可积哈密顿系统及其代数结构
|
摘要
本文研究内容主要涉及可积系统的四个方面:与连续谱问题相联系的无穷维和有限维Hamilton系统;与离散谱问题相联系的Hamilton系统和无穷守恒律;可积耦合系统零曲率方程的代数结构;非线性方程族的Darboux变换和精确解.
第一章,作为与本文相关的研究背景,简要综述了孤立子理论的产生和发展过程,特别针对性地介绍了近年来国内外在可积系统方面的研究成果和发展状况.
第二章,从一个广义Kaup-Newell谱问题出发,导出广义Kaup-Newell方程族,并利用迹恒等式建立该方程族的双Hamilton结构.借助非线性化方法,将广义Kaup-Newell谱问题非线性化为有限维完全可积的Hamilton系统,进而利用可换流的对合解给出孤子方程族解的对合表示.利用李代数的半直和思想构造了广义Kaup-Newell方程族的可积耦合系统,并得到了耦合系统的拟Hamilton结构.
第三章,从一个离散谱问题出发,导出离散的正负发展方程族,利用离散形式的迹恒等式建立了这两个方程族的双Hamilton结构,并进一步获得了相应方程族的无穷守恒律.
第四章,基于一种特殊的李代数半直和,通过定义Lax算子的交换关系,研究了连续和离散耦合系统的零曲率表示的代数结构,并将这种结构分别应用到AKNS方程族和Volterra离散族所生成的等谱族的τ—对称代数.
第五章,从另一个广义Kaup-Newell谱问题出发,导出了相应的非线性发展方程族,并进一步借助Lax对的规范变换,统一地构造了整个方程族的Darboux变换和精确解.
The focus of this dissertation is on the following four topics in the theory of integrable system:infinite-dimensional and finite-dimensional Hamiltonian systems related to a continuous spectral problem;the Hamiltonian systems and infinite conservation laws associated with a discrete spectral problem;the algebraic structure of the zero curvature for an integrable coupling system;the Darboux transformation and exact solutions for a hierarchy of nonlinear equations.
Chapter 1 is devoted to the research background in connection with the dissertation. We briefly outline the origination and development of the soliton theory. Subsequently,we summarize the recent development and achievement in the integrable systems at home and abroad.
In chapter 2,starting from a generalized Kaup-Newell(KN) spectral problem, we derive a generalized KN hierarchy.It is shown that the hierarchy is integrable in Liouville sense and possesses bi-Hamiltonian structure via the trace identity. Moreover,the spectral problem can be nonlineared into a finite-dimensional completely integrable Hamiltonian system through the nonlinearization of the Lax pair. The involutive representation of the solutions for the corresponding soliton equations is given due to the involutive solutions of the commuting flows.Finally,we construct the integrable coupling system of the generalized KN hierarchy and its quasi-Hamiltonian structure by using the conception of semi-direct sums of Lie algebraic.
In chapter 3,by making use of a concrete spectral problem,we deduce a positive and a negative evolution equation hierarchies.It is found that the both hierarchies possess bi-Hamiltonian structure.The infinite conservation laws of the two hierarchies are further obtained based on their Lax pairs.
In chapter 4,based on a kind of special semi-direct sums of Lie algebra,we focus on the algebraic structure of zero curvature representations associated with continuous and discrete integrable couplings(including continue and concrete cases) by defining the commutator of Lax operator.Further we apply such Lie structures in the AKNS and the Velterra integrable couplings to generateτ-symmetry algebras of the corresponding isospectral flows.
In chapter 5,starting from another generalized KN spectral problem,we present a generalized KN hierarchy of nonlinear evolution equations.Following the idea of gauge transformation of Lax pairs,we further provide an uniformly the Darboux transformation and corresponding exact solutions for the whole hierarchy.
第一章,作为与本文相关的研究背景,简要综述了孤立子理论的产生和发展过程,特别针对性地介绍了近年来国内外在可积系统方面的研究成果和发展状况.
第二章,从一个广义Kaup-Newell谱问题出发,导出广义Kaup-Newell方程族,并利用迹恒等式建立该方程族的双Hamilton结构.借助非线性化方法,将广义Kaup-Newell谱问题非线性化为有限维完全可积的Hamilton系统,进而利用可换流的对合解给出孤子方程族解的对合表示.利用李代数的半直和思想构造了广义Kaup-Newell方程族的可积耦合系统,并得到了耦合系统的拟Hamilton结构.
第三章,从一个离散谱问题出发,导出离散的正负发展方程族,利用离散形式的迹恒等式建立了这两个方程族的双Hamilton结构,并进一步获得了相应方程族的无穷守恒律.
第四章,基于一种特殊的李代数半直和,通过定义Lax算子的交换关系,研究了连续和离散耦合系统的零曲率表示的代数结构,并将这种结构分别应用到AKNS方程族和Volterra离散族所生成的等谱族的τ—对称代数.
第五章,从另一个广义Kaup-Newell谱问题出发,导出了相应的非线性发展方程族,并进一步借助Lax对的规范变换,统一地构造了整个方程族的Darboux变换和精确解.
The focus of this dissertation is on the following four topics in the theory of integrable system:infinite-dimensional and finite-dimensional Hamiltonian systems related to a continuous spectral problem;the Hamiltonian systems and infinite conservation laws associated with a discrete spectral problem;the algebraic structure of the zero curvature for an integrable coupling system;the Darboux transformation and exact solutions for a hierarchy of nonlinear equations.
Chapter 1 is devoted to the research background in connection with the dissertation. We briefly outline the origination and development of the soliton theory. Subsequently,we summarize the recent development and achievement in the integrable systems at home and abroad.
In chapter 2,starting from a generalized Kaup-Newell(KN) spectral problem, we derive a generalized KN hierarchy.It is shown that the hierarchy is integrable in Liouville sense and possesses bi-Hamiltonian structure via the trace identity. Moreover,the spectral problem can be nonlineared into a finite-dimensional completely integrable Hamiltonian system through the nonlinearization of the Lax pair. The involutive representation of the solutions for the corresponding soliton equations is given due to the involutive solutions of the commuting flows.Finally,we construct the integrable coupling system of the generalized KN hierarchy and its quasi-Hamiltonian structure by using the conception of semi-direct sums of Lie algebraic.
In chapter 3,by making use of a concrete spectral problem,we deduce a positive and a negative evolution equation hierarchies.It is found that the both hierarchies possess bi-Hamiltonian structure.The infinite conservation laws of the two hierarchies are further obtained based on their Lax pairs.
In chapter 4,based on a kind of special semi-direct sums of Lie algebra,we focus on the algebraic structure of zero curvature representations associated with continuous and discrete integrable couplings(including continue and concrete cases) by defining the commutator of Lax operator.Further we apply such Lie structures in the AKNS and the Velterra integrable couplings to generateτ-symmetry algebras of the corresponding isospectral flows.
In chapter 5,starting from another generalized KN spectral problem,we present a generalized KN hierarchy of nonlinear evolution equations.Following the idea of gauge transformation of Lax pairs,we further provide an uniformly the Darboux transformation and corresponding exact solutions for the whole hierarchy.
引文
[1] Ablowitz M.J. et al., The inverse scattering transform-Fourier analysis for nonlinear problems[J]. Studies in Appl. Math., 1974, 53: 249-315.
[2] Ablowitz M.J., Ramani A. and Segur H., A connection between nonlinear evolution equations and ordinary differential equations of p-type[J]. J. Math. Phys., 1980, 21(5): 1006-1015.
[3] Ablowitz M.J. and Segur H., Soliton and the Inverse Scattering Transform[M]. Philadelphia: SIAM, 1981.
[4] Ablowitz M.J.and Clarkson P.A., Solitons, Nonlinear Evolution Equation and Inverse Scattering[M]. Cambridge: Cambridge University Press, 1991.
[5] Ahmad S. and Chowdhury A.R., On the quasi-periodic solutions to the discrete nonlinear Schrodinger equation[J]. J. Phys. A: Math. Gen., 1987, 20: 293-303.
[6] Alber S.I., Investigation of equations of KdV type by the method of recurrence relations[J]. J. London Math. Soc, 1979, 19: 467-480.
[7] Andrew P. and Zhu Z.N., New integrable lattice hier archies [J]. Phys. Lett. A, 2006, 349: 439-445.
[8] Arnold V.I., Mathematical Methods of classical Mechanics[M]. New York: Springer-Verlag, 1978.
[9] Belokolos E. et al., Algebro-geometrical approach to nonlinear integrable equa-tions[Mj. Berlin: Springer-Verlag, 1994.
[10] Bogoyavlenskii O.I. and Novikov S.P., The connection between the Hamiltonian formulalisms of stationary and nonstationary problems [J]. Functional Anal. Appl., 1976, 10: 9 - 13.
