] Lou S Y. Similarity solutions of the Kadomtsev-Petviashvili equation. J. Phys. A: Math. Gen., 1990, 23: 649-654;
Lou S Y. Symmetries and similarity reductions of the Dym equayion. Physica Scripta, 1996, 54: 428-435;
Lou S Y, Ruang H Y, Chen D F, Chen W Z. Similarity reductions of the KP equation by a direct method. J. Phys. A, 1991, 24: 1455-1467;
Lou S Y, Tang X Y, Lin J. Similarity and conditional similarity reductions of a (2+1)- dimensional KdV equation via a direct method. J. Math. Phys., 2000, 41: 8286-8303.
[18] Pucci E. J. Potential symmetries and solutions by reduction of partial differential equations. Phys. A: Math. Gen., 1993, 26: 681-690.
[19] Saccomandi G. Potential symmetries and direct reduction methods of order two. J. Phys. A. 1997, 30: 2211-2217.
[20] Lou S Y, Ni G J. The relations among a special type of solutions in some (D+1) dimensional nonlinear equations. J. Math. Phys. 1989, 30: 1614-1630.
[21] Hereman W, Banerjee P P, Korpel A et al. Exact solitary wave solutions of non-linear evolution and wave equations using a direct algebraic method. J. Phys. A: Math. Gen. 1986, 19: 607-628.
[22] Dolye P W. Separation of variables for scalar evolution equations in one space dimension. J. Phys. A. 1996, 29: 7581-7595.
[23] Dolye P W, Vassiliou P J. Separation of variables for the 1-dimensional non-linear diffusion equation. Int. J. Nonlear Mech. 1998, 33: 315-326.
[24] Alexeyev N N. A multidimensional superposition principle: classical solitons III. Phys. Lett. A. 2005, 335: 197-206.
[25] Chen C L, Tang X Y, Lou S Y. Solutions of (2+l)-dimensional dispersive long wave equation. Phys. Rev. E. 2002, 66: 036605.
[26] Conte R. Invariant Painleve analysis of partial differential equations. Phys. Lett.A. 1989, 140: 383-390.
[27] Lou S Y. Extended Painleve expansion, nonstand truncation and special reductions of nonlinear evolution equations. Z Naturforsch a. 1998, 53: 251-258.
[28] Picking A. A new truncation in Painleve analysis. J. Phys. A: Math. Gen. 1993, 26: 4395- 4406;Picking A. The singular manifold method revisted. J. Math. Phys. 1996, 37: 1894-1927.
[29] Weiss J, Tabor M, Carnevale G. The Painleve property for partial differential equations. J. Math. Phys. 1983, 24: 522-526.
[30] Chen Y, Li B. New exact travelling wave solutions for generalized Zakharov-Kuzentsov equations using general projective Riccati equation method. Commun. Theor. Phys. 2004, 41: 1-6.
[31] Chen Y, Li Y S. The constraint of the Kadomtsev-Petviashvili equation and its special solutions. Phys. Lett. A. 1991, 157: 22-26.
[32] Fan E G. Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A. 2000, 277: 212-218.
[33] Huang G X, Lou S Y, Dai X X. Exact and explicit solitary wave solutions to a model equation for water waves. Phys. Lett. A. 1989, 139: 373-374.
[34] Lan H B, Wang K L. Exact solutions for two nonlinear equations. J. Phys. A: Math. Gen. 1990, 23: 4097-4105.
[35] Zakharov V E, Shabat A B. A scheme for integrating nonlinear equations of mathematical physics by the method of the inverse scattering transform. Funct. Anal. Appl. 1974, 8(3): 43-53;A scheme for integrating nonlinear equations of mathematical physics by the method of the inverse scattering transform. II. Funct. Anal. Appl. 1979, 13(3): 13-23.
[36] Zenchuk A I. A unified dressing method for C- and S-integrable hierarchies; the particular example of a (3 + l)-dimensional n-wave equation. J. Phys. A: Math. Gen. 2004, 37:6557-6571.
