孤子理论若干问题研究
|
摘要
本文围绕孤子理论的Hirota直接方法，扰动，对称，相似解等问题做了以下三方面的主要工作：
1、首先研究了双线性算子的一些性质，得到了在对数变换意义下KdV类的双线性方程所直接相对应的非线性pde(即过渡方程)的一些性质。利用这些性质，本文首次得出了线性项非空的多项式形式的原方程与相应的过渡方程之间的线性项非空的多项式形式的变换的具体可能形式，并构造例子给以证实；得到了任意给定的线性项非空的多项式形式非线性pde具有线性项非空的多项式形式变换的一些必要条件。如果所给定的线性项非空的多项式形式的非线性PDE能化成KdV类的双线性方程，我们可以发现这个原方程线性部分算子与其相对应的KdV类的双线性方程中的双线性算子的关系，从而直接写出该方程所对应的双线性方程。这些结果使我们对多项式形式的线性项非空的KdV类的非线性PDE有直观的了解，从而能直接的改进一些义献的结果。
2、对一阶非线性扰动方程的情形，用泰勒级数关于扰动参数ε在零点附近直接展开方法推导了该扰动方程渐近解析解所应具有的条件等式。通过这个条件等式，我们就有可能获得一些变换，这些变换把给定的未扰动方程的解析解直接的映成扰动方程的渐近解析解。为了获得更多的这类变换，我们引入了Lie-Baclund(LB)对称，从而利用可能存在的未扰动方程的递归算子和LB对称来构造大量的此类变换，最终从未扰动方程的精确解出发，通过这些变换，直接获得—大批的渐近解析解。扰动的KdV方程和扰动的Burgers方程将被作为例子给出。至于高阶的情形，我们仅给出理论上的一些推导结果和一个简单的例子。值得注意的是，一阶情形与高阶情形的理论结果的推导与变换的计算无须用到任何的群论知识。另外我们用Hirota直接方法研究了一种扰动的KdV方稗的N-孤立子解，并研究了扰动情形下N-孤立子解在的图像情况。
3、对Broer-Kaup(BK)方程组所得到的单一方程作了经典的李对称相似约化，并给出部分相似解。对其中一类特殊的情形，通过观察，我们发现了BK方程绢的相似解与著名的Burgers方程之间的关系，从而，我们可以从已知的大量的Burgers方程和热方程的精确解中得到大量的BK方程的相似解。
In this paper, we discuss mainly three aspects about the Hirota's direct method, perturbation, symmetries and similarity solutions of nonlinear partial differential equations, and they read as follows:
1. Firstly the significant characters of the bilinear operator are studied, then we investigate the characters of the nonlinear PDE directly obtained by the logarithmic transformation for the KdV-type bilinear equation. According to these characters, the polynomial transformations with nonempty linear term between the original polynomial equation with nonempty linear term and its KdV-type bilinear equation are obtained in details and also are confirmed with some examples. Some necessary conditions for the given polynomial nonlinear PDE with nonempty linear term, which owns corresponding polynomial transformation with nonempty linear term, are found. If the given nonlinear PDE owns its Kdv-type bilinear form, according to these necessary conditions, we can find the relationships between linear operators of the original equation and the bilinear operators of its bilinear form. And the bilinear form of the given original equation can be obtained directly in virtue of the relationships. These results make us straightforth understanding of the KdV-type nonlinear PDE, which help us improve directly some results appearing in literature.
2、As for the 1~st-order perturbed nonlinear PDE, we obtain the conditional equality that the coefficient of the 1~st-order term of the Taylor series expansion aroundε=0 for any asymptotical analytical solution of the perturbed PDE with perturbing parameterεmust be admitted. By making use of the conditional equality, we may obtain some transformations, which directly map the analytical solutions of the given unperturbed PDE to the asymptotical analytical solutions of the corresponding perturbed one. The Lie-Baclund symmetries is introduced in order to obtaining more transformations. Hence, we can directly create more transformations in virtue of known Lie-Baclund symmetries and recursion operators of corresponding unperturbed equation. The perturbed Burgers equation and the perturbed Korteweg-de Vries(KdV) equation are used as examples. As for the higher-order cases, we only give the main results with a simple example. We should notice that not any group theoretical knowledge is introduced in the deduce process for obtaining these theoretical results and transformations. In addition, the N-solitons solutions of the perturbed KdV equations are investigated by Hirota's direct method approach and the 1-soliton diagram are illustrated.
3. By a known transformation, the Broer-Kaup(BK) equations are combined to a single one. After the classical Lie symmetry analysis and similarity reductions are performed for the single one, new similar solutions are obtained. From the special kind of its similarity reductions, we find the relationship between the BK equations and the famous Burgers equation and the heat equation by observation. By making use of these relationship, more similarity solution can be created directly by abundant known solutions of the Burgers equation and the heat equation.
1、首先研究了双线性算子的一些性质，得到了在对数变换意义下KdV类的双线性方程所直接相对应的非线性pde(即过渡方程)的一些性质。利用这些性质，本文首次得出了线性项非空的多项式形式的原方程与相应的过渡方程之间的线性项非空的多项式形式的变换的具体可能形式，并构造例子给以证实；得到了任意给定的线性项非空的多项式形式非线性pde具有线性项非空的多项式形式变换的一些必要条件。如果所给定的线性项非空的多项式形式的非线性PDE能化成KdV类的双线性方程，我们可以发现这个原方程线性部分算子与其相对应的KdV类的双线性方程中的双线性算子的关系，从而直接写出该方程所对应的双线性方程。这些结果使我们对多项式形式的线性项非空的KdV类的非线性PDE有直观的了解，从而能直接的改进一些义献的结果。
2、对一阶非线性扰动方程的情形，用泰勒级数关于扰动参数ε在零点附近直接展开方法推导了该扰动方程渐近解析解所应具有的条件等式。通过这个条件等式，我们就有可能获得一些变换，这些变换把给定的未扰动方程的解析解直接的映成扰动方程的渐近解析解。为了获得更多的这类变换，我们引入了Lie-Baclund(LB)对称，从而利用可能存在的未扰动方程的递归算子和LB对称来构造大量的此类变换，最终从未扰动方程的精确解出发，通过这些变换，直接获得—大批的渐近解析解。扰动的KdV方程和扰动的Burgers方程将被作为例子给出。至于高阶的情形，我们仅给出理论上的一些推导结果和一个简单的例子。值得注意的是，一阶情形与高阶情形的理论结果的推导与变换的计算无须用到任何的群论知识。另外我们用Hirota直接方法研究了一种扰动的KdV方稗的N-孤立子解，并研究了扰动情形下N-孤立子解在的图像情况。
3、对Broer-Kaup(BK)方程组所得到的单一方程作了经典的李对称相似约化，并给出部分相似解。对其中一类特殊的情形，通过观察，我们发现了BK方程绢的相似解与著名的Burgers方程之间的关系，从而，我们可以从已知的大量的Burgers方程和热方程的精确解中得到大量的BK方程的相似解。
In this paper, we discuss mainly three aspects about the Hirota's direct method, perturbation, symmetries and similarity solutions of nonlinear partial differential equations, and they read as follows:
1. Firstly the significant characters of the bilinear operator are studied, then we investigate the characters of the nonlinear PDE directly obtained by the logarithmic transformation for the KdV-type bilinear equation. According to these characters, the polynomial transformations with nonempty linear term between the original polynomial equation with nonempty linear term and its KdV-type bilinear equation are obtained in details and also are confirmed with some examples. Some necessary conditions for the given polynomial nonlinear PDE with nonempty linear term, which owns corresponding polynomial transformation with nonempty linear term, are found. If the given nonlinear PDE owns its Kdv-type bilinear form, according to these necessary conditions, we can find the relationships between linear operators of the original equation and the bilinear operators of its bilinear form. And the bilinear form of the given original equation can be obtained directly in virtue of the relationships. These results make us straightforth understanding of the KdV-type nonlinear PDE, which help us improve directly some results appearing in literature.
2、As for the 1~st-order perturbed nonlinear PDE, we obtain the conditional equality that the coefficient of the 1~st-order term of the Taylor series expansion aroundε=0 for any asymptotical analytical solution of the perturbed PDE with perturbing parameterεmust be admitted. By making use of the conditional equality, we may obtain some transformations, which directly map the analytical solutions of the given unperturbed PDE to the asymptotical analytical solutions of the corresponding perturbed one. The Lie-Baclund symmetries is introduced in order to obtaining more transformations. Hence, we can directly create more transformations in virtue of known Lie-Baclund symmetries and recursion operators of corresponding unperturbed equation. The perturbed Burgers equation and the perturbed Korteweg-de Vries(KdV) equation are used as examples. As for the higher-order cases, we only give the main results with a simple example. We should notice that not any group theoretical knowledge is introduced in the deduce process for obtaining these theoretical results and transformations. In addition, the N-solitons solutions of the perturbed KdV equations are investigated by Hirota's direct method approach and the 1-soliton diagram are illustrated.
