Schrodinger方程和Boussinesq-Burgers方程的可积性与求解
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摘要
非线性演化方程是描述物理现象的一类重要数学模型,也是非线性物理特别是孤立子理论最前沿的研究课题之一.非线性演化方程精确解和可积性的研究有助于弄清物质在非线性作用下的运动规律,对相应物理现象的科学解释和工程应用起到重要作用.在非线性演化方程的研究中,寻找方程的行波解、构造多孤子解、Painleve可积性质的检验等经常遇到复杂的符号计算和推理,有的是人力难以完成的,因此妨碍了这些问题的深入剖析.近年来,符号计算的蓬勃发展,极大地推动了非线性演化方程的研究,成果不断涌现,尤其是新的求解方法层出不穷.
本文以非线性演化方程为研究对象,借助于符号计算,主要研究了一类Schrodinger方程的Lax对,并对该类方程做了Painleve方程检验,求得了Boussinesq-Burgers的两类N重Darboux变换和几个精确解.主要工作如下：
第一章绪论部分介绍了孤立子理论产生的背景、发展及研究方法,并对Schrodinger方程的研究做了简单的综述.
第二章是"AC=BD"理论,介绍了该理论的基本思想及应用,并用"AC=BD"来描述其他常见的研究孤子方程的方法.
第三章介绍了两类可积,一是Lax可积及求Lax对的方法—延拓结构法.二是Painleve可积及Painleve检验的方法和与之相关的Backlund变换Darboux变换.
第四章研究了一类变系数Schrodinger方程,利用数学机械化先判断方程Painleve可积性,又讨论了方程的Lax可积性.
第五章构造了Boussinesq-Burgers方程的Douboux变换,并在此基础上给出了该方程的两个新解.
最后给出了本文的简短总结.
Nonlinear evolution equation is an important mathematical model for describing physical phenomenon and an important field in the contempary study of nonlinear physics, especially in the study of soliton theory. The research on the explicit solution and integrability are helpful in clarifying the movement of matter under the nonlinear interactivities and plays an important role in scientifically explaining of the corresponding physical phenomenon and engineering ap-plication. Many research topics, such as searching for exact explicit solutions, multi-soliton solution, the Painleve test etc., often involve a large amount of tedious algebra auxiliary reasoning or calculations which can become unmanageable in practice, In recent years, the development of symbolic computation accelerates the research of nonlinear evolution equation greatly. Many new methods for constructing exact solutions of nonlinear evolution equations are proposed.
This dissertation equations with the aid of symbolic computation, mainly studies Lax pairs and Painleve test of a kind of Schrodinger equation. And as for Boussinesq-Burgers equation, two types of N-fold Darboux transformations and some solutions are derived. The article consists of the following parts:
Chapter 1 introduces the backgound, development and research method of soliton theory and gives Schrodinger equations a summery preparing for Chapter 4.
Chapter 2 states the theory of "AC=BD" including babsic thought and its application describing other common research method.
In Chapter 3, two kinds of integrability are introduced. One is Lax integrability including prolongation structure method for deriving Lax pair of equations. The other is Painleve inte-grability which contains Painleve test method, Backlund transformation and Darboux transfor-mation.
Chapter 4 studies a kind of Schrodinger equation. Making use of symbolic computation we study Painleve integrability first, and then Lax integrability is discussed.
Chapter 5 constructs the Darboux transformation of Boussinesq-Burgers equation. And on that base some exact solutions are presented.
At last a short summery of the dissertation is given.
本文以非线性演化方程为研究对象,借助于符号计算,主要研究了一类Schrodinger方程的Lax对,并对该类方程做了Painleve方程检验,求得了Boussinesq-Burgers的两类N重Darboux变换和几个精确解.主要工作如下：
第一章绪论部分介绍了孤立子理论产生的背景、发展及研究方法,并对Schrodinger方程的研究做了简单的综述.
第二章是"AC=BD"理论,介绍了该理论的基本思想及应用,并用"AC=BD"来描述其他常见的研究孤子方程的方法.
第三章介绍了两类可积,一是Lax可积及求Lax对的方法—延拓结构法.二是Painleve可积及Painleve检验的方法和与之相关的Backlund变换Darboux变换.
第四章研究了一类变系数Schrodinger方程,利用数学机械化先判断方程Painleve可积性,又讨论了方程的Lax可积性.
第五章构造了Boussinesq-Burgers方程的Douboux变换,并在此基础上给出了该方程的两个新解.
最后给出了本文的简短总结.
Nonlinear evolution equation is an important mathematical model for describing physical phenomenon and an important field in the contempary study of nonlinear physics, especially in the study of soliton theory. The research on the explicit solution and integrability are helpful in clarifying the movement of matter under the nonlinear interactivities and plays an important role in scientifically explaining of the corresponding physical phenomenon and engineering ap-plication. Many research topics, such as searching for exact explicit solutions, multi-soliton solution, the Painleve test etc., often involve a large amount of tedious algebra auxiliary reasoning or calculations which can become unmanageable in practice, In recent years, the development of symbolic computation accelerates the research of nonlinear evolution equation greatly. Many new methods for constructing exact solutions of nonlinear evolution equations are proposed.
This dissertation equations with the aid of symbolic computation, mainly studies Lax pairs and Painleve test of a kind of Schrodinger equation. And as for Boussinesq-Burgers equation, two types of N-fold Darboux transformations and some solutions are derived. The article consists of the following parts:
Chapter 1 introduces the backgound, development and research method of soliton theory and gives Schrodinger equations a summery preparing for Chapter 4.
Chapter 2 states the theory of "AC=BD" including babsic thought and its application describing other common research method.