[37] Gardner C S et al. Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 1967, 19: 1059-1097.
[38] Lax P D. Integrals of nonlinear equations of evolution and soliton waves. Commun, Pure Appl. Math. 1968, 21: 467-490.
[39] Zabusky N J, Kruskal M D. Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 1965, 15: 240-243.
[40] Ablowitz M J et al. Method for solving the sine-Gordon equation. Phys. Rev. Lett. 1973, 30: 1262-1264.
[41] Gu C H. On the Backlund transformation for the generalized hierarchies of compound MKdV-SG equation. Lett. Math. Phys. 1986, 11: 31-41.
[42] Gu C H. On the Darboux form of Backlund transformations. In Integrable System, Nankai Lectures on Math. Phys. World Scientic, Singapore, 1989, 162-168.
[43] Gu C H. Darboux transformations for a class of integarable systems in n variables. Analyse, Varietes et Physique, Proc. of Colloque International enl'honneur d'Yvonne Choquet- Bruhat, Kluwer, 1992.
[44] Gu C H, Hu H S. A unified explicit form of Backlund transformations for the AKNS system. Lett. Math. Phys. 1986, 11: 325-335.
[45] Gu C H, Zhou Z X. On the Darboux matrices of Backlund transformations for the AKNS system. Lett. Math. Phys. 1987, 13: 179-187.
[46] Gu C H, Zhou Z X. On the Darboux transformations for soliton equations in high dimen- tional space-time. Lett. Math. Phys. 1994, 32: 1-10.
[47] Hu H S. Darboux transformations of Su-chain. Proc. Symposium in honor of Prof. Su Buchin, World Scientific, 1993, 108-113.
[48] Hu H S. Darboux transformations between △α = sinhα and △α = sinα, and the applications to pseuso-spherical congruences in R~(2,1). Lett. Math. Phys. 1999, 48: 187-195.
[49] Zhou Z X. On the Darboux transformations for 1+2 dimentional equations. Lett. math. Phys. 1988, 16: 9-17.
[50] Zhou Z X. Binary Darboux transformations for Manakov triad. Phys. Lett. A. 1994, 195: 339-345.
[51] Zhou Z X. Binary Darboux transformations for Manakov triad (II). Phys. Lett. A. 1996, 211: 191-198.
[52] Liu Q P, M. Manas. Darboux transformation for the Manin-Radul supersymmetric KdV equation. Phys. Lett. B. 1997, 394: 337-342.
[53] Levi D, Ragnisco O. The inhomogeneous Toda lattice: its hierarchy and Darboux-Backlund transformations. J. Phys. A: Math. Gen. 1991, 24: 1729-1739.
[54] Matveev V B, Salle M A. Darboux transformations and Solitons. Springer, Berlin, 1991.
[55] Rogers C, Schief W K. Backlund and Darboux transformations geometry and modern applications
[56] Matveev V B, Sklyanin E K. On Backlund transformations for many-body systems. J. Phys. A. 1998, 31: 2241-2251.
[57] Sklyanin E K. Canonicity of Backlund transformation: 7-matrix approach. Amer. Math. Soc. Transt. Ser. 2000, 2: 201.
[58] Choudhury A G, Chowhury A R. Canonical and Backlund transformations for discrete integrable systems and classical γ-matrix.Phys.Lett.A.2001,280:37-44.
[59]Zeng Y B,et al.Canonical and B(a|¨)cklund transformations with spectrality for constrained flows of soliton hierarchies.Physica A,2002,303:321-338.
[60]Hu X B.Rational solutions of integrable equations via nonlinear superposition formular.J.Phys.A:Math.Gen.1997,30:8225-8240.
[61]Hu X B,Zhu Z N.Some new results on the Blaszak-Marciniak lattice:B(a|¨)cklund transformation and nonlinear superposition formular.J.Math.Phys.1998,39:4766-4772.