[11] Cao C.W., Nonlinearization of the Lax system for AKNS hierarchy [J]. Sci. Chin. A, 1990, 33: 528-536.
[12]Cao C.W.and Geng X.G.,Classical integrable systems generated through nonlinearization of eigenvalue problems[R].Nonlinear Physics,Berlin:Springer,1990:68-78.
[13]Cao C.W.and Geng X.G.,C.Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy[J].J.Phys.A,1990,23(18):4117-4125.
[14]Cao C.W.,Geng X.G.and Wu Y.T.,From the special 2+1 Toda lattice to the Kadomtsev- Petviashvili equation[J].J.Phys.A:Math.Gen.,1999,32(46):8059-8078.
[15]Cao C.W.,Wu Y.T.and Ceng X.G.,Relation between the Kadometsev-Petviashvili equation and the confocal involutive system[J].J.Math.Phys.,1999,40(8):3948-3970.
[16]Cao C.W.,Geng X.G.and Wang H.Y.,Algebro-geometric solution of the 2+1dimensional Burgers equation with a discrete variable[J].J.Math.Phys.,2002,43(1):621-643.
[17]Chen D.Y.,The potential constraints of the complex(1+2)-dimensional soliton systems and related Hamiltonian equations[J].J.Math.Phys.,1998,39(6):3246-3259.
[18]Chen D.Y.,Zhang D.J.and Deng S.F.,The novel multisoliton solutions of the mKdV-sine-Gordon equation[J].J.Phys.Soc.Japan,2002,71:658-659.
[19]Chen D.Y.,Zhang D.J.and Deng S.F.,Remarks on some solutions of soliton equations[J].J.Phys.Soc.Japan,2002,71:2071-2072.
[20]陈登远,孤子引论[M].北京:科学出版社,2006.
[21]程艺、汪存启、田畴、李翊神,Burgers方程簇的对称、Backlund变换和Painlevé性质之间的关系[J].应用数学学报,1991,14:180-184.
[22]Cheng Y.and Li Y.S.,The constraint of the Kadomtsev-Petviashvili equation and its special solutions[J].Phys.Left.A,1991,157(1):22-26.
[23]Cheng Y.and Li Y.S.,Constraints of the 2+1 dimensional integrable soliton systems[J],J.Phys.A:Math.Gen.,1992,25:419-431.
[24]Cheng Y.,Strampp W.and Zhang Y.J.,Bilinear Backlund transformations for the KP and K-constrained KP hierarchy[J].Phys.Lett.A,1993,182:71-76.
[25]Clarkson P.A.and Kruskal M.D.,New similiarity reductions of the Boussinesq equation[J].J.Math.Phys.,1989,30(10):2201-2213.
[26]Conte R.,Invariant painlevé analysis of partial differential equations[J].Phys.Lett.A,1989,140(7-8):383-390.
[27]Deng S.F.and Chen D.Y.,The novel multisoliton solutions of the KP equation [J].J.Phys.Soc.Japan,2001,70:3174-3175.
[28]Dickey L.A.,Integrable nonlinear equations and Liouville's theorem[J].Commun.Math.Phys.,1981,82:345-375.
[29]Dickey L.A.,Hamiltonian structures and Lax equations generated by matrix differential operators with polynomial dependence on a parameter[J].Commun.Math.Phys.,1983,88:27-42.
[30]Dickey L.A.,Soliton equations and Hamiltonian systems[M].Singapo:World Scientific,1991.
[31]Fan E.G.,Integrable evolution systems based on Gerdjikov-Ivanov equations,Bi- Hamiltonian structure,finite-dimensional integrable systems and N-fold Darboux transformation[J].J.Math.Phys.,2000,41(11):7769-7782.
[32]Fan E.G.,The zero curvature representation for hierarchies of nonlinear evolution equations[J].Phys.Lett.A,2000,274(3-4):135-142.
[33]Fan E.G.,Zhang H.Q.,A hierarchy of nonlinear evolution equations,its Bi-Hamiltonian structure,and finite-dimensional integrable systems[J].J.Math.Phys.,2000,41(4):2058-2065.
[34]Fan E.G.,Integrable systems of derivative nonlinearSchrodinger type and their multi- Hamiltonian structure[J].J.Phys.A:Math.Gen.,2001,34(3):513-519.
[35]Fan E.G.and Zhang H.Q.,A new completely integrable Liouville's system,its Lax representation and Bi-Hamiltonian structure[J].Appl.Math.Mech-Eng.edit.,2001,22(5):520-527.
[36]Fan E.G.,Bi-Hamiltonian structure and liouville integrablity for a Gerdjikov-Ivanov equation hierarchy[J].Chin.Phys.Lett.,2001,18(1):1-3.
[37]范恩贵,可积系统与计算机代数[M].北京:科学出版社,2004.
[38]Fan E.G.and Zhang Y.F.,A simple method for generating integrable hierarchies with multi-potential functions[J].Chaos Solitons Fractals,2005,25:425-439.
[39]Flaschka H.,Relations between infinite-dimensional and finite-dimensional isospectral equations[M].Singapore:Kyoto,Japan,World Sci,1983:219-239.
[40]Fordy A.P.and Kulish P.P.,NonlinearSchrodinger equations and simple Lie algebras[J].Comm.Math.Phys.,1983,89(3):427-443.
[41]Freeman N.C.and Nimmo J.C.,Soliton solutions of the KdV and KP equations:the Wronskian technique[J].Phys.lett.A,1983,95:1-3.
[42]Fuchssteiner B.,Application of hereditary symmetries to nonlinear evolution equations[J].Nonlinear Anal.,1979,3(6):849-862.
[43]Fuchssteiner B.and Fokas A.S.,Symplectic structures,their backlund transformations and hereditary symmetries[J].Physica D,1981,4:47-66.
[44]Fuchssteiner B.,Master symmetries,Higher order time-dependent symmetries and conserved densities of nonlinear evoluton equations[J].Prog.Theor.Phys.,1983,70:1508-1522.
[45]Gardner C.S.et al.,Method for solving the Korteweg-de Vries equation[J].Phys.Rev.Lett.1967,19(19):1095-1103.
[46]Gel' fand I.M.and Dikii L.A.,The asympototics of the resolvent of Sturm-Liouville equations and the algebra of the KdV equations[J].Usp.Matem.Nauk.,1975,30(5):67-100.
[47]Gel' fand I.M.and Dikii L.A.,A Lie algebraic structure in a formal varational calculus[J].Funk.Anal.Pril.,1976,10(1):18-25.
[48]Gel' fand I.M.and Dikii L.A.,Fractional powers of operators and Hamiltonian systems[J].Funk.Anal.Pril.,1976,10(4):13-29.
[49]Gel' fand I.M.and Dikii L.A.,Integrable nonlinear equations and the liouville theorem[J].Funk.Anal.Pril.,1979,13(1):8-20.
[50]Geng X.G.,Discrete Bargmann and Neumann systems and finite-dimensional integrable systems[J].Phys.A,1994,212(1-2):132-142.
[51]Geng X.G.and Wu Y.T.,Finite-band solutions of the classical Boussinesq-Burgers equations[J].J.Math.Phys.,1999,40(6):2971-2982.
[52]Geng X.G.,Wu Y.T.and Cao C.W.,Quasi-periodic solutions of the modified Kadomtsev- Petviashvili equation[J].J.Phys.A.,1999,32(20):3733-3742.
[53]Geng X.G.,Cao C.W.and Dai H.H.,Quasi-periodic solutions for some(2+1)-dimensional integrable models generated by the Jaulent-Miodek hierarchy]J].J.Phys.A:Math.Gen.,2001,34(5):989-1004.
[54]Geng X.G.,Dai H.H.and Cao C.W.,Algebro-geometric constructions of the discrete Ablowitz-Ladik flows and applications[J].J.Math.Phys.,2003,44(10):4573-4588.
[55]谷超豪等,孤立子理论与应用[M].杭州:浙江科学技术出版社,1990年.
[56]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[M].上海:上海科学技术出版社,1999.
[57]He J.S.et al.,Determinator representation of Darboux transformation for the AKNS system[J].Sci.China Ser.A 2006,49(12):1867-1878.
[58]Hisakado M.,Breather trapping mechanism in piecewise homogeneous DNA[J].Phys Lett A,1997,227(1-2):87-93.
[59]Hirota R.,Soliton of the KdV equation for multiple collusions of solitons[J].Phys.Rev.Lett.,1971,27(8):1192-1194.
[60]Hirota R.,Exact N-soliton solutions of the wave equation of long waves in shallow water and in nonlinear lattices[J].J.Math.Phys.,1973,14(7):810-814.