In Chapter 3, two kinds of integrability are introduced. One is Lax integrability including prolongation structure method for deriving Lax pair of equations. The other is Painleve inte-grability which contains Painleve test method, Backlund transformation and Darboux transfor-mation.
Chapter 4 studies a kind of Schrodinger equation. Making use of symbolic computation we study Painleve integrability first, and then Lax integrability is discussed.
Chapter 5 constructs the Darboux transformation of Boussinesq-Burgers equation. And on that base some exact solutions are presented.
At last a short summery of the dissertation is given.
引文
[1]Russell J S. Report on waves. Fourteen meeting of the British association for the advance-ment of science, John Murray, London,1844,311—390.
[2]Boussinesq M J. Theorie des ondes et de remous quise propageant le long d'un canal recangu-laier horizonal, et communiquant an liquide contene dans ce cannal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pure Appl.,1872,17(2):55-108.
[3]Korteweg D J, deVries G. On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Phil. Mag.,1895,39(5):422-443.
[4]Fermi A, Pasta J, Ulam I. Studies of Nonlinear Problems. Los Alamos Scient. Lab., Rep., LA,1940,1955.
[5]Perring J K, Skyrme T H R.A model unified field equation. Nucl. Phys.,1962,31:550-555.
[6]Zabusky N J, Kruskal M D. Interaction of solitons in a collisionless plasma and the recur-rence of initial states. Phys. Rev. Lett.,1965,15:240-243.
[7]Wang G S. Optimal control problem for parabolic variational inequalities. Acta Mathemat-ica Scinetia,2001,21:509-525.
[8]Wang G S, Wang L J. Necessary conditions of optimal control governed by some semilinear elliptic differential equations. Advances in Mathematical Sciences and Applications,2001, 11:39—55.
[9]Wang G S, Wang L J. State-constrained optimal control in Hilbert space. Numerical Func-tional Analysis and Optimization,2001,22:255-276.
[10]汪更生,李书刚.受内反馈控制的Phase-Field系统的稳定性.数学物理学报，2004，24：193-199.
[11]Miura M R. Backlund transformation. Berlin:Springer-Verlag,1978.
[12]Wahlquist H D, Estabrook F B. Backlund transformations for solitons of the Korteweg-de Vries equation. Phys. Rev. Lett.,1973,31:1386-1390.
[13]Darboux G. Compts Rendus Hebdomadaires des Seances de l'Academie des Sciences. Pairs, 1882,94:1456-1459.
[14]M. Wadati, H. Sanuki and K.konno, Relationships among inverse method, Backlund trans-formations and infinite number of conservation laws, Prog. Theor. Phys.,1975,53:418.
[15]V. B. matveev and M. A. Salle, Darboux transformations and solitons, Springer, Berlin, 1991.
[16]C. Rogers and W. K. Schief, Backlund and Darboux transformations, Cambridge University Press,2002.
[17]李翊神,一个特征值问题的达布变换与带外场的S-G方程的孤立子解,应用数学学报,1986,9(2)：196-200.
[18]Y. S. Li, W. X. Ma, J. Zhang, Darboux transformations of classical Boussinesq system and its new solutions, Phys. Lett. A,2000,275:60-66.
[19]Y. S. Li, W. T. Han, Deep reduction of Darboux transformations to a 3×3 spectral problem, Chin. Ann. Math.,2001,22B(2):171-176.
[20]E. G. Fan, Darboux transformation and soliton-like solutions for the Gerdjikov-Ivanov equa-tion, J. Phys. A,2000,33:6925-6933.
[21]E. G. Fan, Integrable evolution systems based on Gerdjikov-Ivanov equation, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold Darboux trans-formation, J. Phys. A,2000,41:7769-7782.
[22]E. G. Fan, Explict N-fold Darboux transformations and soliton solutions for nonlinear derivative Schrodinger equations, Commun. Theor. Phys.,2001,36:401-404.
[23]E. G. Fan, A family of integrable multi-Hamiltonian systems explicitly related to some celebrated equations, J. Math. Phys.,2001,42:4337-4344.
[24]E. G. Fan, Solving Kadomtsev-Petviashvili equation via a new decomposition and Darboux transformation, Commun. Theor. Phys.,2002,37:145-148.
[25]E. G. Fan, N-fold Darboux transformation and soliton solutions for a nonlinear Dirac sys-tem, J. Phys. A,2005,38:1063-1069.
[26]Q. P. Liu, Darboux transformations for supersymmetric Korteweg-de Vries equations, Lett. Math. Phys.,1995,35(2):115-122.
[27]Q. P. Liu and M. Manas, Darboux transformations for super-symmetric KP hierarchies, Phys. Lett. B,2000,485:293-300.
[28]谷超豪,胡和生,周子翔.孤立子理论中的Darboux变换及其几何应用.上海：上海科技出版社,1999.
[29]Gu C H. A unified explicit form of Baclund transformations for generalized hierarchies of KdV equations. Lett. Math, Phys.,1986,11:325-335.
[30]Gu C H, Zhou Z X. On the Darboux matrices of Baclund transformations for AKNS systems. Lett. Math. Phys.,1987,13:179-187.
[31]曾云波.带附加项的AKNS方程族的Darboux变换.数学物理学报,1995,15：337-845.
[32]Lou S Y, Tang X Y, Jia M, et al. Vortices, circumfluence, Symmetry groups and Darboux transformations of the Euler equations. arXiv:nlin. PS/0509039.
[33]Geng X G and Tan H W. Darboux transformation and soliton solutions for generalized nonlinear Schrodinger equation. J. Phys. Soc. Jpn.,1998,68:1508.
[34]Weiss J, Tabor M, Carnvale G. The Painleve property for partial differential equa-tions. J. Math. Phys.,1983,24:522—526.