3. By a known transformation, the Broer-Kaup(BK) equations are combined to a single one. After the classical Lie symmetry analysis and similarity reductions are performed for the single one, new similar solutions are obtained. From the special kind of its similarity reductions, we find the relationship between the BK equations and the famous Burgers equation and the heat equation by observation. By making use of these relationship, more similarity solution can be created directly by abundant known solutions of the Burgers equation and the heat equation.
引文
[1] J.S.Russell. "Report on wave", in Report of the 14th meeting of Brithish Association for the Advancement of Science. 1844. London.
[2] Korteweg, D.J. and G.d. Vries, On the change of form of long waves advancing in a rectangular canal,and on a new type of long stationary waves. Phil.Mag., 1895.39: p. 422-443.
[3] A.Fermi, J.Pasta, and S.Ulam, Studies of nonlinear problems. Los Alamos Sci.Lab.Rep.LA-1940, 1955.
[4] N.J.Zabusky and M.D.Kruskal, Interaction of "Solitons" in a collisionless plasma and the recurrence of initial states. Phys.Rev.Lett., 1965.15: p. 240-243.
[5] A.C.Scott, F.V.F.Chu, and D.McLaughlin, The Soliton: A new concept in applied science. Proc.IEEE, 1973.61: p. 1473-1483.
[6] A.R.Bishop, Solitons in condensed matter physics. Phys.Scripta, 1979.20(409-423).
[7] M.J.Ablowitz and H.Segur, Solitons and inverse scattering transform. 1981, Philadelphia: SIAM.
[8] P.G.Drazin, Solitons. London mathematical society Lecture note series. Vol. 85. 1983: Cambridge University Press.
[9] S.P.Novikov, et al., Theory of Solitons: The Inverse Scattering Method. New York:Consultants Bureau, 1984.
[10] A.C.Newell, Solitons in mathematics and physics. 1985, Philadelphia: PA:SIAM.
[11] A.R.Chowdhury and P.Mitra, New coupled Liouville system: prolongation structure,soliton solution,and complete integrability. Inter. J. Theor. Phys., 1989. 28: p. 119-133.
[12] V.G.Makhankov, V.K.Fedyanin, and O.K.Pashaev, Ivth International Workshop:Solitons and Applications. 1989, Dubna: World Scientific.
[13] P.J.Ovler and D.H.Sattinger, Solitons in Physics, Mathematics, and Nonlinear Optics. 1990, New York: Springer-Verlag.
[14] M.J.Ablowitz and P.A.Clarkson, Solitons,nonlinear evolution equations and inverse scattering. London mathematical society Lecture note series. Vol. 149. 1991: Cambridge University Press.
[15] A.S.Davydov, Solitons in Molecular Systems. Mathematics and its applications. 1991: Kluwer Academic Publisher.
[16] V.B.Matveev and M.A.Salle, Darboux transformations and solitons. 1991: Springer-Verlag.
[17] S.P.Novikov, Solitons and Geometry. 1992: Accadimia Nazionale dei Lincei.
[18] M. Remoissenet,Waves called solitons: concepts and experiments. Springer-Vertag. 1995
[19] Remoissenet, M., Waves Called Solitons 2nd ed. 1999,北京：世界图书出版公司北京公司．
[20] L.A.Dickey, Soliton Equations and Hamiltonian systems. Advanced Series in Mathematical Physics. Vol. 26. 2003: World Scientific.
[21] R.Hirota, The Direct Method in Soliton Theory. 2004: Cambridge University Press.
[22] A.S.Fokas, et al. Nonlinear processes in physics, in Proceedings of Ⅲ Potsdam -V Kiev International Workshop. 1991. Clarkson University, Potsdam, NY: Springer-Verlag.
[23] G. Makhankov and O. K. Pashaev. Nonlinear Evolution Equations and Dynamical Systems. in Proceedings of the 6th International Workshop. 1991. Dubna: Springer-Verlag.
[24] M. Boiti, et al. Proceedings of the workshop on Nonlinearity integrability and all that: twenty years after NEEDS'79. 1999. Lecce, Italy: World Scientific.
[25] Y Cheng, S Hu and YS Li, Nonlinear evolution equations and dynamical systems, in Proceedings of the ICM 2002 satellite conference. 2003: World Scientific.
[26] Steeb, W. -H., Problems and solutions in theoretical and mathematical physics. 2003: World Scientific.
[27] 刘式达，刘式适，《孤波与湍流》，上海科技教育出版社，1994
[28] 李翊神，《孤子与可积系统》， 上海科技教育出版社，1999
[29] 黄念宁，《孤子理论和微扰方法》，上海科技教育出版社，1996
[30] 郭柏灵，苏凤秋，《孤立子》，沈阳：辽宁教育出版社，1997
[31] 谷超豪，郭柏灵，李翊神等，《孤立子理论与应用》，浙江科学技术出版社，1990
[32] 陈陆君，梁昌洪，《孤立子理论及其应用-光孤子理论有光孤子通信》，1997：西安电子科技大学出版社．
[33] C. S. Gardner, et al., Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett., 1967. 19: p. 1095-1097.
[34] 李翊神，汪克林，郭光灿，汪秉宏，《非线性科学选讲》，合肥：中国科学技术大学出版社，1994
[35] C. S. Gardner, et al., Comm pure appl. Math., 1974. 27: p. 97.
[36] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves. Commu. Pure Appl. Math., 1968. 21: p. 467-490.
[37] A. R. Chowdhury and R. Mukherjee, Elliptic solutions, recursion operators and complete Lie-Baclund symmetry for the Harry-Dym equation. Phys. Scripta, 1984. 29(293-295).
[38] K. M. Tamizhmani and M. Lakshmanan, Infinitely many Lie-Backlund symmetries for a quasi-linear evolution equation. Phys. Lett. A, 1982. 90: p. 159-161.
[39] Lee, YQ and M. Ge, Prolongation method of finding Backlund transformation and Lax-pair of Ernst equation and Ernst-maxwell equation, communications in Theoretical Physics, 1988. 10: p. 79-103.
[40] ML Wang and Y Wang, A new Backlund transformation and multi-soliton solutions to the KdV equation with general variable coefficients. Phys. Left. A, 2001. 287: p. 211-216.
[41] A. Bekir, Painleve test for some (2+1)-dimensional nonlinear equations. Chaos, Solitons and Fractals, 2007. 32: p. 449-455.
[42] Chowdhury, A. R., Painleve analysis and its applications. CHAPMAN&HALL/CRC monographs and surveys in pure and applied mathematics. Vol. 105. 2000: CHAPMAN&HALL/CRC.
[43] J. D. Gibbon, et al., The Painleve Property and Hirota's Method. Studies in Applied Mathematics, 1986. 72: p. 39-63.
[44] J. Weiss, M. Tabor, and G. Gerneval, The Painleve property for partial differential equations. J. Math. Phys., 1983. 24(522-526).
[45] K. M. Tamizhmani and P. Punithavathi, Similarity reductions and painleve property of the coupled higher dimensional Burgers equation. Int. J. Non-Linear Mech., 1991. 26: p. 427-438.
[46] K. Shivamoggi, B. and D. K. Rollins, Generalized Painleve formulation and Lie group symmetries of the Zakharov-Kuznetsov equation, phys. Lett. A, 1991. 160: p. 263-266.
[47] Karasu, A., Jordan KdV systems and Painleve property. Inter. J. Theor. Phys., 1997. 36: p. 705-713.
[48] M. Lakshmanan and R. Sahadevan, Painleve property of coupled anharmonic oscillators with N degrees of freedom, phys. Lett. A, 1984. 101: p. 189-194.
[49] A. C. Newell, M. Tabor, and Y. B. Zeng, A UNIFIED APPROACH TO PAINLEVE EXPANSION. Physica D, 1987. 29: p. 1-68.
[50] SL Zhang, B Wu and SY Lou, Painleve analysis and special solutions of generalized Broer-Kaup equations. Physics Letters A, 2002. 300(1).
[51] T. Alagesan, Y. Chung, and K. Nakkeeran, Painleve test for the certain (2+1)-dimensional nonlinear evolution equations. Chaos, Solitons and Fractals, 2005. 26: p. 1203-1209.