[61]Hirota R.,A few forms of Backlund transformations and its relation to inverse scattering problem[J].Prog.Theor.Phys.,1974,52:1498-1512.
[62]Hirota R.and Satsuma J.,Nolinear evolution equations generated from the Backlund transformation for the Boussinesq equation[J].Proy.Thoer.Phys.,1977,57:797-807.
[63]Hon Y.C.and Fan E.G.,An algebro-geometric solution for a Hamiitonian system with application to dispersive long wave equation[J].J.Math.Phys.,2005,46:032701,22pp.
[64]Hu X.B.,A powerful approach to generate new integrable systems[J].J.Phys.A:Math.Gen.,1994,27:2497-2514.
[65]Hu X.B.,Rational solutions of integrable equations via nonlinear superposition formulae[J].J.Phys.A,1997,30(23):8225-8240.
[66]Hu X.B.,A approach to generate superextensions of integrable systems[J].J.Phys.A,1997,30:619-632.
[67]Hu X.B.et al.,Lax pairs and Backlundtransformations for a coupled ramani equation and its related system[J].Appl.Math.Lett.,2000,13:45-48.
[68]Hu X.B.and Wang H.Y.,New type of Kadomtsev-Petviashvili equation with self-consistent sources and its bilinear Backlund transformation[J].Inverse Problem,2007,23(4):1433-1444.
[69]Huang G.X.,Lou S.Y.and Dai X.X.,Exact and explicit solitary wave solutions to a model equation for water waves[J].Phys.Lett.A,1989,139:373-374.
[70]Jimbo M.,Kruskal M.D.and Miwa T.,Painlevé test for the self-dual Yang-Mills equation[J].Phys.Lett.A,1982,92(2):59-60.
[71]Kaup D.J.and Newll A.C.,An exact solution for a derivative nonlinear Schrodinger equation[J].J.Math.Phys.1978,19:798-801.
[72]Konno K.,Sanuki H.and Ichikawa Y.H.,Conservation laws of nonlinear evolution equations[J].Progr.Theoret.Phys.1974,52:375-382.
[73]Li C.X.,A hierarchy of coupled Korteweg-de Vries equations and the corresponding finitedimensional integrable system[J].J.Phys.Soc.Jap.,2004,73(2):327-331.
[74]李翊神、朱国诚,一个谱可变演化方程的对称[J].科学通报,1986,31:1449-1453.
[75]Li Y.S.and Cheng Y.,Symmetries and constants of motion for the new KdV hierarchies[J].Sci.Sinica Ser.A,1988,31:769-776.
[76]李翊神,孤子与可积系统[M].上海:上海科技教育出版社,1999年,
[77]Li Y.S.,Ma W.X.and Zhang J.E.,Darboux transformations of classical Boussinesq system and its how solutions[J].Phys.Lett.A,2000,275:60-66
[78]Li Y.S.and Ma W.X.,Hamiltonian structures of binary higher-order constrained flows associated with two 3×3 spectral problems[J].Phys.Lett.A 2000,272:245-256.
[79]Li Y.S.and Zhang J.E.,Darboux transformation of classical Boussinesq system and its multi-soliton solutions[J].Phy.Lett.A,2001,284:253-258
[80]Li Z.B.and Wang M.L.,Travelling wave solutions to the two-dimensional KdV-Burgers equation[J].J.Phys.A,1993,26:6027-6032.
[81]Li Z.B.,Wu method and solitons[A].Proc.of ASCM'95,He Shi and Hidetsune Kobayashi[C].Tokyo:Scientists Incorporated,1995:157.
[82]Lin R.L.,Zeng Y.B.and Ma W.X.,Solving the KdV hierarchy with self-consistent sources by inverse scattering method[J].Phys.Lett.A,2001,291:287-298.
[83]Liu Q.P.,Darboux transformations for supersymmetric Korteweg-de Vries equations[J].Lett.Math.Phys.,1995,35(2):115-122.
[84]Liu Q.P.and Mannas M.,Darboux transformations for super-symmetric KP hierarchies[J].Phys.Lett B,2000,485:293-300.
[85]Liu Q.P.and Hu X.B.,Bilinearization of N=1 supersymmetric Korteweg-de Vries equation revisited[J].J.Phys.A,2005,38:371-6378.
[86]Liu Q.P.,Hu X.B.and Zhang M.X.,Supersymmetric modified Korteweg-de Vries equation:bilinear approach[J].Nonlinearity,2005,18:1597-1603.
[87]Liu Q.P.and Yang X.X.,Supersymmetric two-boson equation:Bilinearization and solutions[J].Phys.Lett.A,2006,351:131-135.
[88]Liu X.J.and Zeng Y.B.,On the Toda Lattice equation with self-consistent sources[J].J.Phys.A,2005,38(41):8951-8965
[89]Lou S.Y.,A note on the new similarity reductions of the Boussinesq equation[J].Phys.Lett.A,1990,151(3-4):133-135.
[90]Lou S.Y.,Similarity reductions of the KP equation by a direct method[J].J.Phys.A,1990,23:L649-L654.
[91]Lou S.Y.,Symmetries of the Kadomtsec-Petviashvili equation[J].J.Phys.A:Math.Gen.,1993,26:4387-4394.
[92]Lou S.Y.,Generalized symmetries and ω_∞ algebras in three dimensional Toda field theory[J].Phys.Rev.Lett.,1993,71:4099-4102.
[93]Lou S.Y.,Abundant symmetries for the 1+1 dimensional classical Liouville field theory[J].J.Math.Phys,1994,35(5):2336-2348.
[94]Lou S.Y.,Symmtries of the KdV equation and Hierarchies of the integrodifferential KdV equations[J].J.Math.Phys.,1994,35(5):2390-2396.
[95]Lou S.Y.and Hu X.B.,Infinitely many Lax pairs and symmetry constraints of the KP equation[J].J.Math.Phys.,1997,39:6401-6427
[96]Lou S.Y.and Hu X.B.,Broer-Kaup system from Darboux transformation refated symmetry constraints of Kadomtsev-Petviashvili equation[J].Commun.Theor.Phys.,1998,29:145-148.
[97]Lou S.Y.,On the coherent structures of Nizhnik-Novikov-Veselov equation[J].Phys.Lett.A,2000,277:94-100.
[98]Lou S.Y.and Tang X.Y.,Conditional similarity reduction approach:Jimbo-Miwa equation[J].Chin.Phys.,2001,10:897-901.
[99]楼森岳、唐晓艳,非线性数字物理方法[M].北京:科学出版社,2006.
[100]Luo L.,Darboux transformation and exact solutions for a hierarchy of nonlinear evolution equations[J].J.Phys.A:Math.Theor.,2007,40(15):4169-4179.
[101]Luo L.and Fan E.G.,Hamiltonian systems and integrable coupling associated with a new discrete spectral problem[J].Phys.Lett.A,2007,370(3-4):234-242.
[102]Luo L.and Fan E.G.,Integrable decomposition of a hierarchy of soliton equations and integrable coupling system by semidirect sums of Lie algebras[J].Nonlinear Analysis,In Press.
[103]Ma W.X.,K-symmetries and r-symmetries of evolution equations and their Lie algebras[J].J.Phys.A,1990,23:2707-2716.
[104]Ma W.X.,The algebraic structure of isospectral Lax operators and application to integrable equations[J].J.Phys.A,1992,25:5329-5343.
[105]Ma W.X.,The algebraic structure of zero curvature representations and application to coupled KdV systems[J].J.Phys.A,1993,26:2573-2582.
[106]Ma W.X.,A simple scheme for generating nonisospectral flows from zero curvature representation[J].Phys.Lett.A,1993,179:179-185.
[107]Ma W.X.,A method of zero curvature representation for constructing symmetry algebras of integrable systems[A].Proceedings of the 21st International Conference on the Differential Geometry Methods in Theoretical Physics[C].Singapore:World Scientific,1993:535-538.
[108]Ma W.X.and Strampp W.,An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems[J].Phys.Lett.A,1994,185(3):277-286.
[109]Ma W.X.,Symmetry constraint of MKdV equations by binary nonlinearization [J].Phys.A,1995,219(3-4):467-481.
[110]Ma W.X.and Fuchssteiner B.,Integrable theory of the perturbation equations [J].Chaos,Solitons Fractals,1996,7:1227-1250.
[111]Ma W.X.and Fuchssteiner B.,The algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations[J].J.Math.Phys.,1999,40(5):2400-2418.
[112]Ma W.X.,Integrable couplings of soliton equations by perturbations,I.A general theory and application to the KdV hierarchy[J].Methods Appl.Anal.,2000,7:21-55.