[35]Ablowitz M J, Ramani A, Segur H. A connection between nonlinear evolution equations and ordinary differential equations of P-type. J. Math. Phys.,1980,21:715-721.
[36]Weiss J. The Painleve property for partial differential equations. Backlund transforma-tion, Lax pairs, and the Schwarzian derivative, J. Math. Phys.,1983,24:1405-1413.
[37]Conte R. Invariant Painleve analysis of partial differential equations. Phys. Lett. A,1989,140:383-390.
[38]Weiss J. Backlund transformation and the Painleve prop-erty. J. Math. Phys.,1986,27:1293-1305.
[39]曾云波.与At联系的半Toda方程的Lax对与Backlund变换.数学学报,1992,35：454-459.
[40]Lou S Y. KdV extensions with Painleve property. J. Math. Phys.,1998,39:2112-2121.
[41]Chen L L, Lou S Y. Painleve analysis of a (2+1)-dimensional Burgers equa-tion. Commun. Theor. Phys.,1998,29:313-316.
[42]楼森岳.推广的Painleve展开及KdV方程的非标准截断解.物理学报,1998,47：1937-1949.
[43]Cariello F. and Tabor M., Painleve expansion for nonintegrable evolution equations, Phys-ica,1989, V.D39,77-94,,
[44]Euler N., Steeb W.-H. and Cyrus K., Polynomial field theories and nonintegrability, Physica Scripta,1990, V.41,298-301.
[45]Webb G.M. and Zank G.P., On the Painleve analysis of the two-dimensional Burg-ers' equation, Nonl. Anal. Theory Meth. Appl., 1992, V.19,167-176.
[46]Euler N., Painleve series for (1+1)-and (1+2)-dimensional discrete-velocity Boltzmann equations, Lulea University of Technology, Deparment of Mathematics. Research Report, V.7, 1997.
[47]Norbert EULER, Ove LINDBLOM, Marianna EULER and Lars-Erik PERSSON, The Higher Dimensional Bateman Equation and Painleve Analysis of Nonintegrable Wave Equa-tions, Symmetry in Nonlinear Mathematical Physics,1997, V.1,185-192.
[48]Lou S Y, Ruan H Y, Huang G X. Exact solitary waves in a convecting fluid. J. Phys. A,1991,24:L587-L590
[49]Wang M L. Solitary wave solutions for variant Boussinesq equa-tions. Phys. Lett. A,1995,199:169-172.
[50]Gao Y T, Tian B. New families of exact solutions to the integrable dispersive long wave equations in(2+1)-dimensional spaces. J. Phys. A,1996,29:2895-2903.
[51]Tian B, Gao Y T. Extension of generalized tanh method and soliton-like solutions for a set of nonlinear evolution equations. Chaos, Solitons and Fractals,1997,8:1651—1653.
[52]165]Tian B, Gao Y T. Variable-coefficient balancing-act method and variable-coefficient KdV equation from fluid dynamics and plasma physics. Eur. Phys. J. B,2001,22:351-360.
[53]Liu S K, Fu Z T, Liu S D, et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A.2001,289:69-74.
[54]Fu Z T, Liu S K, Liu S D, et al. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys. Lett. A,2001,290:72-76.
[55]CHEN Yong, LI Biao and ZHENG Yu, Exact analytical solutions in-Bose-Einsteincondensates with time-dependent atomic scattering length. Commun. Theor. Phys. (Beijing, China) 2007,47:143-148.
[56]LI Biao and CHEN Yong, A generalized method and exact solutions in Bose-Einstein con-densates in an expulsive parabolic potential. Commun. Theor. Phys. (Beijing, China) 2007, 48:391-398.
[57]Guo Wei-Ming, LI Biao and CHEN Yong, An extended subbequation rational expansion method and solutions of (2+1)-dimensional cubic nonlinear Schrodinger equation. Commun. Theor. Phys. (Beijing, China),2007,48:987-992.
[58]CHENG Xue-Ping, YE Li-Jun and LIN Ji, Approximate solutions of peturbed nonlinear Schrodinger equation. Commun. Theor. Phys. (Beijing, China),2007,48:227-231.
[59]Yang Chun-nuan, Yu Jia-lu, Wang Qu-Quan, Huang Nian-Ning, Demonstration of inverse scattering transform for NLSE equation. Commun. Theor. Phys. (Beijing, China),2007,48: 299-303.
[60]XUE Yu-Shan, TIAN Bo, ZHANG Hai-Qiang, Darboux transformation and soliton solu-tions for inhomogeneous coupled nonlinear Schrodinger equations with symbolic computa-tion. Commun. Theor. Phys. (Beijing, China),2009,52:888-896.
[61]Xing Lu, Hong-Wu Zhu, Xiang-Hua Meng, Zai-Chun Yang, Bo Tian, Soliton solutions and a Backlund transformation for a generalized nonlinear Schrodinger equation with variable coefficients from optical fiber communications. J. Math. Anal. Appl.,2007,336:1305-1315.
[62]Tiecheng Xia, Xiaohong Chen, Dengyuan Chen, Darboux transformation and soliton-like solutions of nonlinear Schrodinger equations. Chaos, Solitons and Fractals,2005,26:889-896.
[63]M.A. Abdou, New exact travelling wave solutions for the generalized nonlinear Schroedinger equation with a source. Chaos, Solitons and Fractals,2008,38:949-955.
[64]Alexander Sakhnovich, Nonisospectral integrable nonlinear equations with external poten-tials and their GBDT solutions. J. Phys. A:Math. Theor.,2008,41:15.
[65]Zhengping Wang, Huan-Song Zhou, Positive solution for nonlinear Schrodinger equation with deepening potential well. Journal of the European Mathematical Society,11(2009), 545-573.
[66]张鸿庆,弹性力学方程组一般解的统一理论.大连理工大学学报,1978,18：23-47.