[52] W. H. Steeb and N. Euler, Nonlinear evolution equations and Painleve test. 1988: World Scientific.
[53] S. Kichenassamy, WTC expansions and nonintegrable equations. Studies in Applied Mathematics, 1999. 102: p. 1-26.
[54] P. A. Clarkson and S. Hood, New symmetry reductions and exact solutions of the Davey-Stewartson system. i. Reducitons to ordinary differential equations. J. Math. Phys., 1994. 35: p. 255-283.
[55] P. A. Clarkson and M. D. Kruskal, New similarity reductions of the Bossinisq equation. J. Math. Phys., 1989. 30: p. 2201-2213.
[56] P. A. Clarkson and P. Winternitz, Nonclassical symmetry reductions of the Kadomtsev-Pteviashvili equation. Phys. D, 1991. 49: p. 257-272.
[57] SY Lou, Nonclassical symmetry reductions for the dispersive wave equations in shallow water. J. Math. Phys., 1992. 33: p. 4300-4305.
[58] SY Lou and HC Ma, Non-Lie symmetry groups of (2+1)-dimensional nonlinear systems obtained from a simple direct method. J. Phys. A: Math. Gen., 2005. 38: p. L129-L137.
[59] 谷超豪，胡和生，周子翔，孤立子理论中的达布变换及其几何应用．1999：上海科学技术出版社．
[60] Morries, H. C., Prolongation structures and nonlinear evolution equations in two spatial dimensions: a general class of equations. J. Phys. A: Math. Gen., 1979. 12: p. 261-267.
[61] Lee, Y. and J. Lu, Prolongation structure and infinite-dimensional algebra for the isotropic Landau-Lifshitz equation, phys. Lett. A, 1989. 135: p. 117-119.
[62] H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equation. J. Math. Phys., 1975. 16: p. 1-7.
[63] E. V. Doktorov, Prolongation structure without prolongation. J. Phys. A: Math. Gen., 1980. 13: p. 3599-3604.
[64] Degang, Z., Y. Guoxiang, and L. Jie, Integrable deformations of the Heisenberg model: prolongation structure technique. J. Phys. A: Math. Gen., 1991. 24: p. L219-L223.
[65] A. R. Chowdhury and S. Paul, Prolongation structure of a new integrable system. Inter. J. Theor. Phys., 1985. 24: p. 633-637.
[66] R.Dodd and A.Fordy, On the integrability of a system of coupled KdVequations. phys. Lett. A, 1982. 89: p. 168-170.
[67] P. A.Clarkson and T. J.Priestley, Symmetries of a class of nonliear fourth order partial differential equations. J. Nonl. Math. Phys., 1999.6: p. 66-98.
[68] A.G.Johnpillai and A.H.Kara, Variational formulation of approximate symmetries and conservation laws. Inter. J. Theor. Phys., 2001.40: p. 1501-1509.
[69] E.Alfinito, G. Soliani and L. Solombrino, The symmetry structure of the Heavenly equation. Lett. Math. Phys., 1997.41: p. 379-389.
[70] B.Champagne and P.Winternitz, On the infinite-dimensional symmetry group of the Davey-Stewartson equations. J. Math. Phys., 1988.29: p. 1-8.
[71] V.Boyko, Symmetry classification of the one-dimensional second order equation of a hydrodynamic type. Nonl. Math. Phys., 1995.2: p. 418-424.
[72] C.C.A.Sastri and K.A.Dunn, Lie symmetries of some equations of the Fokker-Planck type. J. Math. Phys., 1985.26: p. 3042-3047.
[73] R.Cherniha, Lie symmetries of nonlinear two-dimensional reaction-diffusion systems. Reports Math. Phys., 2000.46: p. 63-76.
[74] A.R.Chowdhury, A. Ghose, and M. Naskar, Similarity soliton and Lie symmetry for a coupled nonlinear system. Inter. J. Theor. Phys., 1987.26: p. 357-363.
[75] G.Cicogna and G. Gaeta, Partial Lie-point symmetries of differential equations. J. Phys. A: Math. Gen., 2001.34: p. 491-512.
[76] D.David, et al., Symmetry reduction for the Kadomtsev-Petviashvill equation using a loop algebra. J. Math. Phys., 1986.27: p. 1225-1237.
[77] D.J.Arrigo and J.R.Beckham, Nonclassical symmetries of evolutionary partial differential equations and compatibility. J. Math. Anal. Appl., 2004. 289: p. 55-65.
[78] D.Levi and P. Winternitz, Continuous symmetries of discrete equations. Phys. Lett. A, 1991. 152: p. 335-338.
[79] E.M.Vorob'ev, Symmetry analysis of nonlinear PDE with a "Mathematica" program SYMMAN. Nonl. Math. Phys., 1996.3: p. 85-89.
[80] F.Barannyk, L. and B. Kloskowska, On symmetry reduction and invariant solutions to some nonlinear multidimensional heat equations. Reports Math. Phys., 2000. 45: p. 1-22.
[81] F.Finkel, Symmetries of the Fokker-Planck equation with a constant diffusion matrix in 2+1 dimensions. J. Phys. A: Math. Gen., 1999.32: p. 2671-2684.
[82] W. Fushchych, Z. Symenoh and I. Tsyfra, Symmetry of the Shrodinger equation with variable potential, j. Nonl. Math. Phys., 1998.5: p. 13-22.
[83] G. Baumann, Symmetry analysis of differential equations with Mathematica. 2000: Springer.
[84] G.Baumann, MathLie a program of doing symmetry analysis. Math. Comp. Simul., 1998.48: p. 205-223.
[85] G.M.Webb, Lie symmetries of a coupled nonlinear Burgers-Heat equation system. J. Phys. A: Math. Gen., 1990.23: p. 3885-3894.
[86] G. W.Bluman and G. J.Reid, New classes of symmetries for partial differential equations. J. Math. Phys., 1988.29: p. 806-811.
[87] G.W.Bluman and S.Kumei, Symmetries and differential equations. Applied Mathematical Sciences. Vol. 81. 1989.
[88] G.Gaeta and N.R. Quintero, Lie-point symmetries and stochastic differential equations. j. Phys. A: Math. Gen., 1999.32: p. 8485-8505.
[89] J.Goard, Iterating the classical symmetries method to solve initial value problems. IMA J. Appl. Math., 2003.68: p. 313-328.
[90] F.Gungor, Symmetries and invariant solutions of the two-dimensional variable coefficient Burgers equation. J. Phys. A: Math. Gen., 2001.34: p. 4313-4321.
[91] GC Zhu and HH Chen, Symmetries and integrability of the cylindrical Korteweg-de Vries equation. J. Math. Phys., 1986.27: p. 100-103.
[92] Haas, F. and J. Goedert, Lie symmetries for two-dimensional chargerd-particle motion. J. Phys. A: Math. Gen., 2000.33: p. 4661-4677.
[93] I.T.Habibullin, Symmetry approach in boundary value problems. Nonl. Math. Phys., 1996. 3: p. 147-151.
[94] J.Butcher, J.Carminati, and K.T.Vu, A comparative study of some computer algebra packages which determine the Lie point symmetries of differential equations. Compu. Phys. Commu., 2003. 155: p. 92-114.
[95] J.C.Fuchs, Symmetry groups and similarity solutions of MHD equation. J. Math. Phys., 1991.32: p. 1703-1708.
[96] L.O.Tulupova, Symmetry properties and reduction of the generalized nonlinear system of two-phase liquid equations. Nonl. Math. Phys., 1996.3: p. 214-218.
[97] M.A.Almeida, M.E.Magalhaes, and I.C.Moreira, Lie symmetries and invariants of the Lotka-Volterra system. J. Math. Phys., 1995.36: p. 1854-1867.
[98] M.C.Nucci, Iterating the nonclassical symmetries method. Phys. D, 1994. 78: p. 124-134.
[99] M.Euler, et al., Symmetry classification for a coupled nonlinear Shrodinger equations. Nonl. Math. Phys., 1994. 1: p. 358-379.
[100] M.J.Craddock and A.H.Dooley, Symmetry group methods for heat kernels. J. Math. Phys., 2001.42: p. 390-418.
[101] M.L.Gandarias, E.Medina, and C. Muriel, New symmetry reductions for some ordinary differential equations. J. Nonl. Math. Phys., 2002.9 supplement 1: p. 17-58.
[102] M.L,Gandarias and M.S.Bruzon, Nonclassical symmetries for a family of Cahn-Hilliard equations. Phys. Lett. A, 1999. 263: p. 331-337.
[103] M.L.Gandarias and M.S.Bruzon, Classical and nonclassical symmetries of a generalized Boussinesq equation. J. Nonl. Math. Phys., 1998.5: p. 8-12.
