非线性微分方程精确解及振动性
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摘要
本文主要作了以下二方面的研究：首先，借助于符号计算和吴方法研究了非线性发展方程的精确解(定量)，提出和改进了一些求解非线性发展方程的方法，并在符号计算系统Maple上予以机械化实现。其次，利用Riccati变换和积分平均技巧研究了几类非线性微分方程的振动性(定性)。
第一章主要介绍了本文所涉及到的学科(包括孤立子理论、数学机械化、微分方程的振动性等)的起源及发展过程，以及国内外学者在这些方面所做的工作和已经取得的一些成果。最后介绍了本文的主要工作。
第二章介绍了求解非线性发展方程的AC=BD模式，给出了C-D对和C-D可积系统的基本理论以及构造C-D对的方法，并通过实例说明了这一模式的使用方式。
第三章基于非线性发展方程求解代数化、算法化、机械化的指导思想，以吴方法和符号计算为工具，首先提出一种新的广义tanh函数方法，并将其应用于(2+1)-维Kadomtsev-Petviashvili方程的精确解构造。其次，利用Q-变形双曲函数，提出广义的Q-变形双曲函数方法，以浅水长波近似方程为例验证了该算法的有效性。最后，通过给出辅助方程更多形式的Jacobi椭圆函数解，进一步改进了求解非线性发展方程的Jacobi椭圆函数展开法，并用其求解了广义Ito方程组、Zakharov-Kuznetsov方程、耦合Drinfel’d-Sokolov-Wilson方程和(2+1)-维Davey-Stewartson方程，得到了丰富的有理形式双周期解。在退化情况下，又可得到有理孤波解。
第四章利用推广的Riccati变换及积分平均技巧，首先，研究了一类二阶非线性微分方程在一般假设条件下的振动性，所得结果推广和改进了前人的相关结果。其次，研究了一类广泛的二阶非线性强迫微分方程的振动性，得到若干区间振动准则。这些准则仅依赖于方程在[t_0，∞)的一个子区间序列上的信息，且避免了前人结果中可能出现的问题，所得结果更有效。最后，借助于核函数Φ(t，s，l)和算子A~ρ[(?)，l，t]研究了一类二阶阻尼泛函微分方程的振动性，得到若干新的振动准则及区间振动准则。这些准则有别于以往的结果，更加简单且易于应用。
This dissertation has mainly done the following two aspects research: First, with the aid of symbolic computation and Wu method, the exact solutions of some nonlinear differential equations have been studied. Some methods for constructing the exact solutions of nonlinear evolution equations are presented and improved. These presented methods are realized on symbolic computation system Maple. Second, by use of Riccati transformation and integral average technique, oscillation of several nonlinear differential equations has been studied.
Chapter 1 mainly introduces the origin and development of several subjects related to this dissertation (including soliton theory, mechanization, oscillation et. al.), as well as the work and achievements of the domestic and foreign scholars which have been obtained in these aspects. Our main works are presented at last.
Chapter 2 is devoted to AC=BD model and its applications in solving of nonlinear equations. Basic notations, basic theory of C-D pair and C-D integrable systems and method of constructing C-D pair are given out. Some illlustrative examples are presented to show how to use the model.
Based on the idea of solving nonlinear evolution equations, algebraic method, algorithm reality and mechanization, in Chapter 3, we firstly present the new generalized tanh function method, and apply it to construct the exact solutions of the (2+1)-dimensional Kadomtsev-Petviashvili equation. Second, the generalized Q-deformed hyperbolic functions method is developed based on the Q-deformed hyperbolic functions. The efficiency of the method can be demonstrated on the Shallow long wave approximate equations. At last, we extend Jacobi elliptic function expansion method, with which more Jacobi elliptic function solutions of auxiliary equations are obtained. Then by use of the method, more new doubly -periodic solutions of a class of nonlinear differential equations such as the generalized Ito system, Zakharov-Kuznetsov equation, the coupled Drinfel'd- Sokolov -Wilson equations and the (2+1)-dimensional Davey-Stewartson equation are obtained. These solutions are degenerated to solition solutions under degenerated conditions.
In chapter 4, using general Riccati transformation and integral average technique: we first investigate the oscillation of a class of nonlinear differential equations under quite general assumptions. The results that we obtained generalize and improve some known oscillation criteria. Second, interval oscillation of a class of nonlinear differential equations with forcing term is concerned. These obtained reults are based on the information only on a sequence of subintervals of [t_0,∞) rather than on the whole half-line and avoid the problem of old ones. So these results are more powerful. At last, some new oscillation criteria are established for the nonlinear damped functional differentional equations: which are different from known ones in the sense that they are based on a class of functionsΦ(t, s, r) and operator A~ρ(·; l, t) defined. Our results are sharper than some previous results and easy to apply.
第一章主要介绍了本文所涉及到的学科(包括孤立子理论、数学机械化、微分方程的振动性等)的起源及发展过程，以及国内外学者在这些方面所做的工作和已经取得的一些成果。最后介绍了本文的主要工作。
第二章介绍了求解非线性发展方程的AC=BD模式，给出了C-D对和C-D可积系统的基本理论以及构造C-D对的方法，并通过实例说明了这一模式的使用方式。
第三章基于非线性发展方程求解代数化、算法化、机械化的指导思想，以吴方法和符号计算为工具，首先提出一种新的广义tanh函数方法，并将其应用于(2+1)-维Kadomtsev-Petviashvili方程的精确解构造。其次，利用Q-变形双曲函数，提出广义的Q-变形双曲函数方法，以浅水长波近似方程为例验证了该算法的有效性。最后，通过给出辅助方程更多形式的Jacobi椭圆函数解，进一步改进了求解非线性发展方程的Jacobi椭圆函数展开法，并用其求解了广义Ito方程组、Zakharov-Kuznetsov方程、耦合Drinfel’d-Sokolov-Wilson方程和(2+1)-维Davey-Stewartson方程，得到了丰富的有理形式双周期解。在退化情况下，又可得到有理孤波解。
第四章利用推广的Riccati变换及积分平均技巧，首先，研究了一类二阶非线性微分方程在一般假设条件下的振动性，所得结果推广和改进了前人的相关结果。其次，研究了一类广泛的二阶非线性强迫微分方程的振动性，得到若干区间振动准则。这些准则仅依赖于方程在[t_0，∞)的一个子区间序列上的信息，且避免了前人结果中可能出现的问题，所得结果更有效。最后，借助于核函数Φ(t，s，l)和算子A~ρ[(?)，l，t]研究了一类二阶阻尼泛函微分方程的振动性，得到若干新的振动准则及区间振动准则。这些准则有别于以往的结果，更加简单且易于应用。
This dissertation has mainly done the following two aspects research: First, with the aid of symbolic computation and Wu method, the exact solutions of some nonlinear differential equations have been studied. Some methods for constructing the exact solutions of nonlinear evolution equations are presented and improved. These presented methods are realized on symbolic computation system Maple. Second, by use of Riccati transformation and integral average technique, oscillation of several nonlinear differential equations has been studied.