[67]张鸿庆,王震宇.胡海昌,解的完备性和逼近性.科学通报,1986,30：342-344.
[68]张鸿庆,吴方向.一类偏微分方程组的一般解及其在壳体理论中的应用.力学学报,1992,24：700-707.
[69]张鸿庆,冯红.构造弹性力学位移函数的机械化算法.应用数学和力学,1995,16：315-322.
[70]张鸿庆.偏微分方程组的一般解与完备性.现代数学与力学.1991,Ⅳ.
[71]Zhang H Q, Chen Y F. Proceeding of the 3rd ACM, Lanzhou University Press,1998,147.
[72]Zhang H Q. C-D integrable system and computer aided solver for differential equa-tions. Proceeding of the 5rd ACM, World Scientific Press,2001,221—226.
[73]陈登远.孤子引论.北京：科学出版社,2006.
[74]Liu Q M. Double Wronskian solutions of the AKNS and the classical Bonssinesq hierar-chies. J. Phys. Soc. Jpn.,1990,59:3520-3527.
[75]邓淑芳.孤子方程的新解(博士学位论文).上海；上海大学,2004.
[76]M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problem. Stud. Appl. Math.,1974,53:249-315.
[77]A. C. Scott, F. Y. Chu, D.W.McLaughlin, The soliton:a new concept in applied science. IEEE Proc.,1973,61:1443.
[78]H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equa-tions, J. Math. Phys.,1975,16:1-7.
[79]F. B. Estabrook and H. D. Wahlquist, Prolongation structures of nonlinear evolution equa-tions Ⅱ, J. Math. Phys.,1976,17:1293-1297.
[80]J. P. Corones and F. Testa. Proceedings of the NSF workshop on contact transformation. Vanderbilt,1974.
[81]K. Harrison, Proceeding of the third grossmann meeting on general relativity. Ed. Nu. Ning.. North Holland,1983.
[82]A. Roy Chowdhury and T. Roy, Prolongation structure for Langmuir solitons, J. Math. Phys.,1979,20:1559.
[83]Scheen C. Implementation of the Painleve test for ordinary differential equations. Theor. Comp. Sci.,1997,187:87.
[84]Hajee G. The investigation of movable critical points of ordinary differential equations and systems of ordinary differential equations using formula manipulaiton. Memorandem Nr.386 Department of Applied Mathematics Twentc University of Technology.7500 AE Enschede, 1982.
[85]Rand D W, Winternitz P. ODEPAINLEVE-A MACSYMA package for painleve analysis of ordinary differential equations. Comput. Phys. Comm.,1986,42:359.
[86]Hlavaty L. Test of resonances in the Painleve anlysis. Comput. Phys. Comm.,1986,42: 427.
[87]Renner F. A constructive REDUCE package based upon the Painleve analysis of nonlinear evolution equations in Hamiltonian and normal form. Comp. Phys. Comm.,1992,70:409.
[88]Conte R. Painleve analysis of nonlinear PDE and related topics:a computer algebra pro-gram, in:E.Tournier(Ed.), Computer algebra and differential equations, A cademic Press, New York,1989,219.
[89]Pickering A. Testing nonlinear evolution equations for complete integrability. Ph. D. Thesis, University of Leeds,1992.
[90]Hearn A C. Reduce:User's Manual ver.3.5, The RAND Corporation, Santa Monica, RAND publication CP78,1993.
[91]朱思铭,施齐焉,现代数学和力学(MMM-VII),上海大学出版社,1997,482.
[92]Xie F D, Chen Y. An algorithmic method in Painleve analysid of PDE, Comput. Phys. Commun.,2003,154:197.
[93]Hereman W, Angenent S. The Painleve test for nonlinear ordinary and partial differential equations, Macsyma Newsletter,1989,6:11.
[94]Hereman W, Algorithmic integrability tests of nonlinear differential and lattice equations, Comput. Phys. Commun.,1998,115:428.
[95]J. Weiss, M. Tabor and Georage. Carnevale, J. Math. Phys.,24(3),522-526,1983.
[96]J. Weiss, J. Math. Phys.,27(5),1293-1305,1986.
[97]M Musette and R Conte, J. Phys. A:Math. Gen.,27,3895-3913,1994
[98]R. Conte, M. Musette and A. Pickering, J. Phys. A:Math. Gen.,28,179-187,1995.
[99]G. M. Webb and G. P. Zank, J. Phys. A:Math. Gen.,23,5465-5477,1990.
[100]M. J. Ablowitz, A. Ramani and H. Segur, J. Math. Phys.,21(4),715-721; 21(5),1006-1015, 1980.
[101]G.X.Huang and M.G.Velarde, Phys.Rev. E 54(1996)3048
[102]R.Y.Chiao, E. Garmire, and C.H.Townes, Phys. Rev. Lett.13(1964)479
[103]K.B.Dysthe and H.L. Pecscli, Plasma Physics 19(1977)931.
[104]Y.H.Ichikawa, Physica Scripta 20(1979)296.
[105]T.Sakuma and Y.Kawanami, Phys.Rev.B 29(1984)880.
[106]Li Biao and Chen Yong, Commun.Theor.Phys.(Beijing, China) 48 (2007)391
[107]Z.X.Liang, Z.D.Zhang and W.M.Liu, Phys.Rev.Lett.94(2005)050402
[108]C.H.Gu, H.S.Hu and Z.X.Zhou, Darboux Transformation in Soliton Theory and Its Geo-metric Applications, Shanghai Scientific and Technical Pulishers, Shanghai China(1999)
[109]J.F.Zhang and Q.Yang, Chin. Phys.Lett.22(2005)1855.