[113]Ma W.X.and Zhou Z.X.,Binary symmetry constraints of N-wave interaction equations in 1+1 and 2+1 dimensions[J].J.Math.Phys.,2001,42(9):4345-4382.
[114]Ma W.X.and Zhou R.G.,Binary nonlinearization of spectral problems of the perturbation AKNS systems[J].Chaos,Solitons Fractals,2002,13(7):1451-1463.
[115]Ma W.X.and Zhou R.G.,Adjoint symmetry constraints of multicomponent AKNS equations[J].Chin.Ann.of Math.,2002,23(3):373-384.
[116]Ma W.X.,A bi-Hamiltonian formulation for triangular systems by perturbations [J].J.Math.Phys.,2002,43(3):1408-1421.
[117]Ma W.X.and Xu X.X.,Positive and negative hierarchies of integrable lattice models associated with a Hamiltonian pair[J].Internat.J.Theoret.Phys.,2004,43(1):219-235.
[118]Ma W.X.and Xu X.X.,A modified Toda spectral problem and its hierarchy of bi-Hamiltonian lattice equations[J].J.Phys.A,2004,37(4):1323-1336.
[119]Ma W.X.,Integrable couplings of vecter AKNS soliton equations[J].J.Math.Phys.,2005,46:033507,19pp.
[120]Ma W.X.and Chen M.,Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras[J].J.Phys.A:Math.Gen.,2006,39:10787-10801.
[121]Ma W.X.and Zhang Y.F.,Semidirect sums of Lie algebars and discrete integrable couplings[J].J.Math.Phys.,2006,47:053501-16.
[122]Ma W.X.,Xu X.X.and Zhang Y.F.,Semi-direct sums of Lie algebras and continuous integrable couplings[J].Phys.Lett.A,2006,351:125-130.
[123]Ma W.X.,A discrete varitional identity on semi-direction sums of Lie algebras [J].J.Phys.A,2007,40:15055-15069.
[124]Malanyuk T.M.,Finite-gap solutions of the Dasey-Stewartson equations[J].J.Nonlinear Sci.,1994,4(1):1-21.
[125]Marsden J.E.,Lectures on Geometric Methods in mathematical Physics[R],philadelphia:SAIM,1981.
[126]Matveev V.B.and Salle M.A.,Darboux transformations and solitons,Springer Series in Nonlinear Dynamics[M].Berlin:Springer-Verlag,1991.
[127]Merola I.,Ragnisco O.and Tu G.Z.,A novel hierarchy of integrable lattices[J].Inverse Problems,1994,10:11315-1334.
[128]Moser J.,6 Lectures on integrable Hamiltonian systems and spectral thoery,Report on DD4 Beijing Symposium[R].Eeijing:Science Publishing House,1984.
[129]Novikov S.P.,Period problem for the KdV equation[J].Funk.Anal.Pril.,1974,8(3):54-56.
[130]Nimmo J.C.,A bilinear Backlund transformation for the nonlinear Schrodinger equation[J].Phys.Lett.A,1983,99:279-281.
[131]Novikov S.P.et al.,Theory of solitons,The inverse scattering method[M],Moscow:Nauka,1983.
[132]Nucci M.C.and Clarkson P.A.,The nonclassical method is more general than the direct method for symmetry reductions:An example of the Fitzhugh-Nagumo equation[J].Phys.Lett.A,1992,164:49-56.
[133]Olver P.J.,Application of Lie groups to differential equaitons[M].New York:Springer-Verlag,1986.
[134]Olver P.J.,Direct reduction and differential constraints[J].Proc.Roy.Soc.London Ser.A,1994,444:509-523.
[135]Qiao Z.J.,A new completely integrable Liouville's system produced by the Kaup-Newell eigenvalue problem[J].J.Math.Phys.,1993,34:3110-3120.
[136]Qiao Z.J.,Two new integrable systems in Liouville's sense[J].Phys.Lett.A,1993,172:224-228.
[137]Qiao Z.J.,A finite-dimensional integrable system and the involutive solutions of the higherorder Heisenberg spin equations[J].Phys.Lett.A,1994,186(1-2):97-102.
[138]Qiao Z.J.,Generalized r-matrix structure and algebro-geometric solution for integrable system[J].Rev.Math.Phys.,2001,13(5):545-586.
[139]Qiao Z.J.,The Camassa-Holm hierarchy,N-dimensional integrable systems,and algebro-geometric solution on a symplectic submanifold[J].Comm.Math.Phys.,2003,239:309-341.
[140]Qin Z.Y.,A finite-dimensional integrable system related to a new coupled KdV hierarchy[J].Phys.Lett.A,2006,355(6):452-459.
[141]Qu C.Z.and Yao R.X.,Multi-component WKI equations and their conservation laws[J].Phys.Lett.A,2004,331(5):325-331.
[142]Satsuma J.,Matsukdaira J.and Strampp W.Soliton equations expressed by trilinear forms[J].Phys.Lett.A,1990,147(8-9):467-471.
[143]Sun Y.P.,Chen D.Y.and Xu X.X.,Positice and negative hierarchy of nonlinear integrable lattice models and three integrable coupling systems associated with a discrete spectral problem[J].Nonlinear Analysis,2006,64:2604-2618.
[144]Suris Y.B.,New integrable systems related to the relaticistic Toda lattice[J].J.Phys.A:Math.Gen.,1997,30:1745-1761.
[145]Tam H.W.and Zhu Z.N.,(2+1)-dimensional integralbe lattice hierarchies related to discrete fourth-order nonisospectral problems[J].J.Phys.A:Math.Theor.,2007,40:13031-13045.
[146]Tang X.Y.,Lou S.Y.and Lin J.,Similarity and conditional similarity reductions of a(2+1)- dimensional KdV equation via a direct method[J].J.Math.Phys.,2000,41(12):8286-8303.
[147]Tian C.,New strong symmetry,symmetries and Lie algebra of Burgers' equation [J].Sci.Sinica Ser.A,1988,31(2):141-151.
[148]Tu G.Z.,A new hierarchy of integrable systems and its Hamiltonian structure [J].Sci.China Ser.A,1989,32(2):142-153.
[149]Tu G.Z.,The trace identity,a powerful tool for constructing the Hamiltonian structure of integrable systems[J].J.Math.Phys.,1989,30(2):330-338.
[150]Tu G.Z.,The trace identity and its applications to the theory of discrete integrable systems[J].J.Phys.A:Math.Gen.,1990,23:3903-3922.
[151]Uhlenbeck K,Harmonic maps into Lie group(classical solutions of the chiral model)[J].J.Diff.acorn.,1989,30:1-50.
[152]Veselov A.P.,On Hamiltonian formalism for the Novikov-Kritchever equation on commutativity of two operators[J].Funk.Anal.Pril.,1979,13(1):1-7.
[153]Wahlquist H.D.and Estabrook F.B.,Backlund transformations for solitons of the Kortewegde Vries equation[J].Phys.Rev.Lett.,1973,31:1386-1390.
[154]Wang J.,Algebro-geometrie solutions for some(2+1)-dimensional discrete systems[J].Nonlinear Analysis,In Press.
[155]Wang M.L.,Zhou Y.B.and Li Z.B.,Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics[J].Phys.Lett.A,1996,216:67-75.
[156]Wadati M.,Sanuki H.J.and Konno K.,Relationships among inverse method,Backlundtransformation and an infinite number of conservation laws[J].Progr.Theoret.Phys.,1975,53:419-436.
[157]Xiao T.and Zeng Y.B.,Generalized Darboux transformations for the KP equation with self-consistent sources[J].J.Phys.A,2004,37:7143-7162
[158]Xiao T.and Zeng Y.B.,A new constrained mKP hierarchy and the generalized Darboux transformation for the mKP equation with self-consistent sources[J].Physica A,2005,353:38-60
[159]Xia T.C.and Fan E.G.,The multicomponent generalized Kaup-Newell hierarchy and its multieomponent integrable couplings system with two arbitrary functions[J].J.Math.Phys.,2005,46(4):043510,8pp.
[160]Xu X.X.,A generalized Wadati- Konno- Ichikawa hierarchy and its binary nonlinearization by symmetry constraints[J].Chaos,Solitons and Fractals,2003,15:475-486.
[161]Xu X.X.,Factorization os a hierarchy of the lattice soliton equations from a binary Bargmann symmetry constraint[J].Nonlinear Analysis,2005,61:1225-1240.
[162]Yan Z.Y.and Zhang H.Q.,A Lax integrable hierarchy,N-Hamiltonian structure,r-matrix,finite-dimensional Liouville integrable involutive systems,and involutive solutions[J].Chaos Soliton Fractals,2002,13:1439-1450.