Chapter 1 mainly introduces the origin and development of several subjects related to this dissertation (including soliton theory, mechanization, oscillation et. al.), as well as the work and achievements of the domestic and foreign scholars which have been obtained in these aspects. Our main works are presented at last.
Chapter 2 is devoted to AC=BD model and its applications in solving of nonlinear equations. Basic notations, basic theory of C-D pair and C-D integrable systems and method of constructing C-D pair are given out. Some illlustrative examples are presented to show how to use the model.
Based on the idea of solving nonlinear evolution equations, algebraic method, algorithm reality and mechanization, in Chapter 3, we firstly present the new generalized tanh function method, and apply it to construct the exact solutions of the (2+1)-dimensional Kadomtsev-Petviashvili equation. Second, the generalized Q-deformed hyperbolic functions method is developed based on the Q-deformed hyperbolic functions. The efficiency of the method can be demonstrated on the Shallow long wave approximate equations. At last, we extend Jacobi elliptic function expansion method, with which more Jacobi elliptic function solutions of auxiliary equations are obtained. Then by use of the method, more new doubly -periodic solutions of a class of nonlinear differential equations such as the generalized Ito system, Zakharov-Kuznetsov equation, the coupled Drinfel'd- Sokolov -Wilson equations and the (2+1)-dimensional Davey-Stewartson equation are obtained. These solutions are degenerated to solition solutions under degenerated conditions.
In chapter 4, using general Riccati transformation and integral average technique: we first investigate the oscillation of a class of nonlinear differential equations under quite general assumptions. The results that we obtained generalize and improve some known oscillation criteria. Second, interval oscillation of a class of nonlinear differential equations with forcing term is concerned. These obtained reults are based on the information only on a sequence of subintervals of [t_0,∞) rather than on the whole half-line and avoid the problem of old ones. So these results are more powerful. At last, some new oscillation criteria are established for the nonlinear damped functional differentional equations: which are different from known ones in the sense that they are based on a class of functionsΦ(t, s, r) and operator A~ρ(·; l, t) defined. Our results are sharper than some previous results and easy to apply.
引文
[1] Russell J S. Report on waves. Fourteen meeting of the British association for the advancement of science. John Murray. London. 1844, 311-390.
[2] Airy G.B. Tides and waves. Encycl. Metrop. London Art. 1845,241-396.
[3] Boussinesq M J. Theorie des ondes et de remous quise propageant le long d'un canal recangnlaier horizonal, et communiquant au liquide contene dans ce cannal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. Set. 1876, 2:55-108.
[4] Rayleigh L. On waves. Philos. Mag. Ser. 1876, 5: 257-279.
[5] Fermi A, Ulam J P. Studies of Nonlinear Problem. I.Los Alamous Rep. LA. 1940, 1955.
[6] 谷超豪等，孤立子理论与应用．杭州：浙江科技出版社， 1990．
[7] Petting J K, Skyrme T H R. A Model Uniform Field Equation. Nuci. Phys. 1962, 31: 550-555.
[8] 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．
[9] 郭柏灵，庞小峰．孤立子．北京：科学出版社， 1987．
[10] 李翊神．孤子与可积系统．上海：上海科技出版社，1999．
[11] Ablowitz M J, Clarkson P A. Solitons, nonlinear evolution equations and inverse scattering. Cam- brideg: Cambridge University. Press, 1991.
[12] Newell A C. Soliton in mathematics and physics. SIAM: Philadelphia, 1985.
[13] Faddeev L D, Takhtajan L A. Hamiltonian method in the theory of solitons. Berlin:Springer-Verlag, 1987.
[14] Matveev V B, Salle M A. Darboux transformation and Solutions. Berlin: Springer-Verlag, 1991.
[15] Baklund A V. Lund Universitests Arsskrift. 1885, 10.
[16] Wahlquist H D, Estabrook F B. Backlund transformations for solitons of the Korteweg-de Vriu equation. Phys. Rev Lett. 1973, 31:1386-1390.
[17] Weiss J, Tabor M. Carnvale.G. The Painlevé property for partial differential equations. J. Math. Phys. 1983, 24(3): 522-526.
[18] Weiss J. The Painlevé property for partial differential equations. Ⅱ. Backlund transformation, Lax pairs, and the Schwarzian derivative. J. Math. Phys. 1983, 24(6): 1405-1413.
[19] Wadati M, et al. Simple derivation of Backlund transformation from Riccati form of inverse method, Prog. Theor.Phys. 1975, 53: 418.
[20] Gu C H. A unified explicit form of Backlund transformations for generalized hierarchies of KdV equations. Lett. Math. Phys. 1986, 11(4): 325-335.
[21] Gu C H, Zhou Z X. On the Darboux matrices of Backlund transformations for AKNS systems. Let. Math. Phys. 1987, 13(3): 179-187.
[22] 胡星标，李勇．DJKM方程的Backlund变换及非线性叠加公式．数学物理学报．1991，11：164-172．
[23] Hu X B, Clarkson P A, Backlund transformations and nonlinear superposition formulae of a differential-difference KdV equation. J. Phys. A. 1998, 31:1405-1414.
[24] Hu X B, Zhu Z N, A Backlund transformation and nonlinear superposition formulae for the Belov- Chaltikian lattice. J. Phys. A. 1998, 31:4755-4716.