[110]Y.Chen,B.Li and Y.Zheng, Commun.Theor. Phys.(Beijing, China)47 (2007)143
[111]U.Al Khawaja, Phys.Lett.A 373 (2009)2710
[112]Abdul-Majid and Wazwaz, Chaos, Solitons and Fractals 37 (2008)1136
[113]R.A.Kraenkel, K.Nakkeeran and K.W.Chow, Commun Nolinear Sci Numer Simulat xxx.(2010)xxx(in press)
[114]AO Sheng-Mei and YAN Jia-Ren, Commun.Theor.Phys. (Beijing, China)47(2007)15
[115]M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge (1991).
[116]R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press. Cambridge (2004).
[117]E.D. Belokolos, A.I. Bobenko, V.Z. Enolskii, A.R. Its and V.B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin (1994).
[118]V.B. Matveev and M.A. Salle, Darboux Transformation and Solitons, Springer, Berlin (1991).
[119]C.H.Gu, H.S.Hu, and Z.X.Zhou, Darboux transformations in Soliton Theory and its Ge-ometric Applications. Shanghai Scientific and Technical Publishers, Shanghai (2005).
[120]G. Neugebauer and R. Meinel, General N-soliton solutions of the AKNS class on arbitrary background, Phys. Lett. A,1984,100:467-470.
[121]E.G. Fan, Integrable evolution systems based on Gerdjikov-Ivanov equations, bi-Hamiltonian structure,finite-dimensional integrable systems and n-fold Darboux transfor-mation, J. Math. Phys.,2000,41:7769-7782.
[122]D. Levi, G. Neugebauer and R. Meinel, A new nonlinear Schrodinger equation. itshierarchy and N-soliton solutions, Phys. Lelt. A,1984,102:1-8.
[123]D.J. Huang, H.Q. Zhang, Vandermonde-like determinants'representations of Darboux transformations and explicit solutions for the modified Kadomtsev-Petviashvili equation, Physica A,2008,387:4565-4580.
[124]Sachs.R.L, On the Integrable Variant of the Boussinesq System:Painleve Property, Retional Solutions, a Related Many-body System and Equation with the AKNS Hierarchy, Physics D,1988,30:1-27.
[125]Zakharow.V.E and Shabat.A.B,A Scheme for Integrating the Nonlinear Equations of Math-ematical Physics by the Method of the Inverse Scattering Transform, Funct.Anal. Appl., 1974,8:226-235.
[126]Zhang J., Wu Y., Li X, Quasi-periodic solution of the (2+1)-dimensional Boussinesq-Burgers soliton equation, Physica A,2003,319:213-232.
[127]Xu Rui, Darboux Transformations and Soliton Solutions for Classical Boussinesq-Burgers Equation, Commun..Theor. Phys.,2008,50:579-582.
[128]Liu Yuxiao, Darboux transformations of Boussinesq-Burgers equation, Journal of Pingdingshan Institute of Technology,2005,14(2):57-60.
[129]张渊渊,博士学位论文,大连理工大学,2007.
[130]刘晓蕾,硕士学位论文,北京邮电大学,2008.
[131]徐桂琼,李志斌.非线性演化方程的Painleve检验及其在Maple系统上的实现.中国科技论文在线.
[2]Boussinesq M J. Theorie des ondes et de remous quise propageant le long d'un canal recangu-laier horizonal, et communiquant an liquide contene dans ce cannal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pure Appl.,1872,17(2):55-108.
[3]Korteweg D J, deVries G. On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Phil. Mag.,1895,39(5):422-443.
[4]Fermi A, Pasta J, Ulam I. Studies of Nonlinear Problems. Los Alamos Scient. Lab., Rep., LA,1940,1955.
[5]Perring J K, Skyrme T H R.A model unified field equation. Nucl. Phys.,1962,31:550-555.
[6]Zabusky N J, Kruskal M D. Interaction of solitons in a collisionless plasma and the recur-rence of initial states. Phys. Rev. Lett.,1965,15:240-243.
[7]Wang G S. Optimal control problem for parabolic variational inequalities. Acta Mathemat-ica Scinetia,2001,21:509-525.
[8]Wang G S, Wang L J. Necessary conditions of optimal control governed by some semilinear elliptic differential equations. Advances in Mathematical Sciences and Applications,2001, 11:39—55.
[9]Wang G S, Wang L J. State-constrained optimal control in Hilbert space. Numerical Func-tional Analysis and Optimization,2001,22:255-276.
[10]汪更生,李书刚.受内反馈控制的Phase-Field系统的稳定性.数学物理学报，2004，24：193-199.
[11]Miura M R. Backlund transformation. Berlin:Springer-Verlag,1978.
[12]Wahlquist H D, Estabrook F B. Backlund transformations for solitons of the Korteweg-de Vries equation. Phys. Rev. Lett.,1973,31:1386-1390.
[13]Darboux G. Compts Rendus Hebdomadaires des Seances de l'Academie des Sciences. Pairs, 1882,94:1456-1459.
[14]M. Wadati, H. Sanuki and K.konno, Relationships among inverse method, Backlund trans-formations and infinite number of conservation laws, Prog. Theor. Phys.,1975,53:418.
[15]V. B. matveev and M. A. Salle, Darboux transformations and solitons, Springer, Berlin, 1991.
[16]C. Rogers and W. K. Schief, Backlund and Darboux transformations, Cambridge University Press,2002.
[17]李翊神,一个特征值问题的达布变换与带外场的S-G方程的孤立子解,应用数学学报,1986,9(2)：196-200.
[18]Y. S. Li, W. X. Ma, J. Zhang, Darboux transformations of classical Boussinesq system and its new solutions, Phys. Lett. A,2000,275:60-66.