[163]Yao R.X.,Qu C.Z.and Li Z.B.,Painlevéproperty and conservation laws of multicomponent mKdV equations[J].Chaos,Solitons Fractals,2004,22(3):723-730.
[164]Zakharov V.E.,The inverse scattering method in soliton[M].New York:Springer-verlag Berlin Heideberg,1980:243-28.
[165]Zeng Y.B.,Li Y.S.and Chen D.Y.,A hierarchy of integrable Hamiltonian-systems with Neumann type constraint[J].Chinese Ann.Math.Ser.B,1992,13(3):327-340.
[166]Zeng Y.B.,The higher-order constraints and integrable systems associtated with the Boussinesq equation[J].Chinese Science Bulletin of Mathematics Series,1992,37(23):1937-1942.
[167]Zeng Y.B.and Li Y.S.,The deduction of the Lax representation for constrained flows from the adjoint representation[J].J.Phys.A,1993,26:L273-L278.
[168]Zeng Y.B.,New facorization of the Kaup-Newell hierarchy[J].Physica D,1994,73:171-188.
[169]Zhang D.J.and Chen D.Y.,The conservation laws of some discrete soliton systems[J].Chaos Solitons Fractals,2002,14:573-579.
[170]Zhang D.J.et.al.,New symmetries for the Ablowitz-Ladik hierarchies[J].Phys.Lett.A,2006,359:458-466.
[171]Zhang Y.F.and Fan E.G.,An approach for generating enlarging integrable systems[J].Phys.Lett.A,2007,365:89-96.
[172]Zhou L.J.,Darboux transformation for the nonisospectral AKNS system[J].Phys.Lett.A,2005,345(4-6):314-322.
[173]Zhou R.G.and Qiao Z.J.,On the restricted Toda and cKdV flows of Neumann type[J].Commun.Theor.Phys.,2000,34(2):229-234.
[174]Zhou R.G.,The finite-band solution of the Jaulent-Miodek equation[J].J.Math.Phys.,1997,38(5):2535-2546.
[175]Zhou R.G.,Nonlinearizations of spectral problems of the nonlinear Schrodinger equation and the real-valued modified Korteweg-de Vries equation[J].J.Math.Phys.,2007,48(1):013510,9 pp.
[176]Zhou Z.X.,Ma W.X.and Zhou R.G.,Finite-dimensional integrable systems associated with the Davey-Stewartson I equation[J].Nonlinearity,2001,14(4):701-717.
[177]Zhou Z.X.,Liouville integrability of the finite dimensional Hamiltonian systems derived from principal chiral field[J].J.Math.Phys.,2002,43(10):5002-5012.
[178]Zhou Z.X.,Darboux transformations and exact solutions of two-dimensional C_l~((1)) and D_(l+1)~((2)) Toda equations[J].J.Phys.A,2006,39(20):5727-5737.
[179]Zhou Z.X.,Finite dimensional integrable systems related to two dimensional A_(2l)~((2)),C_l~((1)) and D_(l+1)~((2)) Toda equations[J]J.Geom.Phys.,2007,57:1037-1053.
[180]Zhu Z.N.et ai.,Infinitely many conservation laws for the Blaszak-Marciniak four-field integrable lattice hierarchy[J].Phys.Lett.A,2002,296:280-288.
[181] Zhu Z.N. and Xue W.M., Two new integrable lattice hierarchies associated with a discrete Schrodinger nonisospectral problem and their infinitely many conservation laws[J]. Phys. Lett. A, 2004, 320: 396-407.
[182] Zhu Z.N. and Tam H.W., Nonisospectral negative Volterra flows and mixed Volterra flows: Lax pairs, infinitely many conservation laws and integrable time discretization[J]. J. Phys. A, 2004, 37: 3175-3187.
[2] Ablowitz M.J., Ramani A. and Segur H., A connection between nonlinear evolution equations and ordinary differential equations of p-type[J]. J. Math. Phys., 1980, 21(5): 1006-1015.
[3] Ablowitz M.J. and Segur H., Soliton and the Inverse Scattering Transform[M]. Philadelphia: SIAM, 1981.
[4] Ablowitz M.J.and Clarkson P.A., Solitons, Nonlinear Evolution Equation and Inverse Scattering[M]. Cambridge: Cambridge University Press, 1991.
[5] Ahmad S. and Chowdhury A.R., On the quasi-periodic solutions to the discrete nonlinear Schrodinger equation[J]. J. Phys. A: Math. Gen., 1987, 20: 293-303.
[6] Alber S.I., Investigation of equations of KdV type by the method of recurrence relations[J]. J. London Math. Soc, 1979, 19: 467-480.
[7] Andrew P. and Zhu Z.N., New integrable lattice hier archies [J]. Phys. Lett. A, 2006, 349: 439-445.
[8] Arnold V.I., Mathematical Methods of classical Mechanics[M]. New York: Springer-Verlag, 1978.
[9] Belokolos E. et al., Algebro-geometrical approach to nonlinear integrable equa-tions[Mj. Berlin: Springer-Verlag, 1994.
[10] Bogoyavlenskii O.I. and Novikov S.P., The connection between the Hamiltonian formulalisms of stationary and nonstationary problems [J]. Functional Anal. Appl., 1976, 10: 9 - 13.
[11] Cao C.W., Nonlinearization of the Lax system for AKNS hierarchy [J]. Sci. Chin. A, 1990, 33: 528-536.
[12]Cao C.W.and Geng X.G.,Classical integrable systems generated through nonlinearization of eigenvalue problems[R].Nonlinear Physics,Berlin:Springer,1990:68-78.
[13]Cao C.W.and Geng X.G.,C.Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy[J].J.Phys.A,1990,23(18):4117-4125.
[14]Cao C.W.,Geng X.G.and Wu Y.T.,From the special 2+1 Toda lattice to the Kadomtsev- Petviashvili equation[J].J.Phys.A:Math.Gen.,1999,32(46):8059-8078.
[15]Cao C.W.,Wu Y.T.and Ceng X.G.,Relation between the Kadometsev-Petviashvili equation and the confocal involutive system[J].J.Math.Phys.,1999,40(8):3948-3970.
[16]Cao C.W.,Geng X.G.and Wang H.Y.,Algebro-geometric solution of the 2+1dimensional Burgers equation with a discrete variable[J].J.Math.Phys.,2002,43(1):621-643.
[17]Chen D.Y.,The potential constraints of the complex(1+2)-dimensional soliton systems and related Hamiltonian equations[J].J.Math.Phys.,1998,39(6):3246-3259.
[18]Chen D.Y.,Zhang D.J.and Deng S.F.,The novel multisoliton solutions of the mKdV-sine-Gordon equation[J].J.Phys.Soc.Japan,2002,71:658-659.
[19]Chen D.Y.,Zhang D.J.and Deng S.F.,Remarks on some solutions of soliton equations[J].J.Phys.Soc.Japan,2002,71:2071-2072.
[20]陈登远,孤子引论[M].北京:科学出版社,2006.
[21]程艺、汪存启、田畴、李翊神,Burgers方程簇的对称、Backlund变换和Painlevé性质之间的关系[J].应用数学学报,1991,14:180-184.
[22]Cheng Y.and Li Y.S.,The constraint of the Kadomtsev-Petviashvili equation and its special solutions[J].Phys.Left.A,1991,157(1):22-26.
[23]Cheng Y.and Li Y.S.,Constraints of the 2+1 dimensional integrable soliton systems[J],J.Phys.A:Math.Gen.,1992,25:419-431.
[24]Cheng Y.,Strampp W.and Zhang Y.J.,Bilinear Backlund transformations for the KP and K-constrained KP hierarchy[J].Phys.Lett.A,1993,182:71-76.
[25]Clarkson P.A.and Kruskal M.D.,New similiarity reductions of the Boussinesq equation[J].J.Math.Phys.,1989,30(10):2201-2213.
[26]Conte R.,Invariant painlevé analysis of partial differential equations[J].Phys.Lett.A,1989,140(7-8):383-390.
[27]Deng S.F.and Chen D.Y.,The novel multisoliton solutions of the KP equation [J].J.Phys.Soc.Japan,2001,70:3174-3175.
[28]Dickey L.A.,Integrable nonlinear equations and Liouville's theorem[J].Commun.Math.Phys.,1981,82:345-375.
[29]Dickey L.A.,Hamiltonian structures and Lax equations generated by matrix differential operators with polynomial dependence on a parameter[J].Commun.Math.Phys.,1983,88:27-42.
[30]Dickey L.A.,Soliton equations and Hamiltonian systems[M].Singapo:World Scientific,1991.
[31]Fan E.G.,Integrable evolution systems based on Gerdjikov-Ivanov equations,Bi- Hamiltonian structure,finite-dimensional integrable systems and N-fold Darboux transformation[J].J.Math.Phys.,2000,41(11):7769-7782.