[25] Zeng Y B, et al. Canonical explicit Backlund transformations with spectrality for constrained flows of soliton hierarchies. Physica A. 2002, 303(3-4): 321-338.
[26] Li Z B, Wang M L. Travelling wave solutions to the two-dimensional KdV-Burgers equation. J. Phy. A. 1993, 26: 602-6031.
[27] Wang M L, Li Z B. Proc. of the 1994 Beijing Symposium on nonlinear evolution equations and infinite dimensional dynamics symtems, Zhongshan University Press, 1995, 181-185.
[28] 李志斌，张善卿．非线性波方程准确孤立子解的符号计算．数学物理学报．1997，17(1)：81-89．
[29] Fan E G, Zhang H Q. New exact solutions to a system of coupled KdV equations. Phys. Lett. A. 1998, 245: 389-392.
[30] 范恩贵．孤立子和可积系统： (博士学位论文)．大连：大连理工大学， 1998．
[31] 闫振亚．非线性波与可积系统： (博士学位论文)．大连：大连理工大学，2002．
[32] 陈勇．孤立子理论中的若干问题的研究及机械化实现：(博士学位论文)．大连：大连理工大学，2003．
[33] 李彪．孤立子理论中若干精确解方法的研究及应用：(博士学位论文)．大连：大连理工大学，2005．
[34] Clifford S G, Greene J M, Kruskal M D, Miura R M. Method for Solving the Korteweg-deVries Equation. Phys. Rev. Lett. 1967, 19:1095-1097.
[35] Lax P D. Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math. 1968, 21:467-490.
[36] Zabusky N J, Kruskal M D. Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. rev. Lett. 1965, 15:240-243.
[37] Wadati M. The modified Korteweg-de Vries equation, J. Phys. Soc. Japan. 1973, 34:1289-1296.
[38] Ablowitz M J, Kaup Z J, Newell A C. et.al. Nonlinear-evolution equations of physical significance. Phys. Rev. Lett. 1973, 31: 125-127.
[39] Wahlquist H D, Estabrook F B. Prolongation structures of nonlinear evolution equations. J. Math. Phys. 1975, 16: 1-7.
[40] Wu K, Guo H Y, Wang S K. Prolongation structures of nonlinear evolution systems in higher dimensions. Commun.Theoret.Phys. 1983, 2:1425-1437.
[41] Guo H Y, Wu K, Wang S K. Prolongation structure, Backlund transformation snd principal homo- geneous Hilbert problem in general relativity, Commun. Theoret. Phys. 1983, 2:883-898.
[42] 屠规彰，非线性方程的逆散射方法，应用数学与计算数学．1979，24(20)：913-917．
[43] 李翊神．一类发展方程和谱的变形，中国科学，A．1982，5：386-390．
[44] 曹策问．孤立子与反散射，郑州大学出版社， 1983，35-89．
[45] 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.
[46] Cunningham E. The principle of relativity in electrodynamics and an extension thereof. Proc. London Math. Soc. 1909, 8: 77-97.
[47] Nother E. Invariante varlationsprobleme. Nachr Koning. Cesell. Wissen. Gottingen, Meth. Phys. kl, 1918, 235-257.
[48] Ovsiannikov L V. Group properties of differential .equations. Novosibirsl, 1962.
[49] Sedov L I. Similarity and dimensional method, 4th ed. New York: Academic Press, 1959.
[50] Venikov V A. Theory of Similarity and Simulation. London: Mac Donald Technical and Scientific, 1969.
[51] Bluman G W, Cole J D. The general similarty solution of the heat equation, J.Math.Mech. 1969, 18:1025-1042.
[52] Levi D, Winternitz P. Group theoretical analysis of a rotating shallow liquid in a rigid container. J. Phys. A, 1989, 22:4743-4767.
[53] Vorob'er E M. Reduction and quotient, equations for differential equations with symmetries. Acta Appl. Math. A. 1991, 23: 1-24.
[54] Olver P J. Evolution equations possessing infinitely many symmetries. J. Math. Phys. 1977, 18:1212- 1215.
[55] Olver P J. On the Hamiltonian structure of evolution equations. Math. Proc. Camb. Phill Soc. 1980, 88:71-88.
[56] Fuchsseiner B, Fokas A S. Symplectic structures, their Backlund transformations and hereditary symmetries. Physica D. 1981, 4:47-66.
[57] Chen H H, et.al. In Advance in_nonlinear wave, ed L.Debnath. 1982, 233.
[58] 李翊神，朱国城．一个谱可变演化方程的对称．科学通报．1986，19：1449-1453；可积方程新的对称，李代数及可变演化方程．中国科学A．1987，30：1243-1250；”C可积”非线性方程的代数性质Burgers方程和Calogero方程．中国科学A．1990，33：513-520．
[59] 田踌，Burgers方程的新的强对称，对称和李代数．中国科学A．1987，31：141—151．
[60] Clarkson P A, Kruskal M D. New similarity reductions of the Boussinesq equation. J. Math. Phys., 1989, 30(10): 2210- 2213.
[61] Clark.son P A. New similarity reductions of the modified Bonssinesq equation. J. Phys.A. 1989, 22(13): 2355- 2367.
[62] Lou S Y. A note on the new similarity reductions of the Bonssinesq equation. Phys. Lett. A. 1990, 151:133- 135.
[63] Lou S Y. Twelve sets of symmetries of the Caudrey-Dodd-Gibbon- Sawada-Sawada-Kotera equation; Generalized Boussinesq equation and KdV equation-Painlevé properties, Backlund transformations and Lax pairs. Sci.China A. 1991, 34(9):1098-1108.
[64] Zhdanov R Z, Fushchych W I. Conditional symmetry and new classical solutions of the Yang-Mills equations. J. Phys. A: Math. Gen. 1995,28:6253-6264.
[65] Ablowitz M J. et al., A connection between nonlinear evolution equations and ordinary differential equations of P-type. J. Phys. 1980, 21(4): 715-721.
[66] Weiss J. The Painlevé property for partial differential equations. Ⅱ. Backlund transformation, Lax pairs, and the Schwarzian derivative. J. Math. Phys. 1983, 24: 1405-1413.
[67] Jimbo M. et al., Painlevé test for the self-dual Yang-Mills equation. Phys. Lett. A. 1982, 92(2):59-60.