[19]Y. S. Li, W. T. Han, Deep reduction of Darboux transformations to a 3×3 spectral problem, Chin. Ann. Math.,2001,22B(2):171-176.
[20]E. G. Fan, Darboux transformation and soliton-like solutions for the Gerdjikov-Ivanov equa-tion, J. Phys. A,2000,33:6925-6933.
[21]E. G. Fan, Integrable evolution systems based on Gerdjikov-Ivanov equation, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold Darboux trans-formation, J. Phys. A,2000,41:7769-7782.
[22]E. G. Fan, Explict N-fold Darboux transformations and soliton solutions for nonlinear derivative Schrodinger equations, Commun. Theor. Phys.,2001,36:401-404.
[23]E. G. Fan, A family of integrable multi-Hamiltonian systems explicitly related to some celebrated equations, J. Math. Phys.,2001,42:4337-4344.
[24]E. G. Fan, Solving Kadomtsev-Petviashvili equation via a new decomposition and Darboux transformation, Commun. Theor. Phys.,2002,37:145-148.
[25]E. G. Fan, N-fold Darboux transformation and soliton solutions for a nonlinear Dirac sys-tem, J. Phys. A,2005,38:1063-1069.
[26]Q. P. Liu, Darboux transformations for supersymmetric Korteweg-de Vries equations, Lett. Math. Phys.,1995,35(2):115-122.
[27]Q. P. Liu and M. Manas, Darboux transformations for super-symmetric KP hierarchies, Phys. Lett. B,2000,485:293-300.
[28]谷超豪,胡和生,周子翔.孤立子理论中的Darboux变换及其几何应用.上海：上海科技出版社,1999.
[29]Gu C H. A unified explicit form of Baclund transformations for generalized hierarchies of KdV equations. Lett. Math, Phys.,1986,11:325-335.
[30]Gu C H, Zhou Z X. On the Darboux matrices of Baclund transformations for AKNS systems. Lett. Math. Phys.,1987,13:179-187.
[31]曾云波.带附加项的AKNS方程族的Darboux变换.数学物理学报,1995,15：337-845.
[32]Lou S Y, Tang X Y, Jia M, et al. Vortices, circumfluence, Symmetry groups and Darboux transformations of the Euler equations. arXiv:nlin. PS/0509039.
[33]Geng X G and Tan H W. Darboux transformation and soliton solutions for generalized nonlinear Schrodinger equation. J. Phys. Soc. Jpn.,1998,68:1508.
[34]Weiss J, Tabor M, Carnvale G. The Painleve property for partial differential equa-tions. J. Math. Phys.,1983,24:522—526.
[35]Ablowitz M J, Ramani A, Segur H. A connection between nonlinear evolution equations and ordinary differential equations of P-type. J. Math. Phys.,1980,21:715-721.
[36]Weiss J. The Painleve property for partial differential equations. Backlund transforma-tion, Lax pairs, and the Schwarzian derivative, J. Math. Phys.,1983,24:1405-1413.
[37]Conte R. Invariant Painleve analysis of partial differential equations. Phys. Lett. A,1989,140:383-390.
[38]Weiss J. Backlund transformation and the Painleve prop-erty. J. Math. Phys.,1986,27:1293-1305.
[39]曾云波.与At联系的半Toda方程的Lax对与Backlund变换.数学学报,1992,35：454-459.
[40]Lou S Y. KdV extensions with Painleve property. J. Math. Phys.,1998,39:2112-2121.
[41]Chen L L, Lou S Y. Painleve analysis of a (2+1)-dimensional Burgers equa-tion. Commun. Theor. Phys.,1998,29:313-316.
[42]楼森岳.推广的Painleve展开及KdV方程的非标准截断解.物理学报,1998,47：1937-1949.
[43]Cariello F. and Tabor M., Painleve expansion for nonintegrable evolution equations, Phys-ica,1989, V.D39,77-94,,
[44]Euler N., Steeb W.-H. and Cyrus K., Polynomial field theories and nonintegrability, Physica Scripta,1990, V.41,298-301.
[45]Webb G.M. and Zank G.P., On the Painleve analysis of the two-dimensional Burg-ers' equation, Nonl. Anal. Theory Meth. Appl., 1992, V.19,167-176.
[46]Euler N., Painleve series for (1+1)-and (1+2)-dimensional discrete-velocity Boltzmann equations, Lulea University of Technology, Deparment of Mathematics. Research Report, V.7, 1997.
[47]Norbert EULER, Ove LINDBLOM, Marianna EULER and Lars-Erik PERSSON, The Higher Dimensional Bateman Equation and Painleve Analysis of Nonintegrable Wave Equa-tions, Symmetry in Nonlinear Mathematical Physics,1997, V.1,185-192.
[48]Lou S Y, Ruan H Y, Huang G X. Exact solitary waves in a convecting fluid. J. Phys. A,1991,24:L587-L590
[49]Wang M L. Solitary wave solutions for variant Boussinesq equa-tions. Phys. Lett. A,1995,199:169-172.
[50]Gao Y T, Tian B. New families of exact solutions to the integrable dispersive long wave equations in(2+1)-dimensional spaces. J. Phys. A,1996,29:2895-2903.
[51]Tian B, Gao Y T. Extension of generalized tanh method and soliton-like solutions for a set of nonlinear evolution equations. Chaos, Solitons and Fractals,1997,8:1651—1653.
[52]165]Tian B, Gao Y T. Variable-coefficient balancing-act method and variable-coefficient KdV equation from fluid dynamics and plasma physics. Eur. Phys. J. B,2001,22:351-360.
[53]Liu S K, Fu Z T, Liu S D, et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A.2001,289:69-74.
[54]Fu Z T, Liu S K, Liu S D, et al. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys. Lett. A,2001,290:72-76.
