摘要
本文根据数学机械化思想,以计算机符号数值计算软件为工具,研究了孤立子理论,分数微分方程和混沌系统中的若干问题：
1.构造非线性发展方程的精确解及其机械化实现；
2.构造非线性分数微分方程的数值解；
3.混沌同步及其自动推理算法.
第一章介绍了数学机械化,孤立子理论,分数阶微积分和混沌同步研究的历史发展及现状,同时介绍了国内外学者在这些学科领域所取得的成果.
第二章介绍了Wu-Ritt微分消元理论,分数阶微积分理论中的几个基本的定义及其性质,AC=BD的基本理论及在这一统一理论框架下考虑非线性发展方程(组)精确解的构造问题.
第三章基于将非线性发展方程精确求解代数化,算法化,机械化的指导思想,首次提出了求解非线性发展方程精确解的有理形式展开法,然后以符号计算软件为工具,运用吴消元法,研究了有理形式展开法的各种具体形式及其应用：
(1)利用一个新的辅助方程研究了高维耦合Burgers方程,得到了许多新的complexiton解；
(2)基于椭圆函数,提出椭圆函数有理展开法,得到了(2+1)-维色散长波方程许多新的有理形式椭圆函数解；
(3)基于Riccati方程,提出Riccati方程有理展开法,获得了(2+1)-维Broer-Kaup-Kupershmidt方程许多新的精确解；
(4)提出广义椭圆方程有理展开法并研究了Whitham-Broer-Kaup方程,得到了许多新的有理形式精确解；
(5)推广了Riccati方程有理展开法,并应用其求解(2+1)-维Burgers方程；
(6)提出了多辅助方程有理展开法,应用其构造新型complexiton解；
(7)扩展了Riccati方程有理展开法,求得了一类非线性发展方程的非行波解；
(8)针对随机微分方程,改进了Riccati方程有理展开法,从而得到了许多新的随机精确解；
(9)将有理展开法推广到微分-差分情形,并用其求解了两类Toda方程许多新的精确解.
第四章将原本用于求解整数阶微分方程精确解及数值解的Adomian分解法和同伦扰动法推广到分数微分的情形,运用它们研究了一些有重要物理意义的非线性分数阶偏微分方程如分数阶KdV-mKdV方程,分数阶Boussinesq方程,分数阶KdV-Burgers方程,分数阶Kuramoto-Sivashinsky方程,首次获得了一些有实际物理意义的数值解,且给出了一个简单的收敛性判定定理.
第五章首先改进了混沌和超混沌系统的广义Q-s(滞后,完全,提前)同步机理和自动推理算法.在符号数值计算软件的帮助下,获得了两个初值不同的Chua系统,超混沌Rossler系统和超混沌Tamasevicius-Namajunas-Cenys系统,以及两个初值不同的广义Henon映射,Henon-like映射和广义Henon映射之间的广义Q-s同步,并使用数值模拟说明了该算法的有效性.然后给出了双向广义部分(滞后,完全,提前)同步定义,并给出了获得该同步的一个机械化算法,以Rossler系统,统一的Lorenz-Chen-Lii系统,超混沌Tamasevicius-Namajunas-Cenys系统为例,通过数值模拟说明了该算法的有效性.
In this dissertation, under the guidance of mathematics mechanization and by means of symbolic-numeric computation software, some problems in the theory of soliton, fractional dif-ferential equation and chaotic system are discussed as follows:
1. the realization of mechanization for constructing the exact solutions for nonlinear evo-lution equations;
2. constructing the numerical solutions for nonlinear fractional differential equations;
3. chaos synchronization and automatic reasoning scheme developed for them.
Chapter 1 is devoted to reviewing the history and development of the mathematics mech-anization, soliton theory, fractional calculous and chaos synchronization, with an emphasis on some achievements on the subjects involved in this dissertation are presented at home and abroad.
Chapter 2 introduces Wu-Ritt differential elimination theory, some basic notations and properties on fractional calculous, basic theories on AC=BD as well as the construction of exact solutions of nonlinear evolution equation(s) under the instruction of this theory.
Based on the idea of solving nonlinear evolution equations, algebraic method, algorithm reality, mechanization, Chapter 3 firstly presents the rational expansion method to uniformly construct the exact solutions for nonlinear evolution equations, and then by means of Wu elimina-tion theory and symbolic computation software, considers some concrete forms and applications of the rational expansion method:
(1) the high dimension coupled Burgers equation is considered by a new auxiliary equation method and some new complexiton solutions are found;
(2) based on the elliptic functions, the elliptic function rational expansion method is pre-sented and some new rational formal elliptic function solutions of (2+1)-dimensional dispersive long wave equation are found;
(3) Riccati equation rational expansion method is presented based on the Riccati equa-tion, and some new exact solutions of (2+1)-dimensional Broer-Kaup-Kupershmidt equation are obtained;
(4) Whitham-Broer-Kaup equation is studied by using the generalized elliptic rational expansion method and some new rational formal exact solutions are found;
(5) the Riccati equation rational expansion method is extended to study the (2+1)-dimensiona Burgers equation;
(6) multiple auxiliary equations rational expansion method and its application to construct new complexiton solutions;
(7) generalized Riccati equation rational expansion method and some non-travelling wave solutions of nonlinear evolution equations;
(8) for stochastic differential equations, the Riccati equation rational expansion method is further improved and some new stochastic exact solutions are obtained;
(9) the rational expansion method is further improved for differential-difference equations and some new exact solutions of two Toda equations are found.
In chapter 4, the Adomian decomposition method and homotopy perturbation method which traditionally developed for differential equations of integer order are directly extended to derive numerical solutions of the nonlinear fractional differential equations. Some nonlinear fractional differential equations with physical significance, such as fractional KdV-mKdV equa-tion, fractional Boussinesq equation, fractional KdV-Burgers equation and fractional Kuramoto-Sivashinsky equation are investigated. Some available numerical solutions are firstly obtained. A simple theorem is also given to determine convergence of these methods.
Chapter 5 first improves the automatic reasoning scheme for generalized Q-S (lag, com-plete and anticipated) synchronization of chaotic and hyperchaotic system. Based on the symbolic-numeric computation software, the generalized Q-S synchronization between two iden-tical Chua's circuit with different initial values, hyperchaotic Rossler system and hyperchaotic Tamasevicius-Namajunas-Cenys system, two identical generalized Henon map with different ini-tial values, Henon-like map and generalized Henon map are obtained. Numerical simulations verify the effectiveness of the proposed scheme. Then we present the definition of bidirectional partial generalized (lag, complete and anticipated) synchronization and an automatic reason-ing scheme to obtain it. The Rossler system, a new unified Lorenz-Chen-Lu system as well as the hyperchaotic Tamasevicius-Namajunas-Cenys system are chosen to illustrate the proposed scheme. Numerical simulations are used to verify the effectiveness of the proposed scheme.