[68] Fordy A. Picketing A, Analysing negative resonances in the Painlevé test. Phys. Lett. A. 1991, 160 (4):347-354.
[69] 曾云波．与A_l相联系的半Toda方程的Lax对与Backlund变换．数学学报．1992，35(4)：454—459．
[70] 曾云波．递推算子与Painlevé性质．数学年刊，1991，12A(1)：78—88．
[71] Hirota R. Exact solution of the KdV equation for multiple collisions of solutions, Phys. Rev. Lett. 1971, 27: 1192-1194.
[72] Hu X B, Tam H W. New integral differential -difference systems: Lax pairs, bilinear forms and solition solutions. 2000, 46: 99-105.
[73] Hu X B, Rational solutions of Integral equations via nonlinear superposition formulae. J. Phys. A. 1997, 30:8225-8240.
[74] Hu X B, et. al., Lax pairs and B acklund transformations for a coupled Ramani equation and its related system. Apll. Math. Lett. 2000, 13:45-48.
[75] Boiti M, et. al. On the spectral transform of a Korterweg-de Vries equation in two spatial dimensions. Inv. Probl. 1986, 2:271-280.
[76] Lou S Y, Generalized dromion solutions of the (2+1)-dimensiona] KdV equation. J. Phys. A. 1995, 28: 7227-7232.
[77] Zhang J F. Abundant dromion-like structures to the (2+1)-dimensional KdV equation. Chin. Phys. 2000, 9:1-4.
[78] Lou S Y, Ruan H Y.Revisitation of the localized excitations of the (2+1)-dimensional KdVequation. J. Phys. A, 2001, 34:305-316.
[79] Rosenau P, Hyman M. Compactons: Solitions with finite wavelength, Phys. Rev. Lett. 1993, 70:564- 567.
[80] Lou S Y, Wang Q X. Painleve integrability fo two sets of nonlinear evolution equations with nonlinear dispersions. Phys. Lett. A, 1999, 262:344-349.
[81] Wazwaz A M. Exact special solutions with solitary patterns for the nonlinear dispersive K(m,n) equations. Chaos, Solitons and Fractals. 2002, 13:161-170;
[82] Olver P J, Rosenau P. TRi-Hamiltonian duality between solitions and solitary-wave solutions having compact support. Phys. Rev. E. 1996, 53:1990-1906.
[83] Grammaticos B, Ramani A, Hietarinta J. Multilinear operators: the natural extension of Hirota's bilinear formalism. Phys. Lett. A. 1994, 190:65-70.
[84] Liu Q P, Hu X B, Zhang M X. Supersymmetric modified Korteweg-de Vries equation: bilinear approach. Nonlinearity. 2005, 18(4):1597-1603.
[85] Liu Q P, Hu X B. Bilinearization of N = 1 supersymmetric Korteweg-de Vries equation revisited. J. Phys. A. 2005, 38(28):6371-6378.
[86] Wang M L. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. Lett. A. 1996, 216(1-5): 67-75.
[87] Wang M L, Wang Y M, Zhou Y B. An auto-Backlund transformation and exact solutions to a generalized KdV equation with variable coefficients and their applications. Phys. Lett: A. 2002, 303(1): 45-51.
[88] Gao Y T, Tian B. New families of exact solutions to the integrable dispersive long wave equaitons in 2+1-dimensional spaces. J. Phys. A, 1996, 29(11): 2895-2903.
[89] Fan E G, Zhang H Q. A note on the homogeneous balance method. Phys. Lett. A. 1998, 246: 403-406.
[90] Fan E G, Zhang H Q. New exact solutions to a system of coupled KdV equations. Phys. Lett. A. 1998, 245: 389-392.
[91] Tian B; Gao Y T. Variable-coefficientbalancing-actmethodandvariable-coe?cientKdVequation from- fluiddynamicsandplasmaphysics. Eur. Phys. J. B. 2001,22:351-360.
[92] 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.
[93] 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.
[94] 吕卓生．大连大连大连理工大学博士学位论文，2004．
[95] Barnett M P. Symbolic calculation inchemistry: Selected examples. Int. J. uantumChem. 2004, 100: 80-104.
[96] Lou S Y, Ruan H Y, Huang G X. Exact solitary wave sina convecting fluid. J. Phys. A, 1991, 24: 587-590.
[97] Conte R, Musette M. Link between solitary wave sand projective Riccatiequations. J. Phys. A. Math. Gen. 1992,25:5609-5623.
[98] 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.
[99] 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.
[100] Fan E G. Soliton solutions for ageneralized Hirota-Satsuma coupled KdV equation and acoupled mKdV equation. Phys. Lett. A. 2001, 282: 18-22.
[101] Yan C T. A simple transformation for nonlinear waves. Phys. Lett. A, 1996, 224: 77-84.
[102] Yah Z Y, Zhang H Q. New explicit and exact travelling wave solutions for asystem of variant Boussinesq equations in mathematical physics. Phys. Lett. A. 1999, 252: 291-296.
[103] Rosenau P, Hyman M. Compactons: Solitons with finite wavelength. Phys. Rev. Lett. 1993,70: 564-567.
[104] 张鸿庆，弹性力学方程组一般解的统一理论，大连工学院学报， 1978，3：23-47．
[105] 张鸿庆，杨光．变系数偏微分方程组一般解的构造．应用数学和力学， 1991，12：135—139．
[106] 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(4)：315—322．
[107] 张鸿庆，Maxwell方程的多余阶次与恰当解．应用数学与力学．1981，2(3)：321-331．
[108] Lou S Y, Ni G J. The relations among a special type of solutions in some (D+1) dimensional nonlinear equations. J. Math. Phys. 1989, 30: 1614-1620.
[109] Zakharov V E, Shabat A B. A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. Funct. Anal. Appl. 1974, 8: 226-235.
[110] Dolye P W. Separation of variables for scalar evolution equations in one space dimension. J. Phys. A. 1996, 29: 7581-7596.