[104] M.Lakshmanan and M.S. Velan, Lie symmetries,infinite-dimensional Lie algebra and similarity reductions of certain (2+1)-dimensional nonlinear evolution equations. Nonl. Math. Phys., 1996.3: p. 24-39.
[105] M.S.Bruzon, et al., The symmetry reductions of a turbulence model. J. Phys. A: Math. Gen., 2001.34: p. 3751-3760.
[106] R.A.Kraenkel and M.Senthilvelan, Symmetry analysis of an integrable reaction-diffusion equation. Chaos, Solitons & Fractals, 2001. 12: p. 463-474.
[107] R.A.Kraenkel, M.Senthilvelan, and A.I.Zenchuk, Lie symmetry analysis and reductions of a two-dimensional integrable generalization of the Camassa-Holm equation. Phys. Lett. A, 2000.273: p. 183-193.
[108] R. Gordoa, P., Symmetries, exact solutions and nonlinear superposition formulas for two integrable partial differential equations. J. Math. Phys., 2000. 41: p. 4713-4731.
[109] R. J. Wiltshire, Perturbed Lie symmetry and systems of nonlinear diffusion equation. Nonl. Math. Phys., 1996. 3: p. 130-138.
[110] R. K. Gazizov and N. H. Ibragimov, Lie symmetry analysis of differential equations in Finance. Nonlinear Dynamics, 1998. 15: p. 115-136.
[111] R. Z. Zhdanov and V. I. Lahno, Conditional symmetry of a porous medium equation. Phys. D, 1998. 122: p. 178-186.
[112] Repeta, V., Conditional and Lie symmetry of nonlinear wave equation. Nonl. Math. Phys., 1996. 3: p. 414-416.
[113] Roman, O., Symmetry analysis of the multidimensional polywave equation. Nonl. Math. Phys., 1996. 3: p. 435-440.
[114] S. Lafortune, S. Tremblay, and P. Winternitz, Symmetry classification of diatomic molecular chains. J. Math. Phys., 2001. 42: p. 5341-5357.
[115] Sen, T. and M. Tabor, Lie symmetries of the Lorenz model. Phys. D, 1990. 44: p. 313-339.
[116] Senthilvelan, M. and M. Torrisi, Potential symmetries and new solutions of a simplified model for reacting mixtures. J. Phys. A: Math. Gen., 2000. 33: p. 405-415.
[117] SY Lou, Symmetry analysis and exact solutions of the 2+1 dimensional sine-Gordon system. J. Math. Phys., 2000. 41: p. 6509-6524.
[118] SY Lou and ZJ Lian, Searching for Infinitely Many Symmetries and Exact Solutions Via Repeated Similarity Reductions. Chinese Physical letter, 2005. 22(1): p. 1-4.
[119] S. Spickak and V. Stognii, symmetry classification and exact solutions o the one-dimensional Fokker-Planck equation with arbitrary coefficients of drift and diffusion. J. Phys. A: Math. Gen., 1999. 32: p. 8341-8353.
[120] V. Stognii, Lie and Non-Lie symmetry and some exact solutions of the self-dual equations. Reports Math. Phys., 1999. 44: p. 241-245.
[121] V. A.Tychynin, Nonlocal symmetry of nonlinear wave equations. Nonl. Math. Phys., 1996. 3: p. 409-413.
[122] W. I. Fushchich and O. V. Roman, Symmetry reduction and some exact solutions of nonlinear biwave equation. Reports Math. Phys., 1996. 37: p. 267-281.
[123] W. I. Fushchich, W. M. Shtelen, and N. I. Serov, Symmetry analysis and exact solutions of equations of nonlinear mathematical physics. 1989: Kluwer academic.
[124] XH Niu and ZL Pan, A NEW APPROACH TO SOLVE PERTURBED NONLINEAR EVOLUTION EQUATIONS THROUGH LIE-BACKLUND SYMMETRY METHOD. Appl. Math. J. Chinese Univ. Ser. B, 2006. 21(1): p. 45-51.
[125] Z. J. Zawistowski, Symmetries of integro-differential equations. Reports Math. Phys., 2001. 48: p. 269-276.
[126] 魏诺，《非线性科学基础与应用》，科学出版社，2004
[127] 王明亮，《非线性发展方程与孤立子》，兰州大学出版社，1990
[128] 沈守枫，非线性系统若干问题研究，浙江大学博士学位论文，2005
[129] 潘祖梁，《非线性问题的数学方法及其应用》，浙江大学出版社，杭州，2001
[130] P. J. Olver, Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics. Vol. 107. 1998: Springer.
[131] O. Stormark, Lie structural approach to PDE systems. 2000: Cambridge University Press.
[132] N. H. Ibragimov, CRC handbook of Lie group analysis of differential equations, Vol. 1, 2, 3: CRC Press.
[133] 张解放，郭冠平，(2+1)维破裂孤子方程的新多孤子解．物理学报，2003．52(10)：p．2359-04．
[134] 郭福奎，否定Hirota条件的一个简单方法．应用数学学报，1991．14(1)：p．111-114．
[135] ZJ Zhou, JZ Fu and ZB Li, An implementation for the algorithm of Hirota bilinear form of PDE in the Maple system. Applied Mathematics and Computation, 2006. 183: p. 872-877.
[136] SJ Yu, et al., N soliton solutions to the Bogoyavlenskii-Schiff equation and a quest for the soliton solution in (3+1)dimensions. Journal of Physics A: Mathematical and General, 1998. 31(14): p. 3337-3347.
[137] R. Hirota, Exact solution of the Korteweg-de Vries Equation for multple collishions of solitons. Phys. Rev. Lett., 1971. 27: p. 1192.
[138] Y. Matsuno, Bilinear Transformation Method. 1984.
[139] 钱贤民，张叶，双线性算子与非线性方程的双线性形式，绍兴文理学院学报，2002．22(4)：p．10-15．
[140] 张翼，基于双线性方法的孤子可积系统，上海大学博士学位论文，2004
[141] A. C. Newell, Solitons in Mathematics and physics. Society for Industurial and Applied mathematics. 1985, Philadelphia.
[142] C. Rogers and W. E. Shadwick, Backlund Transformtions and Their Applications. 1982, New York: Academic Press.
[143] R. Hirota, A New Form of Backlund Transformations and Its Relation to the Inverse Scattering Problem. Progress of Theoretical Physics, 1974. 52: p. 1948.
[144] S. Oishi, J. Phys. Soc. Jpn, 1980. 48: p. 639.
[145] A. C. Newell, M. Tabor, and Y. B. Zeng, A UNIFIED APPROACH TO PAINLEVE EXPANSIONS. Phys. D, 1987. 29: p. 1-68.
[146] XH Niu and ZL Pan, New approximate solutions fo the perturbed KdV equation. Physics Letters A, 2006. 349: p. 192-197.
[147] XH Niu, Transformations from the unperturbed equation to the perturbed equation with nth order perturbed. Communications in Nonlinear Science and Numerical Simulation, 2006.
[148] GQ Xu, The soliton solutions, dromions of the Kadomtsev-Petviashvili and Jimbo-Miwa equations in (3+1)-dimensions. Chaos, Solitons & Fractals, 2006. 30: p. 71-76.
[149] B Li, Y Chen and HQ Zhang, Exact travelling wave solutions for a generalized Zakharov-Kuznetsov equation. Appl. Math. Compu, 2003. 146: p. 653-666.
[150] A. M. Wazwaz, Nonlinear dispersive special type of the Zakharov-Kuznetsov equation ZK(n, n) with compact and noncompact structures. Appl. Math. Compu, 2005. 2005: p. 577-590.
[151] 林麦麦，段文山，吕克璞，(3+1)维ZK方程的N孤子解．两北师范大学学报(自然科学版)，2006．42(1)：p．41-45．
[152] 范恩贵，《可积系统与计算机代数》．数学机械化丛书．Vol．6．2004，北京：科学 出版社．
[153] A. Korpel,, Proc IEEE, 1984. 72: p. 1109.
[154] R. Hirota, Phys. Lett. A, 1980. 76: p. 95.
[155] A. Kundu, J. Math. Phys., 1984. 25: p. 3433.
[156] G. W. Bluman and J. D. Cole, J. Math. Phys., 1969. 18: p. 1025.
[157] M. A. Allen and G. Rowlands, On the transverse instabilities of solitary waves. Phys. Lett. A, 1997. 235: p. 145-146.
[158] SY Lou, Dromion-like structures in a (3+1)-dimensional KdV-type equation. J. Phys. A: Math. Gen., 1996. 29: p. 5989-6001.