引文
[1]吴文俊.吴文俊论数学机械化.山东：山东教育出版社,1996.
[2]高小山.数学机械化进展综述.数学进展,2001,30(5)：385-404.
[3]石赫.机械化数学引论.湖南：湖南教育出版社,1998.
[4]吴文俊.几何定理机器证明的基本原理.北京：科学出版社,1984.
[5]Wu W T. On the decision problem and the mechanization of theorem-proving in elementary ge-ometry. Scientia Sinica,1978,21:159-172.
[6]Wu W T. On zeros of algebraic equations:an application of Ritt principle. Kexue Tongbao,1986, 31:1-5.
[7]Ritt J F. Differential Algebra. New York:American mathematical Society,1950.
[8]Wu W T. On the foundation of algebraic differential geometry. MMRP.1989.3.
[9]Gao X S, Zhang J Z, Chou S C. Geometry Expert (in Chinese), Nine Chapters Pub. Taiwan,1998.
[10]Chou S C, Gao X S, Zhang J Z. Machine Proofs in Geometry. Singapore:World Scientific,1994.
[11]Chou S C. Automated reasoning in differential geometry and mechanics using the characteristic set method. Ⅰ. An improved version of Ritt-Wu's decomposition algorithm. J. Auto. Reasoning,1993, 10:161-172.
[12]Gao X S, Chou S C, Zhang J Z. Automated production of traditional proofs for constructive geometry theorems. Proceedings of ISSAC-93,48.
[13]Gao X S. An Introduction to Wu's Method of Mechanical Geometry Theorem Proving. IFIP Trans-action on, Automated Reasoning, North-Holland,1993,13.
[14]Wang D M, Gao X S. Geometry theorems proved mechanically using Wu's method. MM-Reseach Preprints,1987(2).
[15]Wang D M, Hu S. Mechanical proving system for constructible theorems in elementry geometry. J. Sys. Sci. Math. Scis.1995,7.
[16]Lin D D, Liu Z J. Some results in theorem proving in finite geometry. Proc. of IWMM. Inter Academic Pub,1992.
[17]Wu J Z, Liu Z J. Proceedings of 1st Asian Symposium on Computer Mathematics, Beijing of China,1995.
[18]Wu J Z, Liu Z J. On first-order theorem proving using generalized odd-superpositions Ⅱ. Science in China(Series E),1996,39(6):50-61.
[19]Chou S C, Gao X S. Ritt-Wu's decomposition algorithm and geometry theorem proving. CADE'10. Stikel M E, Ed. Springer-Verlag,1990,207-220.
[20]Kapur D, Wan H K. Refutational proofs of geometry theorems via characteristic sets. Proc. of ISSAC-90, Tokyo, Japan,1990,277-284.
[21]Ko H. Geometry theorem proving by decomposition of quasi-algebraic sets:an application of the Ritt-Wu principle. Artif. Intell.1988,37:95-122.
[22]Li H B, Cheng M T. Proving theorems in elementary geometry with Clifford algebraic method. Chinese Math Progress.1997,26(4):357-371.
[23]Li H B. Automated reasoning with differential forms. Proc. of ASCM'92,1992.
[24]Li H B, Cheng M T. Clifford algebraic reduction method for mechanical theorem proving in differ-ential geometry. J. Auto. Reasoning,1998,21:1-21.
[25]Wang D M. Clifford algebraic calculus for geometric reasoning with applications to computer vision. In:D. Wang. Automated Deduction in geometry LNAI.1996,1360:115.
[26]Shi H. Proceeding 1992. International work Math. Mech., International Academic,1992,79. MM-Preprints,1997,1.
[27]Sun X D, Wang K, Wu K. Solutions of Yang-Baxter equation with spectral parameters for a six-vertex model. Acta Phys. Sinica(Chinese),1995,44(1):1-8.
[28]Sun X D, Wang K, Wu K. Classification of six-vertex-type solutions of the colored Yang-Baxter equation. J. Math. Phys.1995,36(10):6043-6063.
[29]Wang M L, Li Z B. Proc. Of the 1994 Beijing Symposium on nonlinear evolution equations and infinite dimensional dynamics systems. Zhongshan University Press,1995,181.
[30]Li Z B, Wang M L. Travelling wave solutions to the two-dimensional KdV-Burgers equation. J. Phys. A:Math. Gen.1993,26:6027-6031.
[31]李志斌,张善卿.非线性波方程准确孤立波解的符号计算.数学物理学报.1997,17(1)：81-89.
[32]Li Z B, Shi H. Exact solutions for Belousov-Zhabotinski reaction-diffusion system. Appl. Math. J. Chinese Univ. Ser. B,1996,11(1):1-6.
[33]Li Z B. Proc. Of Asian Symposition on Computer Mathematics. Lanzhou University,1998,153.
[34]Li Z B, Liu Y P. RATH:A Maple package for finding travelling solitary wave solutions to nonlinear evolution equations. Compu. Phys. Commun.2002,148:256-266.
[35]范恩贵.可积系统与计算机代数.北京：科技出版社,2004.
(361闰振亚.大连理工大学博士学位论文.大连：大连理工大学,2002.
[37]郑学东.大连理工大学硕士学位论文.大连：大连理工大学,2002.
[38]朝鲁.大连理工大学博士学位论文.大连：大连理工大学,1997.
[39]陈勇.大连理工大学博士学位论文.大连：大连理工大学,2003.
[40]谢福鼎.大连理工大学博士学位论文.大连：大连理工大学,2003.
[41]李彪.大连理工大学博士学位论文.大连：大连理工大学,2005.
[42]Fan E G, Zhang H Q. Backlund transformation and exact solutions for Whitham-Broer-Kaup equations in shallow water. Appl. Math. Mech.1998,19:713-716.
[43]范恩贵,张鸿庆.非线性波动方程的孤波解.物理学报.1997,46：1254-1258.
[44]Yan Z Y, Zhang H Q. New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water. Phys. Lett. A.2001,285:355-260.
[45]Yan Z Y, Zhang H Q. On a new algorithm of constructing solitary wave solutions for systems of nonlinear evolution equations in mathematical physics. Appl. Math. Mech.2000,21:342-347.
[46]Yan Z Y, Zhang H Q. Multiple soliton-like and periodic-like solutions to the generalization of integrable (2+1)-dimensional dispersive long-wave equations. J. Phys. Soc. Jpn.2002,71:437-442.
[47]闫振亚,张鸿庆. New explicit travelling wave solutions for a class of nonlinear evolution equations.物理学报,1999,48：1-5.
[48]Kaltofen E. Challenges of Symbolic Computation:My Favorite Open Problems, J. Symb. Comput. 2000,29:891-919.
[49]Buchberger B, Collins G E, Loos R. Computer Algebra-Symbolic and Algebraic Computation. Beijing:World Publishing Corporation,1988.
[50]Risch R H. The problem of integration in finite terms. Trans. Ams. Math. Soc.1969,139:167-189.