[111] 李翊神等．非线性科学选讲．中国科学技术大学出版社．1994．
[112] 吴文俊．吴文俊论数学机械化．山东：山东教育出版社．1996．
[113] 高小山，数学机械化进展综述数学进展，2001 30(5)385-401．
[114] 吴文俊．几何定理机器证明的基本原理．北京：科学出版社， 1984．
[115] 陈玉福，高小山．微分多项式系统的对合特征集．中国科学A．33(2003)97-113．
[116] Wu W J. On the decision problem and the mechanization of theorem-proving in elementary geom- etry. Scintia Sinica, 1978,21:159-172.
[117] Wu W T. On zeros of algebraic equations: an application of Ritt principle. Kexue Tongbao, 1986, 31: 1-5.
[118] Ritt J F. Differential Algebra. New York: American mathematical Society, 1950.
[119] Wu W.J. On the foundation of algebraic differential geometry. Systems Sci. Math. Sci. 1989,2(4): 289-312.
[120] Chou S C, Gao X S, Zhang J Z. Machine proofs in geometry. Automated production of readable proofs for geometry theorems, Singapore, 1994.
[121] Gao X S, Zhang J Z. and Chou S C. Geometry Expert(in Chinese), Nine Chaoters Pub. Taiwan, 1998.
[122] Wang D M, Gao X S. Geometry theorems proved mechanically using Wu's method. MM-Research Preprints, 1987, 2.
[123] Wang D M, Hu S. Mechanical proving system for constructible theorems in elementry gepmetry. J. Sys. Sci. Math. Scis, 1995, 7.
[124] Lin D D, Liu Z J. Some results in theorem proving in finite geometry. Proc of IWMMM. Inter Academic Pub. 1992.
[125] Wu J Z, Liu Z J. Proceedings of 1st Asian Symposium on Computer Mathematics, Beijing of China, 1995.
[126] Wu J Z, Liu Z J. Science in China (Series E), 1996, 39(6): 608-619.
[127] 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.
[128] Kaput D, Wan H K. Refutational proofs of geometry theorems via characteristic sets. Proc of ISSAC-90, Tokyo, Japan, 1990, 277-284.
[129] Ko H. Geometry theorem proving by decomposition of quasi-algebraic sets: an application of the Ritt-Wu principle. Artif. Intell. 1988, 37: 95-122.
[130] Li H B, Cheng M T. Proving theorems in elementary geometry with Cliffird algebraic method. Chinese Math Progress, 1997, 26(4):357-371.
[131] Li H B, Cheng M T. Clifford algebraic reduction method for mechanical theorem proving in differ- ential geometry. J. Automated Reasoning, 1998, 21(1):1-21.
[132] 石赫，机械化数学引论，湖南教育出版社， 1998．
[133] Sun X D, Wang K, Wu K. Wu algorithm and solutions of Yang-Baxter equation for eight-vertex model. Comm. Theoret. Phys. 1995, 23(1): 125-128.
[134] 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.
[135] Li Z B, Shi H. Exact solutions for Belousov-Zhabotinskiǐreaction-diffusion system. Appl. Math. J. Chinese Univ. Set. 1996, 11B(1): 1-6.
[136] Li Z B. Proc. Of Asian Symposition on computer Mathematics, Lanzhou University, 1998: 153.
[137] Li Z B, Liu Y P. RATH: A Maple package for finding travelling solitary wave solutions to nonlinear evolutions. Comput. Phys. Comm. 2002, 148(2): 256-266.
[138] Xie F D, Chen Y. An algorithmic method in Painlevé analysis of PDE. Compu. Phys. Commun. 2003, 154: 197-204.
[139] 郑学东．一类非线性偏微分方程组的机械化求解与P-性质的机械化检验： (硕士学位论文)．大连：大连理工大学，2003．
[140] Cohen A M, Davenport J H, Heck J P. An overview of computer algebra. Computer Algebra for Industry. 1994,16:1-25.
[141] 斯特列可夫CII著，何文蛟译．振动理论引论(上册)北京：高等教育出版社， 1956．
[142] Sturm C. Sur les equations differentielles linearies on second order. J. Math. Pure, 1886, 1106-186.
[143] Cesari L. Asymptotic behavior and stability problem in ordinary differential equations. Berlin, Springer 1963.
[144] willett D. Classification of second orders linear differential equations with respect to oscillation, Advance. Math, 1969, 3: 594-623.
[145] Barrett J H. Oscillation theory of ordinary linear differential equations. Advances in Math. 1969, 3: 415-509.
[146] Wong J S W. On second order nonlinear oscillation, Funkcial. Ekvac, 1968, 11(3): 207-234.
[147] Kartsatos A G. Recent results on oscillation of solutions of forced and perturbed nonlinear dif- ferential equations of even order, J. R. Graef. Ed. stability of Dynamical Systems Theory and Applications. 1977, Vol.28, Charpter 3 Marcel. Dekker, N.Y.
[148] Swanson C A. Comparison and oscillation theory of linear differential Equations. New York Acad. Press, 1968.
[149] 张炳根．二阶常微分方程解的有介性．数学学报．1964，1：128—136．
[150] 燕居让．常微分方程振动理论．太原：山西教育出版社．1992．
[151] Ladde G S, Lakshimikantham V, Zhang B G. Oscillation Theory of Differential Equations with Deviating Arguments. New York: Marcel Dekker, 1987.
[152] Bainov D D, Mishev D P. Oscillation Theory for Neutral Differential Equations with Delay. New York: Adam Hilger, 1991.
[153] Gyori I, Ladas G. Oscillation Theory of Delay Differential Equations. Oxford: Clarendon Press, 1991.
[154] Agarwal R P, Grace S R, O'Regan D. Oscillation Theory for Difference and Differential Equa- tions. Dordrecht: Kluwer Academic Pubhshers, 2000.
[155] Wu J h. Theory and Applications of Partial Functional Differential Equations. New York: Springer- Verlag, 1996.
[156] Allegretto W, Erbe L. Oscillation criteria for matrix differential inequalities. Canad. Math. Bull. 1973, 16: 5-10.
[157] Atkinson F V, Kaper H G, Kwong M K. An oscillation for linear second order differential systems. Proc. Amer. Math. Sec. 1985, 94: 91-96.