[159] M. C. Nucci, Painleve property and pseudopotentials for non-linear evolution equations. J. Phys. A: Math. Gen., 1989. 22: p. 2897-2913.
[160] C. R. Gilson and J. J. C. Nimmo, A (2+1)-dimensional generalization of the AKNS shallow water wave equation. Phys. Lett. A, 1993. 180: p. 337-345.
[161] S Zhang, Application of Exp-function method to a KdV equaiton with variable coefficients. Phys. Lett. A, 2007: p. to appear.
[162]E. Mann, J. Math. Phys., 1997. 38: p. 3772.
[163] H. Cai and N. N. Huang, Chin. Phys. Lett., 2002. 20: p. 469.
[164] D. J. Kaup and A. C. Newell, Proc. R. Soc. London Ser. A, 1978. 361: p. 413.
[165] V. I. Karpman and E. M. Maslov, Soviet Phys. JEPT, 1977. 46: p. 281.
[166] V. I. Karpman and E. M. Maslov, Soviet Phys. JEPT, 1978. 48: p. 252.
[167] T. R. Marchant, Chaos, Solitons & Fractals, 2004. 22: p. 261.
[168] Kraenkel, R. A., J. G. Pereira, and E. C. d. R. Nero, Linearizabilty of the perturbed Burgers equation. Physical Revie E, 1998. 58(2): p. 2526-2530.
[169] R. K. Dodd, et al., Solitons and nonlinear wave Equation. 1982, London: Academic.
[170] R. A. Kraenkel, J. G. Pereira, and M. A. Manna, Phys. Rev. A, 1992. 45: p. 838.
[171] F. Calogero, Why Are Certain Nonlinear PDEs Both Widely Applicable and Integrable. What Is Integrability, ed. V. E. Zakharov. 1991, Berlin: Springer.
[172] HZ Liu, ZL Pan and P Li, On Direct Transformation Approach to Asymptotical Analytical Solutions of Perturbed Partial Differential Equation. Communications in Theoretical Physics, 2006. 46(4): p. 587-590.
[173] A. S. Fokas and Q. M. Liu, Phys. Rev. Lett., 1996. 12: p. 2347.
[174] T. R. Marchant, Phys. Rev. E, 1999. 59: p. 3745.
[175] Y. Kodama, Phys. Lett. A, 1985. 107: p. 245.
[176] R. A. Kraenkel, Phys. Rev. E, 1998. 57: p. 4775.
[177] Kerf, F. d., Asymptotic Analysis of a Class of Perturbed Korteweg-de Vries Initial Value Problems. 1988: Centrum voor Wiskunde en Informatica.
[178] 牛晓花，非线性PDE的对称、扰动和约化，浙江大学硕士学位论文，2005
[179] SY Lou and XB Hu, Infinitely many Lax pairs and symmetry constraints of the KP equation. J. Math. Phys., 1997. 38(12): p. 6401-6427.
[180] HY Ruan and YX Chen, STUDY OF A (2+1)-DIMENSIONAL BROER-KAUP EQUATION. ACTA PHYSICA SINICA(Overseas Edition), 1998. 7(4): p. 241.
[181] 张解放，韩平，(2+1)维Broer-Kaup方程的局域相干结构．物理学报，2002．51(4)：p．705-710．
[182] 张解放，刘宇陆，高阶(2+1)维的Broer-Kaup方程的局域相干结构．应用数学与力 学，2002．23(5)：p．489-496．
[183] GT Liu and TY Fan, New Exact Solutions of Broer-Kaup Equations. Communications in Theoretical Physics, 2004. 42(4): p. 488.
[184] CL Bai and H Zhao, Compacton, peakon and folded localized excitations for the (2+1)-dimensional Broer-Kaup system. Chaos, Solitons & Fractals, 2005. 23: p. 777-786.
[185] TC Xia and HQ Zhang, Dromion and other exact solutions of (2+1)-dimensional Broer-Kaup equation. Chaos, Solitons & Fractals, 2003. 16: p. 167-172.
[186] JQ Mei, DS Li and HQ Zhang, New soliton-like and periodic solution of (2+1)-dimensional higher order Broer-Kaup system. Chaos, Solitons & Fractals, 2004. 22: p. 669-674.
[187] 赵熙强，赵熙强，唐登斌，王利民，张耀明，高阶(2+1)维Broer-KauP方程的孤波解．物理学报，2003．52(8)：p．1927-1931．
[188] S Wang, XY Tang and SY Lou, Soliton fission and fusion: Burgers equation and Sharma-Tasso-Olver equation. Chaos, Solitons & Fractals, 2004. 21: p. 231-239,
[189] HZ Liu and ZL Pan, New Exact Solutions of Broer-Kaup Equations. Communications in Theoretical Physics, 2005. 44(1): p. 15-18.
[2] Korteweg, D.J. and G.d. Vries, On the change of form of long waves advancing in a rectangular canal,and on a new type of long stationary waves. Phil.Mag., 1895.39: p. 422-443.
[3] A.Fermi, J.Pasta, and S.Ulam, Studies of nonlinear problems. Los Alamos Sci.Lab.Rep.LA-1940, 1955.
[4] N.J.Zabusky and M.D.Kruskal, Interaction of "Solitons" in a collisionless plasma and the recurrence of initial states. Phys.Rev.Lett., 1965.15: p. 240-243.
[5] A.C.Scott, F.V.F.Chu, and D.McLaughlin, The Soliton: A new concept in applied science. Proc.IEEE, 1973.61: p. 1473-1483.
[6] A.R.Bishop, Solitons in condensed matter physics. Phys.Scripta, 1979.20(409-423).
[7] M.J.Ablowitz and H.Segur, Solitons and inverse scattering transform. 1981, Philadelphia: SIAM.
[8] P.G.Drazin, Solitons. London mathematical society Lecture note series. Vol. 85. 1983: Cambridge University Press.
[9] S.P.Novikov, et al., Theory of Solitons: The Inverse Scattering Method. New York:Consultants Bureau, 1984.
[10] A.C.Newell, Solitons in mathematics and physics. 1985, Philadelphia: PA:SIAM.
[11] A.R.Chowdhury and P.Mitra, New coupled Liouville system: prolongation structure,soliton solution,and complete integrability. Inter. J. Theor. Phys., 1989. 28: p. 119-133.
[12] V.G.Makhankov, V.K.Fedyanin, and O.K.Pashaev, Ivth International Workshop:Solitons and Applications. 1989, Dubna: World Scientific.
[13] P.J.Ovler and D.H.Sattinger, Solitons in Physics, Mathematics, and Nonlinear Optics. 1990, New York: Springer-Verlag.
[14] M.J.Ablowitz and P.A.Clarkson, Solitons,nonlinear evolution equations and inverse scattering. London mathematical society Lecture note series. Vol. 149. 1991: Cambridge University Press.
[15] A.S.Davydov, Solitons in Molecular Systems. Mathematics and its applications. 1991: Kluwer Academic Publisher.
[16] V.B.Matveev and M.A.Salle, Darboux transformations and solitons. 1991: Springer-Verlag.
[17] S.P.Novikov, Solitons and Geometry. 1992: Accadimia Nazionale dei Lincei.
[18] M. Remoissenet,Waves called solitons: concepts and experiments. Springer-Vertag. 1995
[19] Remoissenet, M., Waves Called Solitons 2nd ed. 1999,北京：世界图书出版公司北京公司．
[20] L.A.Dickey, Soliton Equations and Hamiltonian systems. Advanced Series in Mathematical Physics. Vol. 26. 2003: World Scientific.
[21] R.Hirota, The Direct Method in Soliton Theory. 2004: Cambridge University Press.
[22] A.S.Fokas, et al. Nonlinear processes in physics, in Proceedings of Ⅲ Potsdam -V Kiev International Workshop. 1991. Clarkson University, Potsdam, NY: Springer-Verlag.
[23] G. Makhankov and O. K. Pashaev. Nonlinear Evolution Equations and Dynamical Systems. in Proceedings of the 6th International Workshop. 1991. Dubna: Springer-Verlag.
[24] M. Boiti, et al. Proceedings of the workshop on Nonlinearity integrability and all that: twenty years after NEEDS'79. 1999. Lecce, Italy: World Scientific.
[25] Y Cheng, S Hu and YS Li, Nonlinear evolution equations and dynamical systems, in Proceedings of the ICM 2002 satellite conference. 2003: World Scientific.
[26] Steeb, W. -H., Problems and solutions in theoretical and mathematical physics. 2003: World Scientific.