[62]Hu X B,Wu Y T.A new integrable differential-difference system and its explicit solutions.J.Phys.A:Math.Gen.1999,32:1515-1521.
[63]Hu X B,Tam H W.Some new results on the Blaszak-Marcinizk,3-field and 4-field lattices.Rep.Math.Phys.2000,46:99-105.
[64]Tam H W et al.The Hirota-Satsuma coupled KdV equation and a coupled Ito system revisited.J.Phys.Soc.Jpn.2000,69:45-52.
[65]Hu X B et al.Lax pairs and B(a|¨)cklund transformation for a coupled Ramani equation and its related system.Appl.Math.Lett.2000,13:45-48.
[66]Lou S Y.Generalized dromion solutions of the(2+1)-dimensional KdV equation.J.Phys.A:Math.Gen.1995,28:7227-7232.
[67]Liu Q P,Hu X B,Zhang M X.Supersymmetric modified Korteweg-de Vries equation:bininear approach.Nonlinearity,2005,18:1597-1603.
[68]Liu Q P,Hu X B.Blilinearization of N=1 supersymmetric Korteweg-de Vries equation revisited.J.Phys.A:Math.Gen.2005,38:6371-6378.
[69]曾云波。与A_l相联系的半Toda方程的Lax对与B(a|¨)cklund变换.数学学报,1992,35:454-459.
[70]曾云波.递推算子与Painleve性质.数学年刊,1991,12A:78-88.
[71]郑学东.大连理工大学博士学位论文.大连:大连理工大学,2002.
[72]陈勇.大连理工大学博士学位论文.大连:大连理工大学,2003.
[73]谢福鼎.大连理工大学博士学位论文.大连:大连理工大学,2003.
[74]李彪.大连理工大学博士学位论文.大连:大连理工大学,2005.
[75]Xu G Q,Li Z B.PDEPtest:a package for the Painleve test of nonlinear partial differential equations.Appl.Math.Comput.2005,169:1364-1379.
[76]Lou S Y.KdV extensions with Painleve property.J.Math.Phys.1998,39:2112-2121.
[77]Chen L L,Lou S Y.Painleve analysis of a(2+1)-dimentional Burgers equation.Commun. Theor.Phys.1998,29:313-316.
[78]楼森岳.推广的Painleve展开及KdV方程的非标准截断解.物理学报,1998,47:1937-1949.
[79]Lie S.(U|¨)ber die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichungen.Arch.Math.1881,6:328-368.
[80]Ovsiannikov L.V.Group Analysis of Differential Equations,New York,Academic(1982).
[81]Lou S Y,Ma H C.Non-Lie symmetry groups of(2+1)-dimensional nonlinear systems obtained from a simple direct method.J.Phys.A:Math.Gen.2005,38:129-137.
[82]Lou S Y,Ma H C.Finite symmetry transformation groups and exact solutions of Lax integrable systems.Chaos,Solitons and Fractals,2006,30:804-821.
[83]Lou S Y,Tang X Y.Conditional similarity reduction approach:Jimbo-Miwa equation.Chin.Phys.2001,10:897-901.
[84]Tang X Y,Lin J,Lou S Y.Conditional similarity solution of the Boussinesq equation.2001,35:399-404.
[85]李翊神,程艺.新KdV方程族的对称与守恒量.中国科学.A,1988,31:1-7.
[86]Zhdanov R Z.Separation of variables in the nonlinear wave equation.J.Phys.A:Math.Gen.,1994,27:291-297;Zhdanov R Z,Revenko I V,Fushchych W I.Orthogonal and non-orthogonal separation of variables in the wave equation u_tt-u_xx+V(x)u=0.J.Phys.A:Math.Gen.1993,26:5959-5562;Zhdanov R Z.On the new approach to variable separation in the time-dependent Schrodinger equation with two space dimensions.J.Math.Phys.1995,36:5506-5521.
[87]Blaszak M,Ma W X.New Liouville integrable noncanonical Hamiltonian systems from the AKNS spectral problem.J.Math.Phys.2002,43:3107-3123.