[158] Butler G J, Erbe L H. Oscillation results for second order differential systems. J. Math. AnaL. 1986, 17:19-29
[159] Butler G J, Erbe L H. Oscillation results for self-adjoint differential systems. J. Math. Anal. Appl. 1986, 115: 470-481.
[160] Butler G J, Erbe L H, Mingarelli A B. Riccati techniques and variational principles in oscil- lation theory for linear systems. Trans. Amer. Math. Soc. 1987, 303: 263-282.
[161] 郑召文，孟凡伟，俞元洪．二阶线性矩阵微分系统的振动性．数学学报．1998，41(6)：1231—1238．
[162] 庄容坤．二阶矩阵微分系统的区间准则．数学学报．2001，44(6)：1037-1044．
[163] Zhang H Q. Proceedings of 5th Asian Symposium on Computer Mathematics, World Scientific, Singapore, 2001.
[164] Yan Z Y, Zhang H Q. Explict and exact solutions for the generalized rection Duffing equation, Commun. in Nonlinear and Numerical Simulation. 1999, 4:146.
[165] Yah Z Y, A new sine-Gordon equation expansion algorithm to investigate some special nonlinear differential equations. Chaos, Solitons and Fractals. 2005, 23:767-775.
[166] Hirota R. The direct method in soliton theory, translated from Japanese and .edited by Atsushi Nagai Jon Nimmo and sClaire Gilson. Cambridge University press, 2002.
[167] Parkes E J, Duffy, B R. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput. Phys. Commun, 1996, 98: 288-300.
[168] Hereman W. Exact solitary wave solutions of coupled nonlinear evolution equations using MAC- SYMA. Comput. Phys. Commun. 1991, 65: 143-150.
[169] Yah Z Y, Zhang H Q. New explicit and exact travelling wave solutions for system of variant Boussinessq equations in mathematical physics. Phys. Lett. A. 996,224:77.
[170] Ma W X. et. al. Explicit and exact solutions to a Komog0rov-Petrovskii-Piskunov equation. Int. J. Non-linear Mech. 1996, 31: 329.
[171] Chert H T, Zhang H Q. New mutliple solition-like solutions to the generalized (2+1)-dimensional KP equation. Applied Mathmatics and Computation. 2004, 157: 765-773.
[172] Fu Z T, Liu S K, Liu S D. Lame Function and Its Application to Some Nonlinear Evolution Equations. Commun. Theor. Phys. (Beijing, China) 2003, 40: 53-56.
[173] Arai A. Exactly solvable supersymmetric quantum mechanics. J. Math. Anal. Appl. 1991, 158: 63-79.
[174] Arai A. Exact solutions of multi-component nonlinear Schrodinger and Klein- Gordon equations in two-dimensional space-time. J.Phys. A. 2001, 34: 4281-4288.
[175] Liu X Q, Jiang S, The SeC_q - tanh_q method and its applications. Phys. Lett. A. 2002, 298: 253-258.
[176] Wang M L, Zhou Y B. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Physics. lett. A. 1996, 216:67-75.
[177] Zhang J F, Multiple solition solutions for the approrimate equations of long wave water. Acta Physics Sinica, 1998, 47: 1416(in Chinese).
[178] Yah Z Y. Zhang H Q. Explicit exact solutions for nonlinear approrimate equations with long waves in shallow water. Acta Physica sinica. 1999, 48: 1962(in Chinese).
[179] Liu S K. et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave .equations. Phys. Lett. A. 2001, 289: 69-74.
[180] Liu S K, Fu Z T. et. al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 2001, 289(1-2):69-74.
[181] Fan E G, Hon B Y. Double periodic solutions with Jacobi elliptic functions for two generalized Hirota-Satsuma coupled KdV systems. Phys. Lett. A. 2002, 292: 335-337.
[182] Fan E G, Zhang J. Applications of the Jacobi elliptic function method to special-type nonlinear equations. Phys. Lett. A. 2002, 305: 383-392.
[183] Yah Z Y. Abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional integrable Davey-Stewartson-type equation via a new method. Chaos Solitons Practals. 2003, 18(2):299-309.
[184] Yan Z Y. Extended Jacobian elliptic function algorithm with symbolic computation to construct new doubly-periodic solutions of nonlinear differential equations. Comput. Phys. Comm. 2002, 148(1):30- 42.
[185] Wang M L, Zhou Y B. The periodic wave solutions for the Klein-Gordon- Schrǒinger equations. Physics Letter A. 2003, 318: 84-92.
[186] Zhou Y B, Wang M L, Wang Y M. Periodic wave solutions to a coupled KdV equations with variable coefficients. Physics Letter A. 2003, 308: 31-36.
[187] Chen Y, Wang Q, Li B. Jacobi Elliptic Function Rational Expansion Method with Symbolic Compu- tation to Construct New Doubly-periodic Solutions of Nonlinear Evolution Equations. Z. Naturforsch 2004, 59a: 529-536.
[188] Wang Q, Chen Y, Zhang H Q. A new Jacobi elliptic function rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation. Chaos Solitons Fractals 2005,23 (2):477-483.
[189] Chen H T, Zhang H Q, New double periodic and multiple soliton solutions of the generalized (2+1)-dimentional Boussineq equations, Chaos, Solitions and Fractals. 2004, 20(4): 765-776.
[190] Chen H T, Zhang H Q, New double periodic solutions for a kind of nonlinear wave equation, Progress of Theoretical Physics. 2003, 109(5): 709-716.
[191] Chandrasekharan K. Elliptic Functions, Berlin, Springer, 1978.
[192] Patrich D V. Elliptic Function and Elliptic Curves, Cambridge, At the University Press, 1973.
[193] Wang Z X, Xia X J. Special Functions, Singapone:World Scientific, 1989.
[194] Ito J M. extension of nonlinear evolution equation of the KdV(mKdv) type to higher orders. J. Phys. Soc. Jpn. 1980, 49: 771.
[195] Tam etal. Two Integral couple nonlinear system, J. Phys. Soc. Jpn. 1999, 68: 369-379.
[196] Karasu K etal. Integrallity of generalized Ito system: The Painleve-test. J. phys. soc, Jpn. 2001, 70: 1165-1166.