[51]Berlekamp E R. Factoring polynomials over large finite fields. Math. Comput.1970,24:713-735.
[52]Brown W S. On the partition function of a finite set. J. ACM.1971,18:478-504.
[53]Gosper R W. Decision Procedures for Indefinite Hypergeometric Summation. Proc. Nat. Acad. Sci. USA,1978,75:40-42.
[54]Lenstra A K. Factoring multivariate polynomials over algebraic number fields. SIAM J. Comput. 1987,16:591-598.
[55]Lenstra A K, Lenstra H W Jr, Lovasz L. Factoring polynomials with rational coefficients. Math. Ann.1982,261:515-534.
[56]Kaltofen E, Trager B. Computing with polynomials given by black boxes for their evaluations: greatest common divisors, factorization, separation of numerators and denominators. J. Symb. Comput.1990,9:301-320.
[57]Shoup V. A new polynomial factorization algorithm and its implementation. J. Symb. Comput. 1995,20:363-397.
[58]Russell J S. Report on waves, Fourteen meeting of the British association for the advancement of science, John Murray, London,1844,311-390.
[59]Boussinesq J. Theorie des ondes et de remous qui se propagent. J. Math. Pure Appl.1972,17: 55-108.
[60]Korteweg D J, deVries G. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag.1895; 39:422-433.
[61]Fermi A, Pasta J, Ulam. I. Studies of Nonlinear Problems. Los Alamos Rep. LA,1940,1955.
[62]Perring J K, Skyrme T H R. A model unified field equation. Nucl. Phys.1962,31:550-555.
[63]Zabusky N J, Kruslcal M D. Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett.1965,15:240-243.
[64]Ablowitz M J, Segur H. Solitons and the inverse scattering transformation. Philadelphia:SIAM, 1981.
[65]Ablowitz M J, Clarkson P A. Solitons, nonlinear evolution equations and inverse scattering. Cam-bridge:Cambridge University Press,1991.
[66]Newell A C. Soliton in mathematics and physics. Philadelphia:SIAM,1985.
[67]Faddeev L D, Takhtajan L A. Hamiltonian method in the theory of solitons. Berlin:Springer-Verlag,1987.
[68]Matveev V B, Salle M A. Darboux transformation and Solitons. Berlin:Springer,1991.
[69]郭柏灵,庞小峰.孤立子.北京：科学出版社,1987.
[70]谷超豪等.孤立子理论与应用.杭州：浙江科技出版社,1990.
[71]谷超豪等.孤立子理论中的Darboux变换及其几何应用.上海：上海科技出版社,1999.
[72]李翊神.孤子与可积系统.上海：上海科技出版社,1999.
[73]Hirota R. The direct method in soliton theory. New York:Cambridge University Press,2004.
[74]Backlund A V. Lund Universitets Arsskrift 1885,10.
[75]Wahlquist H D, Estabrook F B. Backlund transformations for solitons of the Korteweg-de Vries equation. Phys. Rev. Lett.1973,31:1386-1390.
[76]Weiss J, Tabor M, Carnvale G. The Painleve property for partial differential equations. J. Math. Phys.1983,24:522-526.
[77]Weiss J. The Painleve property for partial differential equations. Ⅱ. Backlund transformation, Lax pairs, and the Schwarzian derivative. J. Math. Phys.1983,24:1405-1413.
[78]Darboux G. Compts Rendus Hebdomadaires des Seances de l'Academie des Sciences. Pairs,1882, 94:1456-1459.
[79]Wadati M et al. Simple derivation of Backlund transformation from Riccati form of inverse method. Prog. Theor. Phys.1975,53:1652-1656.
[80]Gu. C H. A unified explicit form of Backlund transformations for generalized hierarchies of KdV equations. Lett. Math. Phys.1986,11; 325-335.
[81]Gu C H, Zhou Z X. On the Darboux matrices of Backlund transformations for AKNS systems. Lett. Math. Phys.1987,13:179-187.
[82]Hu X B, Clarkson P A. Backlund transformations and nonlinear superposition formulae of a differential-difference KdV equation. J. Phys. A:Math. Gen.1998,31:1405-1414.
[83]Hu X B, Zhu Z N. A Backlund transformation and nonlinear superposition formula for the Belov-Chaltikian lattice. J. Phys. A:Math. Gen.1998,31:4755-4761.
[84]Matveev V B, Sklyanin E K. On Backlund transformations for many-body systems. J. Phys. A: Math. Gen.1998,31:2241-2251.
[85]Sklyanin E K. Canonicity of Backlund transformation:r-matrix approach. I. L. D. Faddeev's Seminar on Mathematical Physics. Amer. Math. Soc. Transt. Ser.2,2000,201:277-282.
[86]Hone A N W et al. Backlund transformations for many-body systems related to KdV. J. Phys. A: Math. Gen.1999,32:L299-L306.
[87]Choudhury A G, Chowhury A R. Canonical and Backlund transformations for discrete integrable systems and classical r-matrix. Phys. Lett. A,2001,280:37-44.
[88]Zeng Y B et al. Canonical explicit Backlund transformations with spectrality for constrained flows of soliton hierarchies. Phys. A,2002,303:321-338.
[89]Lou S Y et al. Vortices, circumfluence, symmetry groups and Darboux transformations of the Euler equations. arXiv:nlin.PS/0509039.
[90]Lie S. Uber die integration durch bestimmte integrate von einer klasse linear partieller differential-gleichung. Arch for Math.1881,6:328-368; Gesamm elts Abhandlungen, Vol. Ⅲ, Teubner B. G., Leipzig,1922,492-523.
[91]Lie S. Classification und Integration von gewohnlichen Differentialgleichungen zwischenxy, die eine Gruppe von Transformationen gestatten:Die nachstehende Arbeit erschien zum ersten Male im Friihling 1883 im norwegischen Archiv. Math. Ann.1908,32:213-281.
[92]Bateman H. The conformal transformations of a spsce of four dimensions and their applications to geometrical optics. Proc. London Math. Soc.1909,7:70-92.
[93]Cunningham E. The principle of relativity in electrodynamics and an extension thereof. Proc. London Math. Soc.1909,8:77-97.
[94]Nother E. Invariante varlationsprobleme. Nachr Koning. Gesell. Wissen. Gottingen, Meth. Phys. kl,1918,235-257.
[95]Ovsiannikov L V. Groups and group-invariant solutions of differential equations, Dokl. Akad. Nauk. USRR,1958,118:439-442.
[96]Ovsiannikov L V. Group properties of differential equations. Novosibirsl, 1962.
[97]Ovsiannikov L V. Group analysis of differential equations. New York:Academic Press,1982.
[98]Sedov L I. Similarity and dimensional method,4th ed. New York:Academic Press,1959.
[99]Sukharev M G. Invariant solutions of equations describing the motion of fluids and gases in long pipelines. Dokl. Akad. Nauk. USRR,1967,175:781-784.