[32]Fan E.G.,The zero curvature representation for hierarchies of nonlinear evolution equations[J].Phys.Lett.A,2000,274(3-4):135-142.
[33]Fan E.G.,Zhang H.Q.,A hierarchy of nonlinear evolution equations,its Bi-Hamiltonian structure,and finite-dimensional integrable systems[J].J.Math.Phys.,2000,41(4):2058-2065.
[34]Fan E.G.,Integrable systems of derivative nonlinearSchrodinger type and their multi- Hamiltonian structure[J].J.Phys.A:Math.Gen.,2001,34(3):513-519.
[35]Fan E.G.and Zhang H.Q.,A new completely integrable Liouville's system,its Lax representation and Bi-Hamiltonian structure[J].Appl.Math.Mech-Eng.edit.,2001,22(5):520-527.
[36]Fan E.G.,Bi-Hamiltonian structure and liouville integrablity for a Gerdjikov-Ivanov equation hierarchy[J].Chin.Phys.Lett.,2001,18(1):1-3.
[37]范恩贵,可积系统与计算机代数[M].北京:科学出版社,2004.
[38]Fan E.G.and Zhang Y.F.,A simple method for generating integrable hierarchies with multi-potential functions[J].Chaos Solitons Fractals,2005,25:425-439.
[39]Flaschka H.,Relations between infinite-dimensional and finite-dimensional isospectral equations[M].Singapore:Kyoto,Japan,World Sci,1983:219-239.
[40]Fordy A.P.and Kulish P.P.,NonlinearSchrodinger equations and simple Lie algebras[J].Comm.Math.Phys.,1983,89(3):427-443.
[41]Freeman N.C.and Nimmo J.C.,Soliton solutions of the KdV and KP equations:the Wronskian technique[J].Phys.lett.A,1983,95:1-3.
[42]Fuchssteiner B.,Application of hereditary symmetries to nonlinear evolution equations[J].Nonlinear Anal.,1979,3(6):849-862.
[43]Fuchssteiner B.and Fokas A.S.,Symplectic structures,their backlund transformations and hereditary symmetries[J].Physica D,1981,4:47-66.
[44]Fuchssteiner B.,Master symmetries,Higher order time-dependent symmetries and conserved densities of nonlinear evoluton equations[J].Prog.Theor.Phys.,1983,70:1508-1522.
[45]Gardner C.S.et al.,Method for solving the Korteweg-de Vries equation[J].Phys.Rev.Lett.1967,19(19):1095-1103.
[46]Gel' fand I.M.and Dikii L.A.,The asympototics of the resolvent of Sturm-Liouville equations and the algebra of the KdV equations[J].Usp.Matem.Nauk.,1975,30(5):67-100.
[47]Gel' fand I.M.and Dikii L.A.,A Lie algebraic structure in a formal varational calculus[J].Funk.Anal.Pril.,1976,10(1):18-25.
[48]Gel' fand I.M.and Dikii L.A.,Fractional powers of operators and Hamiltonian systems[J].Funk.Anal.Pril.,1976,10(4):13-29.
[49]Gel' fand I.M.and Dikii L.A.,Integrable nonlinear equations and the liouville theorem[J].Funk.Anal.Pril.,1979,13(1):8-20.
[50]Geng X.G.,Discrete Bargmann and Neumann systems and finite-dimensional integrable systems[J].Phys.A,1994,212(1-2):132-142.
[51]Geng X.G.and Wu Y.T.,Finite-band solutions of the classical Boussinesq-Burgers equations[J].J.Math.Phys.,1999,40(6):2971-2982.
[52]Geng X.G.,Wu Y.T.and Cao C.W.,Quasi-periodic solutions of the modified Kadomtsev- Petviashvili equation[J].J.Phys.A.,1999,32(20):3733-3742.
[53]Geng X.G.,Cao C.W.and Dai H.H.,Quasi-periodic solutions for some(2+1)-dimensional integrable models generated by the Jaulent-Miodek hierarchy]J].J.Phys.A:Math.Gen.,2001,34(5):989-1004.
[54]Geng X.G.,Dai H.H.and Cao C.W.,Algebro-geometric constructions of the discrete Ablowitz-Ladik flows and applications[J].J.Math.Phys.,2003,44(10):4573-4588.
[55]谷超豪等,孤立子理论与应用[M].杭州:浙江科学技术出版社,1990年.
[56]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[M].上海:上海科学技术出版社,1999.
[57]He J.S.et al.,Determinator representation of Darboux transformation for the AKNS system[J].Sci.China Ser.A 2006,49(12):1867-1878.
[58]Hisakado M.,Breather trapping mechanism in piecewise homogeneous DNA[J].Phys Lett A,1997,227(1-2):87-93.
[59]Hirota R.,Soliton of the KdV equation for multiple collusions of solitons[J].Phys.Rev.Lett.,1971,27(8):1192-1194.
[60]Hirota R.,Exact N-soliton solutions of the wave equation of long waves in shallow water and in nonlinear lattices[J].J.Math.Phys.,1973,14(7):810-814.
[61]Hirota R.,A few forms of Backlund transformations and its relation to inverse scattering problem[J].Prog.Theor.Phys.,1974,52:1498-1512.
[62]Hirota R.and Satsuma J.,Nolinear evolution equations generated from the Backlund transformation for the Boussinesq equation[J].Proy.Thoer.Phys.,1977,57:797-807.
[63]Hon Y.C.and Fan E.G.,An algebro-geometric solution for a Hamiitonian system with application to dispersive long wave equation[J].J.Math.Phys.,2005,46:032701,22pp.
[64]Hu X.B.,A powerful approach to generate new integrable systems[J].J.Phys.A:Math.Gen.,1994,27:2497-2514.
[65]Hu X.B.,Rational solutions of integrable equations via nonlinear superposition formulae[J].J.Phys.A,1997,30(23):8225-8240.
[66]Hu X.B.,A approach to generate superextensions of integrable systems[J].J.Phys.A,1997,30:619-632.
[67]Hu X.B.et al.,Lax pairs and Backlundtransformations for a coupled ramani equation and its related system[J].Appl.Math.Lett.,2000,13:45-48.
[68]Hu X.B.and Wang H.Y.,New type of Kadomtsev-Petviashvili equation with self-consistent sources and its bilinear Backlund transformation[J].Inverse Problem,2007,23(4):1433-1444.
[69]Huang G.X.,Lou S.Y.and Dai X.X.,Exact and explicit solitary wave solutions to a model equation for water waves[J].Phys.Lett.A,1989,139:373-374.
[70]Jimbo M.,Kruskal M.D.and Miwa T.,Painlevé test for the self-dual Yang-Mills equation[J].Phys.Lett.A,1982,92(2):59-60.
[71]Kaup D.J.and Newll A.C.,An exact solution for a derivative nonlinear Schrodinger equation[J].J.Math.Phys.1978,19:798-801.
[72]Konno K.,Sanuki H.and Ichikawa Y.H.,Conservation laws of nonlinear evolution equations[J].Progr.Theoret.Phys.1974,52:375-382.
[73]Li C.X.,A hierarchy of coupled Korteweg-de Vries equations and the corresponding finitedimensional integrable system[J].J.Phys.Soc.Jap.,2004,73(2):327-331.
[74]李翊神、朱国诚,一个谱可变演化方程的对称[J].科学通报,1986,31:1449-1453.
[75]Li Y.S.and Cheng Y.,Symmetries and constants of motion for the new KdV hierarchies[J].Sci.Sinica Ser.A,1988,31:769-776.
[76]李翊神,孤子与可积系统[M].上海:上海科技教育出版社,1999年,
[77]Li Y.S.,Ma W.X.and Zhang J.E.,Darboux transformations of classical Boussinesq system and its how solutions[J].Phys.Lett.A,2000,275:60-66
[78]Li Y.S.and Ma W.X.,Hamiltonian structures of binary higher-order constrained flows associated with two 3×3 spectral problems[J].Phys.Lett.A 2000,272:245-256.
[79]Li Y.S.and Zhang J.E.,Darboux transformation of classical Boussinesq system and its multi-soliton solutions[J].Phy.Lett.A,2001,284:253-258
[80]Li Z.B.and Wang M.L.,Travelling wave solutions to the two-dimensional KdV-Burgers equation[J].J.Phys.A,1993,26:6027-6032.
[81]Li Z.B.,Wu method and solitons[A].Proc.of ASCM'95,He Shi and Hidetsune Kobayashi[C].Tokyo:Scientists Incorporated,1995:157.
[82]Lin R.L.,Zeng Y.B.and Ma W.X.,Solving the KdV hierarchy with self-consistent sources by inverse scattering method[J].Phys.Lett.A,2001,291:287-298.