[27] 刘式达，刘式适，《孤波与湍流》，上海科技教育出版社，1994
[28] 李翊神，《孤子与可积系统》， 上海科技教育出版社，1999
[29] 黄念宁，《孤子理论和微扰方法》，上海科技教育出版社，1996
[30] 郭柏灵，苏凤秋，《孤立子》，沈阳：辽宁教育出版社，1997
[31] 谷超豪，郭柏灵，李翊神等，《孤立子理论与应用》，浙江科学技术出版社，1990
[32] 陈陆君，梁昌洪，《孤立子理论及其应用-光孤子理论有光孤子通信》，1997：西安电子科技大学出版社．
[33] C. S. Gardner, et al., Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett., 1967. 19: p. 1095-1097.
[34] 李翊神，汪克林，郭光灿，汪秉宏，《非线性科学选讲》，合肥：中国科学技术大学出版社，1994
[35] C. S. Gardner, et al., Comm pure appl. Math., 1974. 27: p. 97.
[36] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves. Commu. Pure Appl. Math., 1968. 21: p. 467-490.
[37] A. R. Chowdhury and R. Mukherjee, Elliptic solutions, recursion operators and complete Lie-Baclund symmetry for the Harry-Dym equation. Phys. Scripta, 1984. 29(293-295).
[38] K. M. Tamizhmani and M. Lakshmanan, Infinitely many Lie-Backlund symmetries for a quasi-linear evolution equation. Phys. Lett. A, 1982. 90: p. 159-161.
[39] Lee, YQ and M. Ge, Prolongation method of finding Backlund transformation and Lax-pair of Ernst equation and Ernst-maxwell equation, communications in Theoretical Physics, 1988. 10: p. 79-103.
[40] ML Wang and Y Wang, A new Backlund transformation and multi-soliton solutions to the KdV equation with general variable coefficients. Phys. Left. A, 2001. 287: p. 211-216.
[41] A. Bekir, Painleve test for some (2+1)-dimensional nonlinear equations. Chaos, Solitons and Fractals, 2007. 32: p. 449-455.
[42] Chowdhury, A. R., Painleve analysis and its applications. CHAPMAN&HALL/CRC monographs and surveys in pure and applied mathematics. Vol. 105. 2000: CHAPMAN&HALL/CRC.
[43] J. D. Gibbon, et al., The Painleve Property and Hirota's Method. Studies in Applied Mathematics, 1986. 72: p. 39-63.
[44] J. Weiss, M. Tabor, and G. Gerneval, The Painleve property for partial differential equations. J. Math. Phys., 1983. 24(522-526).
[45] K. M. Tamizhmani and P. Punithavathi, Similarity reductions and painleve property of the coupled higher dimensional Burgers equation. Int. J. Non-Linear Mech., 1991. 26: p. 427-438.
[46] K. Shivamoggi, B. and D. K. Rollins, Generalized Painleve formulation and Lie group symmetries of the Zakharov-Kuznetsov equation, phys. Lett. A, 1991. 160: p. 263-266.
[47] Karasu, A., Jordan KdV systems and Painleve property. Inter. J. Theor. Phys., 1997. 36: p. 705-713.
[48] M. Lakshmanan and R. Sahadevan, Painleve property of coupled anharmonic oscillators with N degrees of freedom, phys. Lett. A, 1984. 101: p. 189-194.
[49] A. C. Newell, M. Tabor, and Y. B. Zeng, A UNIFIED APPROACH TO PAINLEVE EXPANSION. Physica D, 1987. 29: p. 1-68.
[50] SL Zhang, B Wu and SY Lou, Painleve analysis and special solutions of generalized Broer-Kaup equations. Physics Letters A, 2002. 300(1).
[51] T. Alagesan, Y. Chung, and K. Nakkeeran, Painleve test for the certain (2+1)-dimensional nonlinear evolution equations. Chaos, Solitons and Fractals, 2005. 26: p. 1203-1209.
[52] W. H. Steeb and N. Euler, Nonlinear evolution equations and Painleve test. 1988: World Scientific.
[53] S. Kichenassamy, WTC expansions and nonintegrable equations. Studies in Applied Mathematics, 1999. 102: p. 1-26.
[54] P. A. Clarkson and S. Hood, New symmetry reductions and exact solutions of the Davey-Stewartson system. i. Reducitons to ordinary differential equations. J. Math. Phys., 1994. 35: p. 255-283.
[55] P. A. Clarkson and M. D. Kruskal, New similarity reductions of the Bossinisq equation. J. Math. Phys., 1989. 30: p. 2201-2213.
[56] P. A. Clarkson and P. Winternitz, Nonclassical symmetry reductions of the Kadomtsev-Pteviashvili equation. Phys. D, 1991. 49: p. 257-272.
[57] SY Lou, Nonclassical symmetry reductions for the dispersive wave equations in shallow water. J. Math. Phys., 1992. 33: p. 4300-4305.
[58] SY Lou and HC Ma, Non-Lie symmetry groups of (2+1)-dimensional nonlinear systems obtained from a simple direct method. J. Phys. A: Math. Gen., 2005. 38: p. L129-L137.
[59] 谷超豪，胡和生，周子翔，孤立子理论中的达布变换及其几何应用．1999：上海科学技术出版社．
[60] Morries, H. C., Prolongation structures and nonlinear evolution equations in two spatial dimensions: a general class of equations. J. Phys. A: Math. Gen., 1979. 12: p. 261-267.
[61] Lee, Y. and J. Lu, Prolongation structure and infinite-dimensional algebra for the isotropic Landau-Lifshitz equation, phys. Lett. A, 1989. 135: p. 117-119.
[62] H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equation. J. Math. Phys., 1975. 16: p. 1-7.
[63] E. V. Doktorov, Prolongation structure without prolongation. J. Phys. A: Math. Gen., 1980. 13: p. 3599-3604.
[64] Degang, Z., Y. Guoxiang, and L. Jie, Integrable deformations of the Heisenberg model: prolongation structure technique. J. Phys. A: Math. Gen., 1991. 24: p. L219-L223.
[65] A. R. Chowdhury and S. Paul, Prolongation structure of a new integrable system. Inter. J. Theor. Phys., 1985. 24: p. 633-637.
[66] R.Dodd and A.Fordy, On the integrability of a system of coupled KdVequations. phys. Lett. A, 1982. 89: p. 168-170.
[67] P. A.Clarkson and T. J.Priestley, Symmetries of a class of nonliear fourth order partial differential equations. J. Nonl. Math. Phys., 1999.6: p. 66-98.
[68] A.G.Johnpillai and A.H.Kara, Variational formulation of approximate symmetries and conservation laws. Inter. J. Theor. Phys., 2001.40: p. 1501-1509.
[69] E.Alfinito, G. Soliani and L. Solombrino, The symmetry structure of the Heavenly equation. Lett. Math. Phys., 1997.41: p. 379-389.
[70] B.Champagne and P.Winternitz, On the infinite-dimensional symmetry group of the Davey-Stewartson equations. J. Math. Phys., 1988.29: p. 1-8.
[71] V.Boyko, Symmetry classification of the one-dimensional second order equation of a hydrodynamic type. Nonl. Math. Phys., 1995.2: p. 418-424.
[72] C.C.A.Sastri and K.A.Dunn, Lie symmetries of some equations of the Fokker-Planck type. J. Math. Phys., 1985.26: p. 3042-3047.
[73] R.Cherniha, Lie symmetries of nonlinear two-dimensional reaction-diffusion systems. Reports Math. Phys., 2000.46: p. 63-76.
[74] A.R.Chowdhury, A. Ghose, and M. Naskar, Similarity soliton and Lie symmetry for a coupled nonlinear system. Inter. J. Theor. Phys., 1987.26: p. 357-363.
[75] G.Cicogna and G. Gaeta, Partial Lie-point symmetries of differential equations. J. Phys. A: Math. Gen., 2001.34: p. 491-512.
[76] D.David, et al., Symmetry reduction for the Kadomtsev-Petviashvill equation using a loop algebra. J. Math. Phys., 1986.27: p. 1225-1237.
[77] D.J.Arrigo and J.R.Beckham, Nonclassical symmetries of evolutionary partial differential equations and compatibility. J. Math. Anal. Appl., 2004. 289: p. 55-65.
[78] D.Levi and P. Winternitz, Continuous symmetries of discrete equations. Phys. Lett. A, 1991. 152: p. 335-338.
[79] E.M.Vorob'ev, Symmetry analysis of nonlinear PDE with a "Mathematica" program SYMMAN. Nonl. Math. Phys., 1996.3: p. 85-89.