[100]Ibragimov N H. Transformation groups applied to mathematical Physcics. Boston:Reidel,1985.
[101]Venikov V A. Theory of Similarity and Simulation. London:Mac Donald Technical and Scientific, 1969.
[102]Bluman G W, Cole J D. The general similarity solution of the heat equation. J. Math. Mech.1969, 18:1025-1042.
[103]Bluman G W, Cole J D. Similarity method for differential equation. Appl. Math. Sci. No.13, New York:Springer-Berlag.1974.
[104]Levi D, Winternitz P. Non-classical symmetry reduction:example of the Boussinesq equation. J. Phys. A:Math. Gen.1989,22:2915-2924.
[105]Vorob'er E M. Reduction and quotient equations for differential equations with symmetries. Acta Appl. Math.1991,23:1-24.
[106]Olver P J. Evolution equations possessing infinitely many symmetries. J. Math. Phys.1977,18: 1212-1215.
[107]Olver P J. On the Hamiltonian structure of evolution equations. Math. Proc. Camb. Phill Soc. 1980,88:71-88.
[108]Fuchsseiner B, Fokas A S. Symplectic structures, their Backlund transformations and hereditary symmetries. Phys. D,1981,4:47-66.
[109]Chen H H et al. In Advance in nonlinear wave. ed L. Debnath,1982,233.
[110]李翊神,朱国诚.一个谱可变演化方程的对称.科学通报,1986,19：1449-1453.
[111]李翊神,朱国诚.可积方程新的对称,李代数及谱可变演化方程.中国科学A,1987,30：1243-1250.
[112]李翊神,朱国诚.“C可积”非线性方程的代数性质-(Ⅰ)Burgers方程和Calogero方程.中国科学A,1990,33：513-520.
[113]田畴. Burgers方程的新的强对称,对称和李代数.中国科学A,1987,31：141-151.
[114]Zhu G C, Chen H H. Symmetries and integrability of the cylindrical Korteweg-de Vries equation. J. Math. Phys.1986,27:100-103.
[115]Lou S Y. Twelve sets of symmetries of the Caudrey-Dodd-Gibbon-Sawada-Kotera equation. Phys. Lett. A,1993,175:23-26.
[116]Sun T, Wang W. Symmetries and their Lie algebra properties for the higher-order Burgers equa-tions. J. Phys. A:Math. Gen.1989,22:3743-3751.
[117]Li Y S, Zhu G C. New set of symmetries of the integrable equations, Lie algebra and non-isospectral evolution equations. Ⅱ. AKNS system. J. Phys. A:Math. Gen.1986,19:3713-3725.
[118]Ma W X. K symmetries and tau symmetries of evolution equations and their Lie algebras. J. Phys. A:Math. Gen.1990,23:2707-2716.
[119]Clarkson P A, Kruskal M D. New similarity reductions of the Boussinesq equation. J. Math. Phys. 1989,30:2201-2213.
[120]Lou S Y. A note on the new similarity reductions of the Boussinesq equation. Phys. Lett. A,1990, 151:133-135.
[121]Clarkson P A. New similarity solutions for the modified Boussinesq equation. J. Phys. A:Math. Gen.1989,22:2355-2367.
[122]European H S. Nonclassical Symmetry Reductions of the Boussinesq Equation. Chaos, Solitons & Fractals,1995,5:2261-2301.
[123]Clarkson P A, European H S. Symmetry reductions of a generalized, cylindrical nonlinear Schrodinger equation. J. Phys. A:Math. Gen.1993,26:133-150.
[124]Clarkson P A, Winternitz P. Nonclassical symmetry reductions for the Kadomtsev-Petviashvili equation. Phys. D,1991,49:257-272.
[125]Clarkson P A, Ludlow D K. Symmetry reductions, exact solutions, and Painleve analysis for a generalised Boussinesq equation. J. Math. Anal. Appl.1994,186:132-155.
[126]Lou S Y. Generalized Boussinesq equation and KdV equation-Painleve properties, Backlund trans-formations and Lax pairs. Sci. China A,1991,34:1098-1108.
[127]Lou S Y, Ruan H Y. Nonclassical analysis and Painleve property for the Kupershmidt equations. J. Phys. A:Math. Gen.1993,26:4679-4688.
[128]Qu C Z. Nonclassical symmetry reductions for the integrable super KdV equations. Commun. Theor. Phys.1995,24:177-184.
[129]Zhang J Z, Lin J. Similarity reductions for the Khokhlov-Zabolotskaya equation. Commun. Theor. Phys.1995,24:69-72.
[130]王烈衍.K(m,n)方程的对称约化.物理学报,2000,49：181-185.
[131]Nucci M C, Clarkson P A. The nonclassical method is more general than the direct method for symmetry reductions. An example of the FitzHugh-Nagumo equation. Phys. Lett. A,1992,164: 49-56.
[132]Olver P J. Direct reduction and differential constraints. Proc. R. Soc. Lond. A,1994,444:509-523.
[133]Pucci E. Similarity reductions of partial differential equations. J. Phys. A:Math. Gen.1992,25: 2631-2640.
[134]Arrigo P J et al. Nonclassical symmetry solutions and the methods of Bluman-Cole and Clarkson-Kruskal. J. Math. Phys.1993,34:4692-4703.
[135]Lou S Y. Symmetries of the Kadomtsev-Petviashvili equation. J. Phys. A:Math. Gen.1993,26: 4387-4394.
[136]Lou S Y et al. Similarity and conditional similarity reductions of a (2+1)-dimensional KdV equation via a direct method. J. Math. Phys.2000,41:8286-9303.
[137]Lou S Y, Tang X Y. Conditional Similarity Reduction Approach:Jimbo-Miwa equation. Chin. Phys.2001,10:897-901.
[138]Tang X Y et al. Conditional similarity solutions of the Boussinesq equation. Commun. Theor. Phys. 2001,35:399-404.
[139]Zhaanov R Z, Fuschchych W I. Conditional symmetry and new classical solutions of the Yang-Mills equations. J. Phys. A:Math. Gen.1995,28:6253-6264.
[140]朝鲁.微分方程(组)对称向量的吴-微分特征列算法及其应用.数学物理学报,1999,19(3)：326-332.
[141]Hereman W. Review of symbolic software for Lie symmetry analysis. Math. Comput. Modelling, 1997,25:115-132.
[142]朝鲁,张鸿庆,唐立民.一个计算微分方程(组)对称群的Mathematica程序包及其应用.计算物理,1997,14(3)：375-379.
[143]吕卓生.大连理工大学博士学位论文.大连：大连理工大学,2004.
[144]Gardner C S et al. Method for solving the Korteweg-deVries equation. Phys. Rev. Lett.1967,19: 1095-1097.
[145]Lax P D. Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math.1968,21:467-490.
[146]Wadati M. The modified Korteweg-de Vries equation. J. Phys. Soc. Jpn.1973,32:1289-1296.