[88]Cao C W.Nonlinearization of the Lax system for AKNS hierarchy.Sci.China A,1990,33:528-536.
[89]Cao C W,Geng X G,Wu Y T.From the special(2+1) Toda lattice to the Kadomtsev-Petviashvili equation.J.Phys.A:Math.Gen.1999,32:8059-8078.
[90]Ma W X,Fuchssteiner B,Oevel W.A 3×3 matrix spectral problem for AKNS hierarchy and its binary nonlinearization.Physica A,1996,233:331-354.
[91]Konopelchenko B G,Sidorenko V,Strampp W.Phys.Lett.A,1991,175:17.
[92]Lou S Y,Hu X B.Infinitely many Lax pairs and symmetry constraints of the KP equation.J.Math.Phys.1997,38:6401-6427.
[93]Zeng Y B, Ma W X. Separation of variables for soliton equations via their binary constrained flows. J. Math. Phys. 1999, 40: 6526-6557. Zeng Y B. A family of separable integrable Hamiltonian systems and their classical dynamical r-matrix Poisson structures. Inverse Problem, 1996, 12: 797-809.
[94] Lou S Y, Chen L L. Formally variable separation approach for nonintegrable models. J. Math. Phys. 1999, 40: 6491-6500.
[95] Chou K S, Qu C Z. Symmetry groups and separation of variables of a class of nonlinear diffusion-convection equations. J. Phys. A: Math. Gen. 1999, 32: 6271-6286.
[96] Estevez P G, Qu C Z, Zhang S L. Separation of variables of a generalized porous medium equation with nonlinear source. J. Math. Anal. Appl. 2002, 275: 44-59.
[97] Estevez P G, Qu C Z. Separation of variables in a nonlinear wave equation with a variable wave speed. Theor. Math. Phys. 2002, 133: 1490-1497.
[98]Qu C Z, Zhang S L, Liu R C. Separation of variables and exact solutions to diffusion equations with nonlinear source. Physica D. 2000, 144: 97-123.
[99]Qu C Z, He W L, Dou J H. Separation of variables and exact solutions of generalized nonlinear Klein-Gordon equations. Prog. Theor. Phys. 2001, 105: 379-398.
[100] Zhang S L, Lou S Y, Qu C Z. Variable separation and exact solutions to generalized nonlinear diffusion equations. Chin. Phys. Lett. 2002, 12: 1471-1744.
[101] Zhang S L, Lou S Y. Variable separation and derivative-dependent functional separable solutions to generalized KdV equations. Commun. Theor. Phys. 2003, 40: 401-406.Zhang S L, Lou S Y. Derivative-dependent functional separation solutions for the Kdv- type equations. Physica A. 2004, 335: 430-444. Zhang S L, Lou S Y, Qu C Z. New variable separation approach: application to nonlinear diffusion equations. J. Phys. A: Math. Gen. 2003, 36: 12223-12242.
[102] Lou S Y, Rogers C, Schief W K. Virasoro structure and localized excitations of the LKR system. J. Math. Phys. 2003, 44: 5869-5887. Lou S Y, Rogers C, Schief W K. On a nonlinear discretized LKR system: reductions and underlying Virasoro symmetry algebras. Stud. Appl. Math. 2004, 113: 353-380.
[103] Lou S Y, Tang X Y. Equations of arbitrary order invariant under the Kadomtsev- Petviashvili symmetry group. J. Math. Phys. 2004, 45: 1020-1030.
[104] Darboux G. Comptes Rendus Acad. 1982, 94: 1456-1459.
[105] Nimmo J J C. Darboux Transformations from Reductions of the KP Hierarchy (Singapore: World Scientific). 1995.
[106] Nimmo J J C. Inverse Problem. 1992, 81: 219-243.
[107] Deng S F. Darboux transformations for the isospectral and nonisospectral mKP equation. Physica A. 2007, 382: 487-493.