[55]CHEN Yong, LI Biao and ZHENG Yu, Exact analytical solutions in-Bose-Einsteincondensates with time-dependent atomic scattering length. Commun. Theor. Phys. (Beijing, China) 2007,47:143-148.
[56]LI Biao and CHEN Yong, A generalized method and exact solutions in Bose-Einstein con-densates in an expulsive parabolic potential. Commun. Theor. Phys. (Beijing, China) 2007, 48:391-398.
[57]Guo Wei-Ming, LI Biao and CHEN Yong, An extended subbequation rational expansion method and solutions of (2+1)-dimensional cubic nonlinear Schrodinger equation. Commun. Theor. Phys. (Beijing, China),2007,48:987-992.
[58]CHENG Xue-Ping, YE Li-Jun and LIN Ji, Approximate solutions of peturbed nonlinear Schrodinger equation. Commun. Theor. Phys. (Beijing, China),2007,48:227-231.
[59]Yang Chun-nuan, Yu Jia-lu, Wang Qu-Quan, Huang Nian-Ning, Demonstration of inverse scattering transform for NLSE equation. Commun. Theor. Phys. (Beijing, China),2007,48: 299-303.
[60]XUE Yu-Shan, TIAN Bo, ZHANG Hai-Qiang, Darboux transformation and soliton solu-tions for inhomogeneous coupled nonlinear Schrodinger equations with symbolic computa-tion. Commun. Theor. Phys. (Beijing, China),2009,52:888-896.
[61]Xing Lu, Hong-Wu Zhu, Xiang-Hua Meng, Zai-Chun Yang, Bo Tian, Soliton solutions and a Backlund transformation for a generalized nonlinear Schrodinger equation with variable coefficients from optical fiber communications. J. Math. Anal. Appl.,2007,336:1305-1315.
[62]Tiecheng Xia, Xiaohong Chen, Dengyuan Chen, Darboux transformation and soliton-like solutions of nonlinear Schrodinger equations. Chaos, Solitons and Fractals,2005,26:889-896.
[63]M.A. Abdou, New exact travelling wave solutions for the generalized nonlinear Schroedinger equation with a source. Chaos, Solitons and Fractals,2008,38:949-955.
[64]Alexander Sakhnovich, Nonisospectral integrable nonlinear equations with external poten-tials and their GBDT solutions. J. Phys. A:Math. Theor.,2008,41:15.
[65]Zhengping Wang, Huan-Song Zhou, Positive solution for nonlinear Schrodinger equation with deepening potential well. Journal of the European Mathematical Society,11(2009), 545-573.
[66]张鸿庆,弹性力学方程组一般解的统一理论.大连理工大学学报,1978,18：23-47.
[67]张鸿庆,王震宇.胡海昌,解的完备性和逼近性.科学通报,1986,30：342-344.
[68]张鸿庆,吴方向.一类偏微分方程组的一般解及其在壳体理论中的应用.力学学报,1992,24：700-707.
[69]张鸿庆,冯红.构造弹性力学位移函数的机械化算法.应用数学和力学,1995,16：315-322.
[70]张鸿庆.偏微分方程组的一般解与完备性.现代数学与力学.1991,Ⅳ.
[71]Zhang H Q, Chen Y F. Proceeding of the 3rd ACM, Lanzhou University Press,1998,147.
[72]Zhang H Q. C-D integrable system and computer aided solver for differential equa-tions. Proceeding of the 5rd ACM, World Scientific Press,2001,221—226.
[73]陈登远.孤子引论.北京：科学出版社,2006.
[74]Liu Q M. Double Wronskian solutions of the AKNS and the classical Bonssinesq hierar-chies. J. Phys. Soc. Jpn.,1990,59:3520-3527.
[75]邓淑芳.孤子方程的新解(博士学位论文).上海；上海大学,2004.
[76]M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problem. Stud. Appl. Math.,1974,53:249-315.
[77]A. C. Scott, F. Y. Chu, D.W.McLaughlin, The soliton:a new concept in applied science. IEEE Proc.,1973,61:1443.
[78]H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equa-tions, J. Math. Phys.,1975,16:1-7.
[79]F. B. Estabrook and H. D. Wahlquist, Prolongation structures of nonlinear evolution equa-tions Ⅱ, J. Math. Phys.,1976,17:1293-1297.
[80]J. P. Corones and F. Testa. Proceedings of the NSF workshop on contact transformation. Vanderbilt,1974.
[81]K. Harrison, Proceeding of the third grossmann meeting on general relativity. Ed. Nu. Ning.. North Holland,1983.
[82]A. Roy Chowdhury and T. Roy, Prolongation structure for Langmuir solitons, J. Math. Phys.,1979,20:1559.
[83]Scheen C. Implementation of the Painleve test for ordinary differential equations. Theor. Comp. Sci.,1997,187:87.
[84]Hajee G. The investigation of movable critical points of ordinary differential equations and systems of ordinary differential equations using formula manipulaiton. Memorandem Nr.386 Department of Applied Mathematics Twentc University of Technology.7500 AE Enschede, 1982.
[85]Rand D W, Winternitz P. ODEPAINLEVE-A MACSYMA package for painleve analysis of ordinary differential equations. Comput. Phys. Comm.,1986,42:359.
[86]Hlavaty L. Test of resonances in the Painleve anlysis. Comput. Phys. Comm.,1986,42: 427.
[87]Renner F. A constructive REDUCE package based upon the Painleve analysis of nonlinear evolution equations in Hamiltonian and normal form. Comp. Phys. Comm.,1992,70:409.
[88]Conte R. Painleve analysis of nonlinear PDE and related topics:a computer algebra pro-gram, in:E.Tournier(Ed.), Computer algebra and differential equations, A cademic Press, New York,1989,219.