[147]Ablowitz M J et al. Method for solving the sine-Gordon equation. Phys. Rev. Lett.1973,30: 1262-1264.
[148]Wahlquist H D, Estabrook F B. Prolongation structures of nonlinear evolution equations. J. Math. Phys.1975,16:1-7.
[149]Guo H Y et al. On the prolongation structure of Ernst equation. Commun. Theor. Phys.1982,1: 661-664.
[150]Guo H Y, Wu K, Wang S K. Prolongation structure, Backlund transformation and principal ho-mogeneous Hilbert problem in general relativity. Commun. Theor. Phys.1983,2:883-898.
[151]Guo H Y, Wu K, Wang S K. Inverse scattering transform and regular Riemann-Hilbert problem. Commun. Theor. Phys.1983,2:1169-1173.
[152]Wu K, Guo H Y, Wang S K. Prolongation structures of nonlinear systems in higher dimensions. Commun. Theor. Phys.1983,2:1425-1437.
[153]Case K M, Kac M. A discrete version of the inverse scattering problem. J. Math. Phys.1973,14: 594-603.
[154]Flaschka H. On the Toda lattice. II. Inverse-scattering solution. Prog. Theor. Phys.1974,51: 703-716.
[155]Ablowitz M J, Ladik J F. Nonlinear differential-difference equations. J. Math. Phys.1975,16: 598-603.
[156]Ablowitz M J et al. A connection between nonlinear evolution equations and ordinary differential equations of P-type. J. Math. Phys.1980,21:715-721.
[157]Jimbo M et al. Painleve test for the self-dual Yang-Mills equation. Phys. Lett. A.1982,92:59-60.
[158]Fordy A P, Pickering A. Preprint, University of Leeds,1991.
[159]Musette M, Conte R. The two-singular-manifold method:I. Modified Korteweg-de Vries and sine-Gordon equations. J. Phys. A:Math. Gen.1994,27:3895-3913.
[160]Conte R, Musette M, Pickering A. The two-singular manifold method:II. Classical Boussinesq system. J. Phys. A:Math. Gen.1995,28:179-187.
[161]曾云波.与A1相联系的半Toda方程的Lax对与Backlund变换.数学学报,1992,35：454-459.
[1621曾云波.递推算子与Painleve性质.数学年刊,1991,12A：78-88.
[163]Xu G Q, Li Z B. Symbolic computation of the Painleve test for nonlinear partial differential equations using Maple. Comput. Phys. Commun.2004,161:65-75.
[164]Xu G Q, Li Z B. PDEPtest:a package for the Painleve test of nonlinear partial differential equations. Appl. Math. Comput.2005,169:1364-1379.
[165]Lou S Y. KdV extensions with Painleve property. J. Math. Phys.1998,39:2112-2121.
[166]Chen L L, Lou S Y. Painleve analysis of a (2+1)-dimensional Burgers equation. Commun. Theor. Phys.1998,29:313-316.
[167]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.
[168]Whitham G B. Nonlinear dispersive waves. SIAM J. Appl. Math.1966,14:956-958.
[169]Kruskal M D, Zabusky N J. Progress on the Fermi-Pasta-Ulam. nonlinear string problem. Princeton Plasma Physics Lab. Ann. Rep., MATT-Q-21, Princeton, N. J.1963,301-308.
[170]Kruskal M D. Korteweg-de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws. J. Math. Phys.1970,11:952-960.
[171]屠规彰,秦孟兆.非线性演化方程的不变群与守恒律-对称函数方法.中国科学A,1980,24：13-26.
[172]屠规彰.Boussinesq方程的Backhand变换与守恒律.应用数学学报,1981,4：63-68.
[173]曾云波. Darboux transformations of families of AKNS equations with additional terms.数学学报,1995,15：337-345.
[174]屠规彰,秦孟兆. Relationship between symmetries and conservation laws of nonlinear evolution equations.科学通报,1979,29:913-917.
[175]Hu X B. Rational solutions of integrable equations via nonlinear superposition formulae. J. Phys. A:Math. Gen.1997,30:8225-8240.
[176]Hu X B, Zhu Z N. Some new results on the Blaszak-Marciniak lattice:Backlund transformation and nonlinear superposition formula. J. Math. Phys.1998,39:4766-4772.
[177]Hu X B, Wu Y T. A new integrable differential-difference system and its explicit solutions. J. Phys. A:Math. Gen.1999,32:1515-1521.
[178]Hu X B, Tam H W. Some new results on the Blaszak-Marciniak,3-field and 4-field lattices. Rep. Math. Phys.2000,46:99-105.
[179]Tam H W et al. The Hirota-Satsuma coupled KdV equation and a coupled Ito system revisited. J. Phys. Soc. Jpn.2000,69:45-52.
[180]Hu X B et al. Lax pairs and Backlund transformations for a coupled Ramani equation and its related system. Appl. Math. Lett.2000,13:45-48.
[181]Boiti M et al. On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions. Inv. Probl.1986,2:271-280.
[182]Radha R, Lakshmanan M. Singularity analysis and localized coherent structures in (2+1)-dimensional generalized Korteweg-de Vries equations. J. Math. Phys.1994,35:4746-4756.
[183]Lou S Y. Generalized dromion solutions of the (2+1)-dimensional KdV equation. J. Phys. A:Math. Gen.1995,28:7227-7232.
[184]Zhang J F. Abundant dromion-like structures to the (2+1) dimensional KdV equation. Chin. Phys. 2000,9:1-4.
[185]Lou S Y, Ruan H Y. Revisitation of the localized excitations of the (2+1)-dimensional KdV equa-tion. J. Phys. A:Math. Gen.2001,34:305-316.
[186]Rosenau P, Hyman M. Compactons:Solitons with finite wavelength. Phys. Rev. Lett.1993,70: 564-567.
[187]Rosenau P. Nonlinear dispersion and compact structures. Phys. Rev. Lett.1994,73:1737-1741.
[188]Olver P J, Rosenau P. Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E,1996,53:1900-1906.
[189]Dusuel S et al. Predictability in systems with many characteristic times:The case of turbulence. Phys. Rev. E,1998,57:2320-2326.
[190]Ismail M S, Taha T A. A numerical study of compactons. Math. Computer Simulation,1998,47: 519-530.
[191]Wazwaz A M. Exact special solutions with solitary patterns for the nonlinear dispersive K(m,n) equations. Chaos, Solitons & Fractals,2002,13:161-170.
[192]Lou S Y, Wang Q X. Painleve integrability of two sets of nonlinear evolution equations with nonlinear dispersions. Phys. Lett. A,1999,262:344-349.
[193]Liu Q P, Hu X B, Zhang M X. Supersymmetric modified Korteweg-de Vries equation:bilinear approach. Nonlinearity,2005,18:1597-1603.
[194]Liu Q P, Hu X B. Bilinearization of N=1 supersymmetric Korteweg-de Vries equation revisited. J. Phys. A:Math. Gen.2005,38:6371-6378.