[197] Gottwalld G A. The Zakharov-Kuznetsov equation as a two-dimenlonal model for nonlinear Rossby wave. (arXiv: nlin) 2003, 0312009
[198] Fu Z T, Liu S K, Liu S D. Multiple structures of two-dimensional nonlinear Rossby wave. Chaos, Solitons and Fractals. 2005, 24: 383-390.
[199] Yao R X, Li Z B. New exact solutions for three nonlinear evolution equations. Phys. Lett. A. 2002, 297:196-204.
[200] Fan E G. An algebraic method for finding a series of exact solutions to integral and nonintegral nonlinear evolution equations. J. Phys. A: Math. Gen. 2003, 36: 7009-7026.
[201] Lou S Y, Lu J Z. Special solutions from the variable separation approach: the Davey-Stewartson equation, J. Phys. A. 1996, 29: 4016-4029.
[202] Fan E G, Zhang J. Applications of the Jacobi elliptic function method to special-type nonlinear equations. Physics Letters A. 2002, 305: 383-392.
[203] Winter A. A criteria of oscillation stability. Quart. Appl. Math. 1949, 7: 115-117.
[204] Hartman P . On nonoscillatory linear differential equations of second order, Amer. J. Math. 1952, 74: 389-400.
[205] Coles W J. An oscillation criterion for second order diferential equation[J] Proc. Amer. Math. Soc. 1968, 19:175-194
[206] Willett D. On the oscillatory behavior of the solutions of second order li. 二 diferential equations. Ann. Polon. Math. 1969, 21: 175-19.
[207] Leighton W. The detection of the oscillation of solution of a second order linear diferential equation. Duke Math. J. 1950, 17: 57-61.
[208] Kamenev I V. Integral criteria for oscillation of linear differential equations of second order. Mat. zametki. 1978, 23:249-251.
[209] Philos C G. Oscillation theorems for linear differential equations of second order. Arch. Math. (Basel) , 1989, 53: 482-492.
[210] Yan J. Oscillation theorems for second order linear diferential equations with.damping[J]. Proc. Amer. Math. Soc. 1986, 98: 276-282.
[211] Li H J. Oscillation criteria for second order linear di.erential equations. J. Math. Anal. Appl. 1995, 194: 217-234.
[212] Rogovchenko Y V. Note on 'Oscillation criteria for second order linear di.erential equations'. J. Math. Anal. Appl. 1996, 203: 560-563.
[213] Kirane M, Rogovchenko Y V. On oscillation of nonlinear second order differential equation with damping term. Applied Mathematics and Computation. 2001, 117: 177-192.
[214] Grace S R. Oscillation theorems for nonlinear differential equations of second order. J. Math. Anal. Appl. 1992, 171: 220-241.
[215] Grace S R, Lalli B S. Integral averaging technique for the oscillation of second order nonlinear differential equations. J. Math. Anal. Appl. 1990, 149: 277-311.
[216] Rogovchenko Y V. Oscillation theroems for second order differential equations with damping. Non- linear Anal. 2000, 41: 1005-1028.
[217] Li W T, Agarwal R P. Intercal oscillation criteria for second order nonlinear differential equations with damping. Computers and Math. Appl. 2000, 40: 217-230.
[218] Rogovchenko S P, Rogovchenko Y V. Oscillation of second order differential equations with damping. Dynam. Contin. Discrete Impuls. Systems Ser. A. 2003, 10: 447-461.
[219] Mustafa O G, Rogovchenko S P, Rogovchenko Y. V. On oscillation of nonlinear second-order dif- feretial equations with damping term. J. Math. Anal. Appl. 2004, 298: 604-620.
[220] Wong J S W. On Kamenev-Type oscillation theorems, for second-order differential equations with damping. J. Math. Anal. Appl. 2001, 258: 244-257.
[221] Kirane M, Rogovchenko Y V. Oscillation results for a second order damped differential equation with nonmonotonous nonlinearity, J. Math. Anal. Appl. 2000, 250: 118-138.
[222] Grace S R. Oscillation theorems for second order nonlinear differential equations with damping. Math. Nachr. 1989, 141: 117-127.
[223] Rogovchenko S P, Rogovchenko Y V. Oscillation theorems for differenttial equations with a non- linear damping term. J. Math. Anal. Appl. 2003, 279:121-134.
[224] Tiryaki A, Zafer A. Oscillation of second-order nonlinear differential equations with nonlinear damp- ing. Mathmatical and computer Modelling. 2004, 39:197-208.
[225] Wang Q R. Oscillation criteria for nonlinear second order damped differential equations. Acta Math. Hungar. 2004, 102(1-2): 117-139.
[226] Ayanlar B, Tiryaki A. Oscillation theorems for nonlinear second order differential equations with damping. Acta. Math. Hungar. 2000, 89(1-2): 1-13.
[227] Wong J S W. Second order nonlinear forced oscillations. SIMA J. Math. Anal. 1988, 19(3): 667-675.
[228] Kartsatos A G. Maintenance of oscillations under the effect of a periodic forcing term. Proc. Amer. Math. Soc. 1972, 33: 377-383.
[229] EI-Sayed M A. An oscillation criteria for a forced second-order, linear differential eqaution. Proc. Amer. Math. Soc. 1993, 118(3): 813-817.
[230] Leighton W. Comparison theorems for linear differential equations of second order. Proc. Amer. Math. Soc. 1962, 13: 603-610.
[231] Wong J S W. Oscillation criteria for a forced second-order linear differential eqaution. J. Math. Anal. Appl. 1999, 231: 235-240.
[232] Jaros J, Kusano T. A Picone type identity for second order half-linear differential equation. Acta. Math. Univ. Comeniance. 1999, 68(1): 137-151.
[233] Li W T, Cheng S S. An oscillation criterion for nonhomogeneous half-linear differential equations. Appl. Math. Lett. 2002, 15: 259-263.
[234] Qakmak D, Tiryaki A. Oscillation criteria for certain forced second-order nonlinear differential equations. Appl. Math. Lett. 2004, 17: 275-279.
[235] 李万同．二阶半线性常微分方程的区间振动准则．数学学报．2002，45(3)：509—516．
[236] Nasr A H. Necessary and sufficient conditions for the oscillation of foced nonlinear second order differential eqautions with delayed argument. J. Math. Anal. Appl. 1997, 212: 51-59.