[80] F.Barannyk, L. and B. Kloskowska, On symmetry reduction and invariant solutions to some nonlinear multidimensional heat equations. Reports Math. Phys., 2000. 45: p. 1-22.
[81] F.Finkel, Symmetries of the Fokker-Planck equation with a constant diffusion matrix in 2+1 dimensions. J. Phys. A: Math. Gen., 1999.32: p. 2671-2684.
[82] W. Fushchych, Z. Symenoh and I. Tsyfra, Symmetry of the Shrodinger equation with variable potential, j. Nonl. Math. Phys., 1998.5: p. 13-22.
[83] G. Baumann, Symmetry analysis of differential equations with Mathematica. 2000: Springer.
[84] G.Baumann, MathLie a program of doing symmetry analysis. Math. Comp. Simul., 1998.48: p. 205-223.
[85] G.M.Webb, Lie symmetries of a coupled nonlinear Burgers-Heat equation system. J. Phys. A: Math. Gen., 1990.23: p. 3885-3894.
[86] G. W.Bluman and G. J.Reid, New classes of symmetries for partial differential equations. J. Math. Phys., 1988.29: p. 806-811.
[87] G.W.Bluman and S.Kumei, Symmetries and differential equations. Applied Mathematical Sciences. Vol. 81. 1989.
[88] G.Gaeta and N.R. Quintero, Lie-point symmetries and stochastic differential equations. j. Phys. A: Math. Gen., 1999.32: p. 8485-8505.
[89] J.Goard, Iterating the classical symmetries method to solve initial value problems. IMA J. Appl. Math., 2003.68: p. 313-328.
[90] F.Gungor, Symmetries and invariant solutions of the two-dimensional variable coefficient Burgers equation. J. Phys. A: Math. Gen., 2001.34: p. 4313-4321.
[91] GC Zhu and HH Chen, Symmetries and integrability of the cylindrical Korteweg-de Vries equation. J. Math. Phys., 1986.27: p. 100-103.
[92] Haas, F. and J. Goedert, Lie symmetries for two-dimensional chargerd-particle motion. J. Phys. A: Math. Gen., 2000.33: p. 4661-4677.
[93] I.T.Habibullin, Symmetry approach in boundary value problems. Nonl. Math. Phys., 1996. 3: p. 147-151.
[94] J.Butcher, J.Carminati, and K.T.Vu, A comparative study of some computer algebra packages which determine the Lie point symmetries of differential equations. Compu. Phys. Commu., 2003. 155: p. 92-114.
[95] J.C.Fuchs, Symmetry groups and similarity solutions of MHD equation. J. Math. Phys., 1991.32: p. 1703-1708.
[96] L.O.Tulupova, Symmetry properties and reduction of the generalized nonlinear system of two-phase liquid equations. Nonl. Math. Phys., 1996.3: p. 214-218.
[97] M.A.Almeida, M.E.Magalhaes, and I.C.Moreira, Lie symmetries and invariants of the Lotka-Volterra system. J. Math. Phys., 1995.36: p. 1854-1867.
[98] M.C.Nucci, Iterating the nonclassical symmetries method. Phys. D, 1994. 78: p. 124-134.
[99] M.Euler, et al., Symmetry classification for a coupled nonlinear Shrodinger equations. Nonl. Math. Phys., 1994. 1: p. 358-379.
[100] M.J.Craddock and A.H.Dooley, Symmetry group methods for heat kernels. J. Math. Phys., 2001.42: p. 390-418.
[101] M.L.Gandarias, E.Medina, and C. Muriel, New symmetry reductions for some ordinary differential equations. J. Nonl. Math. Phys., 2002.9 supplement 1: p. 17-58.
[102] M.L,Gandarias and M.S.Bruzon, Nonclassical symmetries for a family of Cahn-Hilliard equations. Phys. Lett. A, 1999. 263: p. 331-337.
[103] M.L.Gandarias and M.S.Bruzon, Classical and nonclassical symmetries of a generalized Boussinesq equation. J. Nonl. Math. Phys., 1998.5: p. 8-12.
[104] M.Lakshmanan and M.S. Velan, Lie symmetries,infinite-dimensional Lie algebra and similarity reductions of certain (2+1)-dimensional nonlinear evolution equations. Nonl. Math. Phys., 1996.3: p. 24-39.
[105] M.S.Bruzon, et al., The symmetry reductions of a turbulence model. J. Phys. A: Math. Gen., 2001.34: p. 3751-3760.
[106] R.A.Kraenkel and M.Senthilvelan, Symmetry analysis of an integrable reaction-diffusion equation. Chaos, Solitons & Fractals, 2001. 12: p. 463-474.
[107] R.A.Kraenkel, M.Senthilvelan, and A.I.Zenchuk, Lie symmetry analysis and reductions of a two-dimensional integrable generalization of the Camassa-Holm equation. Phys. Lett. A, 2000.273: p. 183-193.
[108] R. Gordoa, P., Symmetries, exact solutions and nonlinear superposition formulas for two integrable partial differential equations. J. Math. Phys., 2000. 41: p. 4713-4731.
[109] R. J. Wiltshire, Perturbed Lie symmetry and systems of nonlinear diffusion equation. Nonl. Math. Phys., 1996. 3: p. 130-138.
[110] R. K. Gazizov and N. H. Ibragimov, Lie symmetry analysis of differential equations in Finance. Nonlinear Dynamics, 1998. 15: p. 115-136.
[111] R. Z. Zhdanov and V. I. Lahno, Conditional symmetry of a porous medium equation. Phys. D, 1998. 122: p. 178-186.
[112] Repeta, V., Conditional and Lie symmetry of nonlinear wave equation. Nonl. Math. Phys., 1996. 3: p. 414-416.
[113] Roman, O., Symmetry analysis of the multidimensional polywave equation. Nonl. Math. Phys., 1996. 3: p. 435-440.
[114] S. Lafortune, S. Tremblay, and P. Winternitz, Symmetry classification of diatomic molecular chains. J. Math. Phys., 2001. 42: p. 5341-5357.
[115] Sen, T. and M. Tabor, Lie symmetries of the Lorenz model. Phys. D, 1990. 44: p. 313-339.
[116] Senthilvelan, M. and M. Torrisi, Potential symmetries and new solutions of a simplified model for reacting mixtures. J. Phys. A: Math. Gen., 2000. 33: p. 405-415.
[117] SY Lou, Symmetry analysis and exact solutions of the 2+1 dimensional sine-Gordon system. J. Math. Phys., 2000. 41: p. 6509-6524.
[118] SY Lou and ZJ Lian, Searching for Infinitely Many Symmetries and Exact Solutions Via Repeated Similarity Reductions. Chinese Physical letter, 2005. 22(1): p. 1-4.
[119] S. Spickak and V. Stognii, symmetry classification and exact solutions o the one-dimensional Fokker-Planck equation with arbitrary coefficients of drift and diffusion. J. Phys. A: Math. Gen., 1999. 32: p. 8341-8353.
[120] V. Stognii, Lie and Non-Lie symmetry and some exact solutions of the self-dual equations. Reports Math. Phys., 1999. 44: p. 241-245.
[121] V. A.Tychynin, Nonlocal symmetry of nonlinear wave equations. Nonl. Math. Phys., 1996. 3: p. 409-413.
[122] W. I. Fushchich and O. V. Roman, Symmetry reduction and some exact solutions of nonlinear biwave equation. Reports Math. Phys., 1996. 37: p. 267-281.
[123] W. I. Fushchich, W. M. Shtelen, and N. I. Serov, Symmetry analysis and exact solutions of equations of nonlinear mathematical physics. 1989: Kluwer academic.
[124] XH Niu and ZL Pan, A NEW APPROACH TO SOLVE PERTURBED NONLINEAR EVOLUTION EQUATIONS THROUGH LIE-BACKLUND SYMMETRY METHOD. Appl. Math. J. Chinese Univ. Ser. B, 2006. 21(1): p. 45-51.
[125] Z. J. Zawistowski, Symmetries of integro-differential equations. Reports Math. Phys., 2001. 48: p. 269-276.
[126] 魏诺，《非线性科学基础与应用》，科学出版社，2004
[127] 王明亮，《非线性发展方程与孤立子》，兰州大学出版社，1990
[128] 沈守枫，非线性系统若干问题研究，浙江大学博士学位论文，2005
[129] 潘祖梁，《非线性问题的数学方法及其应用》，浙江大学出版社，杭州，2001
[130] P. J. Olver, Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics. Vol. 107. 1998: Springer.
[131] O. Stormark, Lie structural approach to PDE systems. 2000: Cambridge University Press.
[132] N. H. Ibragimov, CRC handbook of Lie group analysis of differential equations, Vol. 1, 2, 3: CRC Press.