[195]Miura R M. Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys.1968,9:1202-1204.
[196]Ablowitz M J et al. A note on Miura's transformation. J. Math. Phys.1979,20:991-1003.
[197]Wang M L. Solitary wave solutions for variant Boussinesq equations. Phys. Lett. A,1995,199: 169-172.
[198]Lou S Y, Ruan H Y, Huang G X. Exact solitary waves in a convecting fluid. J. Phys. A:Math. Gen.1991,24:L587-L590.
[199]Conte R, Musette M. Link between solitary waves and projective Riccati equations. J. Phys. A: Math. Gen.1992,25:5609-5623.
[200]Liu S K, Fu Z T, Liu S D, Zhao Q. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A,2001,289:69-74.
[201]Fu Z T, Liu S K, Liu S D, Zhao Q. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys. Lett. A,2001,290:72-76.
[202]Fan E G. Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a coupled mKdV equation. Phys. Lett. A,2001,282:18-22.
[203]Yan C T. A simple transformation for nonlinear waves. Phys. Lett. A,1996,224:77-84.
[204]Yan Z Y, Zhang H Q. New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics. Phys. Lett. A,1999,252:291-296.
[205]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.
[206]Tian B, Gao Y T. Extension of generalized tanh method and soliton-like solutions for a set of nonlinear evolution equations. Chaos, Solitons & Fractals,1997,8:1651-1653.
[207]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:Math. Gen.1996,29:2895-2903.
[208]Barnett M P et al. Symbolic calculation in chemistry:Selected examples. Int. J. Quantum Chem. 2004,100:80-104.
[209]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.
[210]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.
[211]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.
[212]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.
[213]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.
[214]Zhang H Q, Chen Y F. Proceeding of the 3rd ACM, Lanzhou University Press,1998,147.
[215]Zhang H Q. C-D integrable system and computer aided solver for differential equations. Proceeding of the 5rd ACM, World Scientific Press,2001,221-226.
[216]Ross B ed. The Fractional Calculus and Its Application. Lecture Notes in Mathematics(457). Berlin: Springer-Verlag,1975.
[217]Spanier J, Oldham K B. The Fractional Calculus:Integrations and Differentiations of Arbitrary Order. New York:Academic Press,1974.
[218]叶其孝等译.索伯列夫空间.北京：人民教育出版社,1981.
[219]Lions J L, Magenes E. Non-Homogeneous Boundary Value Problems and Applications. New York: Springer-verlag,1970.
[220]Riewe F. Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E,1996,53:1890-1899.
[221]Goldstein H. Classical Mechanics. Addison-Wesley, Reading, MA,1950.
[222]Vujanovic B D, Jones S E. Variational Methods in Nonconservative Phenomena. San Diego:Aca-demic,1989.
[223]Morse P M, Feshbach H. Methods of Theoretical Physics. New York:McGraw-Hill,1953.
[224]Ford G W, Lewis J T, O'Connell R F. Quantum Langevin equation. Phys. Rev. A,1988,37: 4419-4428.
[225]Riewe F. Mechanics with Fractional derivatives. Phys. Rev. E,1997,55:3581-3592.
[226]Agrawal O P, Gregory J, PericakSpector K. A Bliss-type multiplier rule for constrained variational problems with time delay. J. Math. Ann. Appl.1997,210:702-711.
[227]Laskin N. Fractional quantum mechanics and Levy path integrals. Phys. Lett. A 2000,268:298-305.
[228]Agrawal O P. Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl.2002,272:368-379.
[229]Eqab M R, Tarbeq S A. Potential of arbitrary forces with fractional derivatives. International Journal of Modern Physics A,2004,19:3083-3092.
[230]Muslih S I, Baleanu D. Formulation of Hamiltonian equations for fractional variational problems. Czech. J. Phys.2005,55:633-642.
[231]Muslih S I, Baleanu D, Rabei E. Hamiltonian formulation of classical fields within Riemann-Liouville fractional derivatives. Phys. Scr.2006,73:436-438.
[232]Carpinteri A, Mainardi F. Fractals and Fractional Calculus in Continuum Mechanics. New York: Springer-Verlag,1997.
[233]Nigmatullin R R. The realization, of the generalized transfer equation in a medium with fractal geometry. Phys. Slat. Sol. (B),1986,133:425-430.
[234]Rocco A, West B J. Fractional calculus and the evolution of fractal phenomena. Phys. A,1999, 265(3):535-546.
[235]Ren F Y, Liang J R, Wang X T. The determination of the diffusion kernel on fractals and fractional diffusion equation for transport phenomena in random media. Phys. Lett. A,1999,252(3):141-150.
[236]Kolwankar K M. Fractas and differential equations of fractional order. J. Indian Inst. Sci.,1998, 78(4):275-291.
[237]Nonnenmacher T F, Metzler R. On the Riemann-Liouville fractional calculus and some recent applications. Fractals,1995,3(3):557-566.
[238]Friendvich C. Relaxation function of rheological constitutive equations with fractional derivatives: Thermodynamical constraints. In:Rheological Modelling:Thermodynamic and Statistical Ap-proaches. Casas-vasque J and Jou D, Berlin:Springer,1991,320-330.
[239]Schiessel H, Blumen A. Hierarchical analogues to fractional relaxation equations. J. Phys. A:Math. Gen.1993,26:5057-5069.
[240]Ovidiu V M et al. Universality classes for asymptotic behavior of relaxation processes in systems with dynamical disorder:Dynamical generalizations of stretched exponential. J. Math. Phys.1996, 37(5):2279-2306.
[241]Mainardi F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Soli-tons & Fractals,1996,7:1461-1477.
[242]Schneider W R, Wyss W. Fractional diffusion and wave equations. J. Math. Phys.1989,30:134-144.
[243]Wyss W. Fractional diffusion equation. J. Math. Phys.1986,27(11):2782-2785.
[244]Hilfer R. Fractional diffusion based on Riemann-Liouville fractional derivatives. J. Phys. Chem. B, 2000.104:3914-3917.
[245]Matzler R, Barkai E, Klafter J. Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equatin approach. Phys. Rev. Lett.1999,82(18):3563-3567.
[246]Glocke W G, Nonenmacher T F. A fractional calculus approach to self-similar protein dynamics. Biophysical Journal,1995,68:46-53.
[247]徐明瑜,谭文长.半规管内流体动力学问题.中国科学(A辑),2000,30(3)：272-280.
[248]Kopf M et al. Anomalous diffusion behavior of water in biological tissues. Biophysical Journal, 1996,70(6):2950-2958.
[249]Srinivas G, Yethiraj A, Bagchi B. Nonexponentiality of time dependent survivial probability and the fractional viscosity dependence of the rate in diffusion controlled reactions in a polymer chain. J. Chem. Phys.2001,114(20):9170-9179.
[250]徐明瑜,谭文长.广义二阶流体分数阶反常扩散速度场、应力场及涡旋层的理论分析.中国科学(A辑),2001,31(7)：626-638.