[83]Liu Q.P.,Darboux transformations for supersymmetric Korteweg-de Vries equations[J].Lett.Math.Phys.,1995,35(2):115-122.
[84]Liu Q.P.and Mannas M.,Darboux transformations for super-symmetric KP hierarchies[J].Phys.Lett B,2000,485:293-300.
[85]Liu Q.P.and Hu X.B.,Bilinearization of N=1 supersymmetric Korteweg-de Vries equation revisited[J].J.Phys.A,2005,38:371-6378.
[86]Liu Q.P.,Hu X.B.and Zhang M.X.,Supersymmetric modified Korteweg-de Vries equation:bilinear approach[J].Nonlinearity,2005,18:1597-1603.
[87]Liu Q.P.and Yang X.X.,Supersymmetric two-boson equation:Bilinearization and solutions[J].Phys.Lett.A,2006,351:131-135.
[88]Liu X.J.and Zeng Y.B.,On the Toda Lattice equation with self-consistent sources[J].J.Phys.A,2005,38(41):8951-8965
[89]Lou S.Y.,A note on the new similarity reductions of the Boussinesq equation[J].Phys.Lett.A,1990,151(3-4):133-135.
[90]Lou S.Y.,Similarity reductions of the KP equation by a direct method[J].J.Phys.A,1990,23:L649-L654.
[91]Lou S.Y.,Symmetries of the Kadomtsec-Petviashvili equation[J].J.Phys.A:Math.Gen.,1993,26:4387-4394.
[92]Lou S.Y.,Generalized symmetries and ω_∞ algebras in three dimensional Toda field theory[J].Phys.Rev.Lett.,1993,71:4099-4102.
[93]Lou S.Y.,Abundant symmetries for the 1+1 dimensional classical Liouville field theory[J].J.Math.Phys,1994,35(5):2336-2348.
[94]Lou S.Y.,Symmtries of the KdV equation and Hierarchies of the integrodifferential KdV equations[J].J.Math.Phys.,1994,35(5):2390-2396.
[95]Lou S.Y.and Hu X.B.,Infinitely many Lax pairs and symmetry constraints of the KP equation[J].J.Math.Phys.,1997,39:6401-6427
[96]Lou S.Y.and Hu X.B.,Broer-Kaup system from Darboux transformation refated symmetry constraints of Kadomtsev-Petviashvili equation[J].Commun.Theor.Phys.,1998,29:145-148.
[97]Lou S.Y.,On the coherent structures of Nizhnik-Novikov-Veselov equation[J].Phys.Lett.A,2000,277:94-100.
[98]Lou S.Y.and Tang X.Y.,Conditional similarity reduction approach:Jimbo-Miwa equation[J].Chin.Phys.,2001,10:897-901.
[99]楼森岳、唐晓艳,非线性数字物理方法[M].北京:科学出版社,2006.
[100]Luo L.,Darboux transformation and exact solutions for a hierarchy of nonlinear evolution equations[J].J.Phys.A:Math.Theor.,2007,40(15):4169-4179.
[101]Luo L.and Fan E.G.,Hamiltonian systems and integrable coupling associated with a new discrete spectral problem[J].Phys.Lett.A,2007,370(3-4):234-242.
[102]Luo L.and Fan E.G.,Integrable decomposition of a hierarchy of soliton equations and integrable coupling system by semidirect sums of Lie algebras[J].Nonlinear Analysis,In Press.
[103]Ma W.X.,K-symmetries and r-symmetries of evolution equations and their Lie algebras[J].J.Phys.A,1990,23:2707-2716.
[104]Ma W.X.,The algebraic structure of isospectral Lax operators and application to integrable equations[J].J.Phys.A,1992,25:5329-5343.
[105]Ma W.X.,The algebraic structure of zero curvature representations and application to coupled KdV systems[J].J.Phys.A,1993,26:2573-2582.
[106]Ma W.X.,A simple scheme for generating nonisospectral flows from zero curvature representation[J].Phys.Lett.A,1993,179:179-185.
[107]Ma W.X.,A method of zero curvature representation for constructing symmetry algebras of integrable systems[A].Proceedings of the 21st International Conference on the Differential Geometry Methods in Theoretical Physics[C].Singapore:World Scientific,1993:535-538.
[108]Ma W.X.and Strampp W.,An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems[J].Phys.Lett.A,1994,185(3):277-286.
[109]Ma W.X.,Symmetry constraint of MKdV equations by binary nonlinearization [J].Phys.A,1995,219(3-4):467-481.
[110]Ma W.X.and Fuchssteiner B.,Integrable theory of the perturbation equations [J].Chaos,Solitons Fractals,1996,7:1227-1250.
[111]Ma W.X.and Fuchssteiner B.,The algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations[J].J.Math.Phys.,1999,40(5):2400-2418.
[112]Ma W.X.,Integrable couplings of soliton equations by perturbations,I.A general theory and application to the KdV hierarchy[J].Methods Appl.Anal.,2000,7:21-55.
[113]Ma W.X.and Zhou Z.X.,Binary symmetry constraints of N-wave interaction equations in 1+1 and 2+1 dimensions[J].J.Math.Phys.,2001,42(9):4345-4382.
[114]Ma W.X.and Zhou R.G.,Binary nonlinearization of spectral problems of the perturbation AKNS systems[J].Chaos,Solitons Fractals,2002,13(7):1451-1463.
[115]Ma W.X.and Zhou R.G.,Adjoint symmetry constraints of multicomponent AKNS equations[J].Chin.Ann.of Math.,2002,23(3):373-384.
[116]Ma W.X.,A bi-Hamiltonian formulation for triangular systems by perturbations [J].J.Math.Phys.,2002,43(3):1408-1421.
[117]Ma W.X.and Xu X.X.,Positive and negative hierarchies of integrable lattice models associated with a Hamiltonian pair[J].Internat.J.Theoret.Phys.,2004,43(1):219-235.
[118]Ma W.X.and Xu X.X.,A modified Toda spectral problem and its hierarchy of bi-Hamiltonian lattice equations[J].J.Phys.A,2004,37(4):1323-1336.
[119]Ma W.X.,Integrable couplings of vecter AKNS soliton equations[J].J.Math.Phys.,2005,46:033507,19pp.
[120]Ma W.X.and Chen M.,Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras[J].J.Phys.A:Math.Gen.,2006,39:10787-10801.
[121]Ma W.X.and Zhang Y.F.,Semidirect sums of Lie algebars and discrete integrable couplings[J].J.Math.Phys.,2006,47:053501-16.
[122]Ma W.X.,Xu X.X.and Zhang Y.F.,Semi-direct sums of Lie algebras and continuous integrable couplings[J].Phys.Lett.A,2006,351:125-130.
[123]Ma W.X.,A discrete varitional identity on semi-direction sums of Lie algebras [J].J.Phys.A,2007,40:15055-15069.
[124]Malanyuk T.M.,Finite-gap solutions of the Dasey-Stewartson equations[J].J.Nonlinear Sci.,1994,4(1):1-21.
[125]Marsden J.E.,Lectures on Geometric Methods in mathematical Physics[R],philadelphia:SAIM,1981.
[126]Matveev V.B.and Salle M.A.,Darboux transformations and solitons,Springer Series in Nonlinear Dynamics[M].Berlin:Springer-Verlag,1991.
[127]Merola I.,Ragnisco O.and Tu G.Z.,A novel hierarchy of integrable lattices[J].Inverse Problems,1994,10:11315-1334.
[128]Moser J.,6 Lectures on integrable Hamiltonian systems and spectral thoery,Report on DD4 Beijing Symposium[R].Eeijing:Science Publishing House,1984.
[129]Novikov S.P.,Period problem for the KdV equation[J].Funk.Anal.Pril.,1974,8(3):54-56.
[130]Nimmo J.C.,A bilinear Backlund transformation for the nonlinear Schrodinger equation[J].Phys.Lett.A,1983,99:279-281.
[131]Novikov S.P.et al.,Theory of solitons,The inverse scattering method[M],Moscow:Nauka,1983.
[132]Nucci M.C.and Clarkson P.A.,The nonclassical method is more general than the direct method for symmetry reductions:An example of the Fitzhugh-Nagumo equation[J].Phys.Lett.A,1992,164:49-56.
[133]Olver P.J.,Application of Lie groups to differential equaitons[M].New York:Springer-Verlag,1986.
[134]Olver P.J.,Direct reduction and differential constraints[J].Proc.Roy.Soc.London Ser.A,1994,444:509-523.
[135]Qiao Z.J.,A new completely integrable Liouville's system produced by the Kaup-Newell eigenvalue problem[J].J.Math.Phys.,1993,34:3110-3120.