[133] 张解放，郭冠平，(2+1)维破裂孤子方程的新多孤子解．物理学报，2003．52(10)：p．2359-04．
[134] 郭福奎，否定Hirota条件的一个简单方法．应用数学学报，1991．14(1)：p．111-114．
[135] ZJ Zhou, JZ Fu and ZB Li, An implementation for the algorithm of Hirota bilinear form of PDE in the Maple system. Applied Mathematics and Computation, 2006. 183: p. 872-877.
[136] SJ Yu, et al., N soliton solutions to the Bogoyavlenskii-Schiff equation and a quest for the soliton solution in (3+1)dimensions. Journal of Physics A: Mathematical and General, 1998. 31(14): p. 3337-3347.
[137] R. Hirota, Exact solution of the Korteweg-de Vries Equation for multple collishions of solitons. Phys. Rev. Lett., 1971. 27: p. 1192.
[138] Y. Matsuno, Bilinear Transformation Method. 1984.
[139] 钱贤民，张叶，双线性算子与非线性方程的双线性形式，绍兴文理学院学报，2002．22(4)：p．10-15．
[140] 张翼，基于双线性方法的孤子可积系统，上海大学博士学位论文，2004
[141] A. C. Newell, Solitons in Mathematics and physics. Society for Industurial and Applied mathematics. 1985, Philadelphia.
[142] C. Rogers and W. E. Shadwick, Backlund Transformtions and Their Applications. 1982, New York: Academic Press.
[143] R. Hirota, A New Form of Backlund Transformations and Its Relation to the Inverse Scattering Problem. Progress of Theoretical Physics, 1974. 52: p. 1948.
[144] S. Oishi, J. Phys. Soc. Jpn, 1980. 48: p. 639.
[145] A. C. Newell, M. Tabor, and Y. B. Zeng, A UNIFIED APPROACH TO PAINLEVE EXPANSIONS. Phys. D, 1987. 29: p. 1-68.
[146] XH Niu and ZL Pan, New approximate solutions fo the perturbed KdV equation. Physics Letters A, 2006. 349: p. 192-197.
[147] XH Niu, Transformations from the unperturbed equation to the perturbed equation with nth order perturbed. Communications in Nonlinear Science and Numerical Simulation, 2006.
[148] GQ Xu, The soliton solutions, dromions of the Kadomtsev-Petviashvili and Jimbo-Miwa equations in (3+1)-dimensions. Chaos, Solitons & Fractals, 2006. 30: p. 71-76.
[149] B Li, Y Chen and HQ Zhang, Exact travelling wave solutions for a generalized Zakharov-Kuznetsov equation. Appl. Math. Compu, 2003. 146: p. 653-666.
[150] A. M. Wazwaz, Nonlinear dispersive special type of the Zakharov-Kuznetsov equation ZK(n, n) with compact and noncompact structures. Appl. Math. Compu, 2005. 2005: p. 577-590.
[151] 林麦麦，段文山，吕克璞，(3+1)维ZK方程的N孤子解．两北师范大学学报(自然科学版)，2006．42(1)：p．41-45．
[152] 范恩贵，《可积系统与计算机代数》．数学机械化丛书．Vol．6．2004，北京：科学 出版社．
[153] A. Korpel,, Proc IEEE, 1984. 72: p. 1109.
[154] R. Hirota, Phys. Lett. A, 1980. 76: p. 95.
[155] A. Kundu, J. Math. Phys., 1984. 25: p. 3433.
[156] G. W. Bluman and J. D. Cole, J. Math. Phys., 1969. 18: p. 1025.
[157] M. A. Allen and G. Rowlands, On the transverse instabilities of solitary waves. Phys. Lett. A, 1997. 235: p. 145-146.
[158] SY Lou, Dromion-like structures in a (3+1)-dimensional KdV-type equation. J. Phys. A: Math. Gen., 1996. 29: p. 5989-6001.
[159] M. C. Nucci, Painleve property and pseudopotentials for non-linear evolution equations. J. Phys. A: Math. Gen., 1989. 22: p. 2897-2913.
[160] C. R. Gilson and J. J. C. Nimmo, A (2+1)-dimensional generalization of the AKNS shallow water wave equation. Phys. Lett. A, 1993. 180: p. 337-345.
[161] S Zhang, Application of Exp-function method to a KdV equaiton with variable coefficients. Phys. Lett. A, 2007: p. to appear.
[162]E. Mann, J. Math. Phys., 1997. 38: p. 3772.
[163] H. Cai and N. N. Huang, Chin. Phys. Lett., 2002. 20: p. 469.
[164] D. J. Kaup and A. C. Newell, Proc. R. Soc. London Ser. A, 1978. 361: p. 413.
[165] V. I. Karpman and E. M. Maslov, Soviet Phys. JEPT, 1977. 46: p. 281.
[166] V. I. Karpman and E. M. Maslov, Soviet Phys. JEPT, 1978. 48: p. 252.
[167] T. R. Marchant, Chaos, Solitons & Fractals, 2004. 22: p. 261.
[168] Kraenkel, R. A., J. G. Pereira, and E. C. d. R. Nero, Linearizabilty of the perturbed Burgers equation. Physical Revie E, 1998. 58(2): p. 2526-2530.
[169] R. K. Dodd, et al., Solitons and nonlinear wave Equation. 1982, London: Academic.
[170] R. A. Kraenkel, J. G. Pereira, and M. A. Manna, Phys. Rev. A, 1992. 45: p. 838.
[171] F. Calogero, Why Are Certain Nonlinear PDEs Both Widely Applicable and Integrable. What Is Integrability, ed. V. E. Zakharov. 1991, Berlin: Springer.
[172] HZ Liu, ZL Pan and P Li, On Direct Transformation Approach to Asymptotical Analytical Solutions of Perturbed Partial Differential Equation. Communications in Theoretical Physics, 2006. 46(4): p. 587-590.
[173] A. S. Fokas and Q. M. Liu, Phys. Rev. Lett., 1996. 12: p. 2347.
[174] T. R. Marchant, Phys. Rev. E, 1999. 59: p. 3745.
[175] Y. Kodama, Phys. Lett. A, 1985. 107: p. 245.
[176] R. A. Kraenkel, Phys. Rev. E, 1998. 57: p. 4775.
[177] Kerf, F. d., Asymptotic Analysis of a Class of Perturbed Korteweg-de Vries Initial Value Problems. 1988: Centrum voor Wiskunde en Informatica.
[178] 牛晓花，非线性PDE的对称、扰动和约化，浙江大学硕士学位论文，2005
[179] SY Lou and XB Hu, Infinitely many Lax pairs and symmetry constraints of the KP equation. J. Math. Phys., 1997. 38(12): p. 6401-6427.
[180] HY Ruan and YX Chen, STUDY OF A (2+1)-DIMENSIONAL BROER-KAUP EQUATION. ACTA PHYSICA SINICA(Overseas Edition), 1998. 7(4): p. 241.
[181] 张解放，韩平，(2+1)维Broer-Kaup方程的局域相干结构．物理学报，2002．51(4)：p．705-710．
[182] 张解放，刘宇陆，高阶(2+1)维的Broer-Kaup方程的局域相干结构．应用数学与力 学，2002．23(5)：p．489-496．
[183] GT Liu and TY Fan, New Exact Solutions of Broer-Kaup Equations. Communications in Theoretical Physics, 2004. 42(4): p. 488.
[184] CL Bai and H Zhao, Compacton, peakon and folded localized excitations for the (2+1)-dimensional Broer-Kaup system. Chaos, Solitons & Fractals, 2005. 23: p. 777-786.
[185] TC Xia and HQ Zhang, Dromion and other exact solutions of (2+1)-dimensional Broer-Kaup equation. Chaos, Solitons & Fractals, 2003. 16: p. 167-172.
[186] JQ Mei, DS Li and HQ Zhang, New soliton-like and periodic solution of (2+1)-dimensional higher order Broer-Kaup system. Chaos, Solitons & Fractals, 2004. 22: p. 669-674.
[187] 赵熙强，赵熙强，唐登斌，王利民，张耀明，高阶(2+1)维Broer-KauP方程的孤波解．物理学报，2003．52(8)：p．1927-1931．
[188] S Wang, XY Tang and SY Lou, Soliton fission and fusion: Burgers equation and Sharma-Tasso-Olver equation. Chaos, Solitons & Fractals, 2004. 21: p. 231-239,
[189] HZ Liu and ZL Pan, New Exact Solutions of Broer-Kaup Equations. Communications in Theoretical Physics, 2005. 44(1): p. 15-18.