[251]Fannjiang A, Komorowski T. Fractional Brownian motions and enhanced diffusion in a unidirec-tional wave-like turbulence. J. Stat. Phys.2000,100(5):1071-1095.
[252]Barkai E, Silbey R J. Fractional Kramers equation. J. Phys. Chem. B,2000,104:3866-3874.
[253]Nonenmacher T F. Fractioanl integral and differential for a class of Levy-type probalility densities. J. Phys. A:Math. Gen.1990,23:697-699.
[254]Metzler R, Klafter J. The random walk's guide to anomalous difusion:a fractional dynamics approach. Phys. Rep.2000,339(1):1-52.
[255]Uchaikin V V. Montroll-Weiss problem, fractional equations and stable distributions. I. J. Theor. Phys.2000,39(8):2087-2105.
[256]Shlesinger M F. Williams-Watts dielectric relaxation:A fractal time stochastic process. J. Stat. Phys.1984,36(5):639-648.
[257]Glockle W G, Nonnenmacher T F. Fractional integral operators and Fox functions in the theory of viscoelasticity. Macromolecules,1991,24(24):6426-6434.
[258]West B J, Bolognab M, Grigolini P. Physics of Fractal Operators. New York:Springer,2003.
[259]Miller K S, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equa-tions. New York:Wiley,1993.
[260]Samko S G, Kilbas A A, Marichev O I. Fractional Integrals and Derivatives:Theory and Applica-tions. Yverdon:Gordon and Breach,1993.
[261]Podlubny I. Fractional Differential Equations. San Diego:Academic Press,1999.
[262]Zaslavsky G M. Chaos, fractional kinetics, and anomalous transport. Phys. Rep.2002,371:461-580.
[263]Huygenii C. Horoloquim Qscilatorium. France:Parisiis,1673.
[264]Blekman II. Synchronization in Science and Technology. Moscow:Nauka,1981 (in Russian); New York:ASME Press,1988 (in English).
[265]胡岗,萧井华,郑志刚.混沌控制.上海：上海科技教育出版社,2000.
[266]Kolmogorov A N, Doklady Akad, Naul. USSR,1954,98:1.
[267]Lorenz E N. Deterministic Nonperiodic Flow. J. Atmos. Sci.1963,20:130-141.
[268]Li T Y, Yorke J A. Period three implies chaos. Amer. Math. Monthly.1975,82:985-992.
[269]Ruelle D, Takens F. On the nature of turbulence. Commun. Math. Phys.1971,20:167-192.
[270]Feigenbaum M J. Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 1978,19:25-52.
[271]Eckmann J P, Ruelle D. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys.1985, 57:617-656.
[272]Guckenbeimer J, Holmes P. Nonlinear oscillations, dynamical system and Bifurcations of vector fields. Spring-Verlay,1983.
[273]郝柏林,从抛物线谈起-混沌动力学引论.上海：上海科技教育出版社,1995.
[274]Hao B L, Zheng W M. Applied symbolic dynamics and chaos. Singapore:World Scientific,1998.
[275]Hubler A. A daptive control of chaotic systems. Phys. Acta.1989,62:343-346.
[276]Ott E, Grebogi C, Yorke J A. Controlling chaos. Phys. Rev. Lett.1990,64:1196-1199.
[277]Pecora L M, Carroll T L. Synchronization in chaotic systems. Phys. Rev. Lett.1990,64:821-824.
[278]Carroll T L, Pecora L M. Synchronizing chaotic circuits. IEEE Trans. Circuits Syst. Ⅰ:Fundam. Theory Appl.1991,38:453-456.
[279]Abarbanel H D I et al. Generalized synchronization of chaos:The auxiliary system approach. Phys. Rev. E,1996,53:4528-4535.
[280]Kocarev L, Parlitz U. Generalized Synchronization, Predictability, and Equivalence of Unidirec-tionally Coupled Dynamical Systems. Phys. Rev. Lett.1996,76:1816-1819.
[281]Boccaletti S et al. The synchronization of chaotic systems. Phys. Rep.2002,366:1-101.
[282]Rulkov N F et al. Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E,1995,51:980-994.
[283]Li C et al. Lag synchronization of Rossler system and Chua circuit via a scalar signal. Phys. Lett. A,2004,329:301-308.
[284]Sivaprakasam S et al. Experimental Demonstration of Anticipating Synchronization in Chaotic Semiconductor Lasers with Optical Feedback. Phys. Rev. Lett.2001,87:154101.
[285]Ho M et al. Phase and anti-phase synchronization of two chaotic systems by using active control. Phys. Lett. A,2002,296:43-48.
[286]Heagy J F et al. Synchronous chaos in coupled oscillator systems. Phys. Rev. E,1994,50:1874-1885.
[287]Hsieh J Y et al. Controlling hyperchaos of the Rossler system. Int. J. Control,1999,72:882-886.
[288]Tanaka K, Wang H O. Fuzzy control of chaotic systems using LMIs:Regulation, synchronization and chaos model following. IEEE World Congress on Fuzzy Systems Proceedings,1998,1:434-439.
[289]Yang T et al. Impulsive control of Lorenz system. Phys. D,1997,110:18-24.
[290]Wang Y et al. Impulsive control for synchronization of a class of continuous systems. Chaos,2004. 14:199-203.
[291]Boccaletti S et al. The control of chaos:theory and applications. Phys. Rep.2000,329:103-197.
[292]Chen G, Dong X. From Chaos to Order. Singapore:World Scientific,1998.
[293]Yang X S. A framework for synchronization theory. Chaos, Solitons & Fractals,2000,11:1365-1368.
[294]Yan Z Y. A new scheme to generalized (lag, anticipated, and complete) synchronization in chaotic and hyperchaotic systems. Chaos,2005,15:013101.
[295]Yan Z Y. Q-S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic systems— A symbolic-numeric computation approach. Chaos,2005,15:023902.
[296]Kolchin E R. Differential Algebra and Algebra Groups. New York:Academic Press,1973.
[297]Leibniz G W. Mathematicsche Schiften. Hildesheim:Georg Olms Verlagsbuchhandlung,1962.
[298]Caputo M. Linear model of dissipation whose Q is almost frequency independent-Ⅱ, Geophys. J. Roy. Astr. Soc.1967,13:529-539.
[299]Caputo M. Elasticitae Dissipazione. Bologna:Zanichelli,1969.
[300]Ochmann M, Makarov S. Representation of the absorption of nonlinear waves by fractional deriva-tives. J. Amer. Acoust. Soc.1993,94:3392-3399.
[301]Ma W X. Complexiton solutions to the Korteweg-de Vries equation. Phys. Lett. A,2002,301: 35-44.
[302]Lou S Y, Hu H C, Tang X Y. Interactions among periodic waves and solitary waves of the (n+1)-dimensional sine-Gordon field. Phys. Rev. E,2005,71:036604.