[136]Qiao Z.J.,Two new integrable systems in Liouville's sense[J].Phys.Lett.A,1993,172:224-228.
[137]Qiao Z.J.,A finite-dimensional integrable system and the involutive solutions of the higherorder Heisenberg spin equations[J].Phys.Lett.A,1994,186(1-2):97-102.
[138]Qiao Z.J.,Generalized r-matrix structure and algebro-geometric solution for integrable system[J].Rev.Math.Phys.,2001,13(5):545-586.
[139]Qiao Z.J.,The Camassa-Holm hierarchy,N-dimensional integrable systems,and algebro-geometric solution on a symplectic submanifold[J].Comm.Math.Phys.,2003,239:309-341.
[140]Qin Z.Y.,A finite-dimensional integrable system related to a new coupled KdV hierarchy[J].Phys.Lett.A,2006,355(6):452-459.
[141]Qu C.Z.and Yao R.X.,Multi-component WKI equations and their conservation laws[J].Phys.Lett.A,2004,331(5):325-331.
[142]Satsuma J.,Matsukdaira J.and Strampp W.Soliton equations expressed by trilinear forms[J].Phys.Lett.A,1990,147(8-9):467-471.
[143]Sun Y.P.,Chen D.Y.and Xu X.X.,Positice and negative hierarchy of nonlinear integrable lattice models and three integrable coupling systems associated with a discrete spectral problem[J].Nonlinear Analysis,2006,64:2604-2618.
[144]Suris Y.B.,New integrable systems related to the relaticistic Toda lattice[J].J.Phys.A:Math.Gen.,1997,30:1745-1761.
[145]Tam H.W.and Zhu Z.N.,(2+1)-dimensional integralbe lattice hierarchies related to discrete fourth-order nonisospectral problems[J].J.Phys.A:Math.Theor.,2007,40:13031-13045.
[146]Tang X.Y.,Lou S.Y.and Lin J.,Similarity and conditional similarity reductions of a(2+1)- dimensional KdV equation via a direct method[J].J.Math.Phys.,2000,41(12):8286-8303.
[147]Tian C.,New strong symmetry,symmetries and Lie algebra of Burgers' equation [J].Sci.Sinica Ser.A,1988,31(2):141-151.
[148]Tu G.Z.,A new hierarchy of integrable systems and its Hamiltonian structure [J].Sci.China Ser.A,1989,32(2):142-153.
[149]Tu G.Z.,The trace identity,a powerful tool for constructing the Hamiltonian structure of integrable systems[J].J.Math.Phys.,1989,30(2):330-338.
[150]Tu G.Z.,The trace identity and its applications to the theory of discrete integrable systems[J].J.Phys.A:Math.Gen.,1990,23:3903-3922.
[151]Uhlenbeck K,Harmonic maps into Lie group(classical solutions of the chiral model)[J].J.Diff.acorn.,1989,30:1-50.
[152]Veselov A.P.,On Hamiltonian formalism for the Novikov-Kritchever equation on commutativity of two operators[J].Funk.Anal.Pril.,1979,13(1):1-7.
[153]Wahlquist H.D.and Estabrook F.B.,Backlund transformations for solitons of the Kortewegde Vries equation[J].Phys.Rev.Lett.,1973,31:1386-1390.
[154]Wang J.,Algebro-geometrie solutions for some(2+1)-dimensional discrete systems[J].Nonlinear Analysis,In Press.
[155]Wang M.L.,Zhou Y.B.and Li Z.B.,Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics[J].Phys.Lett.A,1996,216:67-75.
[156]Wadati M.,Sanuki H.J.and Konno K.,Relationships among inverse method,Backlundtransformation and an infinite number of conservation laws[J].Progr.Theoret.Phys.,1975,53:419-436.
[157]Xiao T.and Zeng Y.B.,Generalized Darboux transformations for the KP equation with self-consistent sources[J].J.Phys.A,2004,37:7143-7162
[158]Xiao T.and Zeng Y.B.,A new constrained mKP hierarchy and the generalized Darboux transformation for the mKP equation with self-consistent sources[J].Physica A,2005,353:38-60
[159]Xia T.C.and Fan E.G.,The multicomponent generalized Kaup-Newell hierarchy and its multieomponent integrable couplings system with two arbitrary functions[J].J.Math.Phys.,2005,46(4):043510,8pp.
[160]Xu X.X.,A generalized Wadati- Konno- Ichikawa hierarchy and its binary nonlinearization by symmetry constraints[J].Chaos,Solitons and Fractals,2003,15:475-486.
[161]Xu X.X.,Factorization os a hierarchy of the lattice soliton equations from a binary Bargmann symmetry constraint[J].Nonlinear Analysis,2005,61:1225-1240.
[162]Yan Z.Y.and Zhang H.Q.,A Lax integrable hierarchy,N-Hamiltonian structure,r-matrix,finite-dimensional Liouville integrable involutive systems,and involutive solutions[J].Chaos Soliton Fractals,2002,13:1439-1450.
[163]Yao R.X.,Qu C.Z.and Li Z.B.,Painlevéproperty and conservation laws of multicomponent mKdV equations[J].Chaos,Solitons Fractals,2004,22(3):723-730.
[164]Zakharov V.E.,The inverse scattering method in soliton[M].New York:Springer-verlag Berlin Heideberg,1980:243-28.
[165]Zeng Y.B.,Li Y.S.and Chen D.Y.,A hierarchy of integrable Hamiltonian-systems with Neumann type constraint[J].Chinese Ann.Math.Ser.B,1992,13(3):327-340.
[166]Zeng Y.B.,The higher-order constraints and integrable systems associtated with the Boussinesq equation[J].Chinese Science Bulletin of Mathematics Series,1992,37(23):1937-1942.
[167]Zeng Y.B.and Li Y.S.,The deduction of the Lax representation for constrained flows from the adjoint representation[J].J.Phys.A,1993,26:L273-L278.
[168]Zeng Y.B.,New facorization of the Kaup-Newell hierarchy[J].Physica D,1994,73:171-188.
[169]Zhang D.J.and Chen D.Y.,The conservation laws of some discrete soliton systems[J].Chaos Solitons Fractals,2002,14:573-579.
[170]Zhang D.J.et.al.,New symmetries for the Ablowitz-Ladik hierarchies[J].Phys.Lett.A,2006,359:458-466.
[171]Zhang Y.F.and Fan E.G.,An approach for generating enlarging integrable systems[J].Phys.Lett.A,2007,365:89-96.
[172]Zhou L.J.,Darboux transformation for the nonisospectral AKNS system[J].Phys.Lett.A,2005,345(4-6):314-322.
[173]Zhou R.G.and Qiao Z.J.,On the restricted Toda and cKdV flows of Neumann type[J].Commun.Theor.Phys.,2000,34(2):229-234.
[174]Zhou R.G.,The finite-band solution of the Jaulent-Miodek equation[J].J.Math.Phys.,1997,38(5):2535-2546.
[175]Zhou R.G.,Nonlinearizations of spectral problems of the nonlinear Schrodinger equation and the real-valued modified Korteweg-de Vries equation[J].J.Math.Phys.,2007,48(1):013510,9 pp.
[176]Zhou Z.X.,Ma W.X.and Zhou R.G.,Finite-dimensional integrable systems associated with the Davey-Stewartson I equation[J].Nonlinearity,2001,14(4):701-717.
[177]Zhou Z.X.,Liouville integrability of the finite dimensional Hamiltonian systems derived from principal chiral field[J].J.Math.Phys.,2002,43(10):5002-5012.
[178]Zhou Z.X.,Darboux transformations and exact solutions of two-dimensional C_l~((1)) and D_(l+1)~((2)) Toda equations[J].J.Phys.A,2006,39(20):5727-5737.
[179]Zhou Z.X.,Finite dimensional integrable systems related to two dimensional A_(2l)~((2)),C_l~((1)) and D_(l+1)~((2)) Toda equations[J]J.Geom.Phys.,2007,57:1037-1053.
[180]Zhu Z.N.et ai.,Infinitely many conservation laws for the Blaszak-Marciniak four-field integrable lattice hierarchy[J].Phys.Lett.A,2002,296:280-288.
[181] Zhu Z.N. and Xue W.M., Two new integrable lattice hierarchies associated with a discrete Schrodinger nonisospectral problem and their infinitely many conservation laws[J]. Phys. Lett. A, 2004, 320: 396-407.
[182] Zhu Z.N. and Tam H.W., Nonisospectral negative Volterra flows and mixed Volterra flows: Lax pairs, infinitely many conservation laws and integrable time discretization[J]. J. Phys. A, 2004, 37: 3175-3187.