[89]Pickering A. Testing nonlinear evolution equations for complete integrability. Ph. D. Thesis, University of Leeds,1992.
[90]Hearn A C. Reduce:User's Manual ver.3.5, The RAND Corporation, Santa Monica, RAND publication CP78,1993.
[91]朱思铭,施齐焉,现代数学和力学(MMM-VII),上海大学出版社,1997,482.
[92]Xie F D, Chen Y. An algorithmic method in Painleve analysid of PDE, Comput. Phys. Commun.,2003,154:197.
[93]Hereman W, Angenent S. The Painleve test for nonlinear ordinary and partial differential equations, Macsyma Newsletter,1989,6:11.
[94]Hereman W, Algorithmic integrability tests of nonlinear differential and lattice equations, Comput. Phys. Commun.,1998,115:428.
[95]J. Weiss, M. Tabor and Georage. Carnevale, J. Math. Phys.,24(3),522-526,1983.
[96]J. Weiss, J. Math. Phys.,27(5),1293-1305,1986.
[97]M Musette and R Conte, J. Phys. A:Math. Gen.,27,3895-3913,1994
[98]R. Conte, M. Musette and A. Pickering, J. Phys. A:Math. Gen.,28,179-187,1995.
[99]G. M. Webb and G. P. Zank, J. Phys. A:Math. Gen.,23,5465-5477,1990.
[100]M. J. Ablowitz, A. Ramani and H. Segur, J. Math. Phys.,21(4),715-721; 21(5),1006-1015, 1980.
[101]G.X.Huang and M.G.Velarde, Phys.Rev. E 54(1996)3048
[102]R.Y.Chiao, E. Garmire, and C.H.Townes, Phys. Rev. Lett.13(1964)479
[103]K.B.Dysthe and H.L. Pecscli, Plasma Physics 19(1977)931.
[104]Y.H.Ichikawa, Physica Scripta 20(1979)296.
[105]T.Sakuma and Y.Kawanami, Phys.Rev.B 29(1984)880.
[106]Li Biao and Chen Yong, Commun.Theor.Phys.(Beijing, China) 48 (2007)391
[107]Z.X.Liang, Z.D.Zhang and W.M.Liu, Phys.Rev.Lett.94(2005)050402
[108]C.H.Gu, H.S.Hu and Z.X.Zhou, Darboux Transformation in Soliton Theory and Its Geo-metric Applications, Shanghai Scientific and Technical Pulishers, Shanghai China(1999)
[109]J.F.Zhang and Q.Yang, Chin. Phys.Lett.22(2005)1855.
[110]Y.Chen,B.Li and Y.Zheng, Commun.Theor. Phys.(Beijing, China)47 (2007)143
[111]U.Al Khawaja, Phys.Lett.A 373 (2009)2710
[112]Abdul-Majid and Wazwaz, Chaos, Solitons and Fractals 37 (2008)1136
[113]R.A.Kraenkel, K.Nakkeeran and K.W.Chow, Commun Nolinear Sci Numer Simulat xxx.(2010)xxx(in press)
[114]AO Sheng-Mei and YAN Jia-Ren, Commun.Theor.Phys. (Beijing, China)47(2007)15
[115]M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge (1991).
[116]R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press. Cambridge (2004).
[117]E.D. Belokolos, A.I. Bobenko, V.Z. Enolskii, A.R. Its and V.B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin (1994).
[118]V.B. Matveev and M.A. Salle, Darboux Transformation and Solitons, Springer, Berlin (1991).
[119]C.H.Gu, H.S.Hu, and Z.X.Zhou, Darboux transformations in Soliton Theory and its Ge-ometric Applications. Shanghai Scientific and Technical Publishers, Shanghai (2005).
[120]G. Neugebauer and R. Meinel, General N-soliton solutions of the AKNS class on arbitrary background, Phys. Lett. A,1984,100:467-470.
[121]E.G. Fan, Integrable evolution systems based on Gerdjikov-Ivanov equations, bi-Hamiltonian structure,finite-dimensional integrable systems and n-fold Darboux transfor-mation, J. Math. Phys.,2000,41:7769-7782.
[122]D. Levi, G. Neugebauer and R. Meinel, A new nonlinear Schrodinger equation. itshierarchy and N-soliton solutions, Phys. Lelt. A,1984,102:1-8.
[123]D.J. Huang, H.Q. Zhang, Vandermonde-like determinants'representations of Darboux transformations and explicit solutions for the modified Kadomtsev-Petviashvili equation, Physica A,2008,387:4565-4580.
[124]Sachs.R.L, On the Integrable Variant of the Boussinesq System:Painleve Property, Retional Solutions, a Related Many-body System and Equation with the AKNS Hierarchy, Physics D,1988,30:1-27.
[125]Zakharow.V.E and Shabat.A.B,A Scheme for Integrating the Nonlinear Equations of Math-ematical Physics by the Method of the Inverse Scattering Transform, Funct.Anal. Appl., 1974,8:226-235.
[126]Zhang J., Wu Y., Li X, Quasi-periodic solution of the (2+1)-dimensional Boussinesq-Burgers soliton equation, Physica A,2003,319:213-232.
[127]Xu Rui, Darboux Transformations and Soliton Solutions for Classical Boussinesq-Burgers Equation, Commun..Theor. Phys.,2008,50:579-582.
[128]Liu Yuxiao, Darboux transformations of Boussinesq-Burgers equation, Journal of Pingdingshan Institute of Technology,2005,14(2):57-60.
[129]张渊渊,博士学位论文,大连理工大学,2007.
[130]刘晓蕾,硕士学位论文,北京邮电大学,2008.
[131]徐桂琼,李志斌.非线性演化方程的Painleve检验及其在Maple系统上的实现.中国科技论文在线.