[303]Tamizhmani K M et al. Similarity reductions and Painleve property of the coupled higher-dimensional Burgers' equation. Int. J. Nonlinear Meth.1991,26:427-438.
[304]Fan E G et al. Double periodic solutions with Jacobi elliptic functions for two generalized Hirota-Satsuma coupled KdV systems. Phys. Lett. A,2002,292:335-337.
[305]Chandrasekharan K. Elliptic Function. Berlin:Springer,1978.
[306]Patrick D V. Elliptic Function and Elliptic Curves. Cambridge, At the University Press,1973.
[307]Yan Z Y. The new extended Jacobian elliptic function expansion algorithm and its applications in nonlinear mathematical physics equations. Comp. Phys. Commun.2003,153:145-154.
[308]Boiti M et al. Spectral transform for a two spatial dimension extension of the dispersive long wave equation. Inverse Problems,1987,3:371-387.
[309]Paquin G, Winternitz P. Group theoretical analysis of dispersive long wave equations in two space dimensions. Phys. D,1990,46:122-138.
[310]Lou S Y. Similarity solutions of dispersive long wave equations in two space dimensions. Math. Methods Appl. Sci.1995,18:789-802.
[311]Lou S Y. Painleve test for the integrable dispersive long wave equation. Phys. Lett. A,1993,176: 96-100.
[312]Tang X Y, Chen C L, Lou S Y. Localized solutions with chaotic and fractal behaviours in a (2+1)-dimensional dispersive long-wave system. J. Phys. A:Math. Gen.2002,35:L293-L301.
[313]Lou S Y, Hu X B. Infinitely many Lax pairs and symmetry constraints of the KP equation. J. Math. Phys.1997,38:6401-6427.
[314]Fan E G. A direct approach with computerized symbolic computation for finding a series of traveling waves to nonlinear equations. Comput. Phys. Commun.2003,153:17-30.
[315]Kaup D J. A higher-order water-wave equation and the method for solving it. Prog. Theor. Phys. 1975,54:396-408.
[316]Kupershmidt B A. Mathematics of dispersive water waves. Commun. Math. Phys.1985,99:51-73.
[317]Tang X Y, Lou S Y. Variable separation solutions for the (2+1)-dimensional Burgers equations. Chin. Phys. Lett.2003,20(3):335-337.
[318]Serkin V N, Hasegawa A. Novel Soliton Solutions of the Nonlinear Schrodinger Equation Model. Phys. Rev. Lett.2000,85:4502-4505.
[319]Serkin V N, Hasegawa A. SPIE Proceedings Vol.3927 (SPIE-Inernational Society for Optical Engineering, Bellingham, WA,2000); ⅹⅹⅹ.lanl.gov/arⅩⅳ:Physics/0002027.
[320]Ruan H Y, Chen Y X. Dromion interactions of (2+1)-dimensional KdV-type equations. J. Phys. Soc. Jpn.2003,72:491-495.
[321]Hong J L, Liu Y. A novel numerical approach to simulating nonlinear Schrodinger equations with varying coefficients. Appl. Math. Lett.2003,16:759-765.
[322]Holden H et al. Stochastic partial differential equtions:a modeling, white noise functional approach. Birhkauser,1996.
[323]Xie Y C. Exact solutions for stochastic mKdV equations. Chaos, Solitons & Fractals,2004,19: 509-513.
[324]Baldwin D et al. Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations. Comput. Phys. Commun.2004,162:203-217.
[325]Yan Z Y. Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres. Chaos, Solitons & Fractals,2003,16:759-766.
[326]Adomian G. Solving Frontier Problems of Physics:The Decomposition Method. Boston:Kluwer Academic Publishers,1994.
[327]Wazwaz A M. Partial Differential Equations:Methods and Applications. The Netherlands: Balkema Publishers,2002.
328] Yan Z Y. Numerical doubly periodic solution of the KdV equation with the initial condition via the decomposition method. Appl. Math. Comput.2005,168:1065-1078.
[329]潘秀德.组合KdV方程的孤立波解与相似解.应用数学和力学,1988,9：281-285.
[330]Cherruault Y. Convergence of Adomian's method. Kybernetes 1989,18(2):31-38.
[331]Mavoungou T, Cherruault Y. Convergence of Adomian method and applications to non-linear partial differential equation. Kybernetes.1992,21:13-25.
[332]Mavoungou T, Cherruault Y. Numerical study of Fisher's equation by Adomian's method. Math. Comput. Model.1994,19:89-95.
[333]Ngarhasta N et al. New numerical study of Adomian method applied to a diffusion model. Kyber-netes,2002,31:61-75.
[334]Abbaoui K, Cherruault Y. New ideas for proving convergence of decomposition methods. Computer Math. Applic.1995,29:103-108.
[335]Comtet L. Advanced Combinatorics-The art of finite and infinite expansions, Revised and enlarged edition. Dordrecht:D. Reidel Publishing Co.1974.
[336]Liao S J. Beyond Perturbation:Introduction to Homotopy Analysis Method. Boca Raton:Chap-man & Hall/CRC Press,2003.
[337]He J H. Perturbation Methods:Basic and Beyond. Amsterdam:Elsevier,2006.
[338]He J H. Homotopy perturbation method for solving boundary value problems. Phys. Lett. A,2006, 350:87-88.
[339]熊树林.Burgers-KdV方程的一类解析解.科学通报,1989,34：26-29.
[340]Kawahara T. Formation of Saturated Solitons in a Nonlinear Dispersive System with Instability and Dissipation. Phys. Rev. Lett.1983,51:381-383.
[341]Tang K S et al. Generation of n-scroll attractors via sine function. IEEE Trans. Circuits Syst. Ⅰ, 2001,48(11):1369-1372.
[342]Rossler O E. An equation for hyperchaos. Phys. Lett. A.1979,71:155-157.
[343]Tamasevicius A et al. Simple 4D chaotic oscillator. Electron. Lett.1996,32:957-958.
[344]Hitzl D L, Zele F. An exploration of the Henon quadratic map. Phys. D,1985,14:305-326.
[345]Stefanski K. Modelling Chaos and Hyperchaos with 3-D Maps. Chaos Solitons & Fractals,1998, 9:83-93.
[346]Hasler M et al. Simple example of partial synchronization of chaotic systems. Phys. Rev. E,1998, 58:6843-6846.
[347]Yanchuk S et al. Partial synchronization and clustering in a system of diffusively coupled chaotic oscillators. Math. Comput. Simu.2001,54:491-508.
[348]Lu J, Xi Y. Linear generalized synchronization of continuous-time chaotic systems. Chaos, Solitons & Fractals,2003,17:825-931.
[349]Rossler O E. An equation for continuous chaos. Phys. Lett. A,1976,57:397-398.
[350]Lii J et al. Bridge the Gap Between the Lorenz System and the Chen System. Int. J. Bifur. Chaos, 2002,12:2917-2926.