[237] Rogovechenko Y V. Oscillation criteria for certain nonlinear differential equation. J. Math. Anal. Appl. 1998, 229:399-416.
[238] Grace S R, Lalli B S. An oscillation criteria for certain second order strongly sublinear differential equation. J. Math. Anal. Appl. 1987, 123:584-588.
[239] Hamedani G G, Krenz G S. Oscillation criteria for certain second order differential equation. J. Math. Anal. Appl. 1990, 149:271-276.
[240] Wong J S W, Burton T A. Some properties of solutions of u″+ a(t)f(u)g(u′) = 0. Monatsh.Math. 1965, 69: 364-374.
[241] Erbe L. Oscillation criteria for second order nonlinear differential equation. Canad. Math. Bull. 1973, 16:49-56.
[242] Kong Q. Interval criteria for oscillation of second order linear ordinar y differential equations. J. Math. Anal. Appl. 1999, 229: 258-270.
[243] Li W T, Agarwal R P. Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations, J. Math. Anal. Appl. 2000, 245:171-188.
[244] Li W T, Agarwal R P. Interval oscillation criteria for second order linear order nonlinear differential equation with damping. Comput. Math. Appl. 2000, 40: 217-230.
[245] Tang Y, Yang Q. Oscillation of even order nonlinear functional differential equations with dampin. Acta Math Hungar. 2004, 102(3):223-238.
[246] Erbe L H. Oscillation criteria for second order nonlinear delay equation. Can. Math. Bull. 1972, 16: 49-59.
[2] Airy G.B. Tides and waves. Encycl. Metrop. London Art. 1845,241-396.
[3] Boussinesq M J. Theorie des ondes et de remous quise propageant le long d'un canal recangnlaier horizonal, et communiquant au liquide contene dans ce cannal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. Set. 1876, 2:55-108.
[4] Rayleigh L. On waves. Philos. Mag. Ser. 1876, 5: 257-279.
[5] Fermi A, Ulam J P. Studies of Nonlinear Problem. I.Los Alamous Rep. LA. 1940, 1955.
[6] 谷超豪等，孤立子理论与应用．杭州：浙江科技出版社， 1990．
[7] Petting J K, Skyrme T H R. A Model Uniform Field Equation. Nuci. Phys. 1962, 31: 550-555.
[8] 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．
[9] 郭柏灵，庞小峰．孤立子．北京：科学出版社， 1987．
[10] 李翊神．孤子与可积系统．上海：上海科技出版社，1999．
[11] Ablowitz M J, Clarkson P A. Solitons, nonlinear evolution equations and inverse scattering. Cam- brideg: Cambridge University. Press, 1991.
[12] Newell A C. Soliton in mathematics and physics. SIAM: Philadelphia, 1985.
[13] Faddeev L D, Takhtajan L A. Hamiltonian method in the theory of solitons. Berlin:Springer-Verlag, 1987.
[14] Matveev V B, Salle M A. Darboux transformation and Solutions. Berlin: Springer-Verlag, 1991.
[15] Baklund A V. Lund Universitests Arsskrift. 1885, 10.
[16] Wahlquist H D, Estabrook F B. Backlund transformations for solitons of the Korteweg-de Vriu equation. Phys. Rev Lett. 1973, 31:1386-1390.
[17] Weiss J, Tabor M. Carnvale.G. The Painlevé property for partial differential equations. J. Math. Phys. 1983, 24(3): 522-526.
[18] Weiss J. The Painlevé property for partial differential equations. Ⅱ. Backlund transformation, Lax pairs, and the Schwarzian derivative. J. Math. Phys. 1983, 24(6): 1405-1413.
[19] Wadati M, et al. Simple derivation of Backlund transformation from Riccati form of inverse method, Prog. Theor.Phys. 1975, 53: 418.
[20] Gu C H. A unified explicit form of Backlund transformations for generalized hierarchies of KdV equations. Lett. Math. Phys. 1986, 11(4): 325-335.
[21] Gu C H, Zhou Z X. On the Darboux matrices of Backlund transformations for AKNS systems. Let. Math. Phys. 1987, 13(3): 179-187.
[22] 胡星标，李勇．DJKM方程的Backlund变换及非线性叠加公式．数学物理学报．1991，11：164-172．
[23] Hu X B, Clarkson P A, Backlund transformations and nonlinear superposition formulae of a differential-difference KdV equation. J. Phys. A. 1998, 31:1405-1414.
[24] Hu X B, Zhu Z N, A Backlund transformation and nonlinear superposition formulae for the Belov- Chaltikian lattice. J. Phys. A. 1998, 31:4755-4716.
[25] Zeng Y B, et al. Canonical explicit Backlund transformations with spectrality for constrained flows of soliton hierarchies. Physica A. 2002, 303(3-4): 321-338.
[26] Li Z B, Wang M L. Travelling wave solutions to the two-dimensional KdV-Burgers equation. J. Phy. A. 1993, 26: 602-6031.
[27] Wang M L, Li Z B. Proc. of the 1994 Beijing Symposium on nonlinear evolution equations and infinite dimensional dynamics symtems, Zhongshan University Press, 1995, 181-185.
[28] 李志斌，张善卿．非线性波方程准确孤立子解的符号计算．数学物理学报．1997，17(1)：81-89．
[29] Fan E G, Zhang H Q. New exact solutions to a system of coupled KdV equations. Phys. Lett. A. 1998, 245: 389-392.
[30] 范恩贵．孤立子和可积系统： (博士学位论文)．大连：大连理工大学， 1998．
[31] 闫振亚．非线性波与可积系统： (博士学位论文)．大连：大连理工大学，2002．
[32] 陈勇．孤立子理论中的若干问题的研究及机械化实现：(博士学位论文)．大连：大连理工大学，2003．
[33] 李彪．孤立子理论中若干精确解方法的研究及应用：(博士学位论文)．大连：大连理工大学，2005．
[34] Clifford S G, Greene J M, Kruskal M D, Miura R M. Method for Solving the Korteweg-deVries Equation. Phys. Rev. Lett. 1967, 19:1095-1097.
[35] Lax P D. Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math. 1968, 21:467-490.
[36] Zabusky N J, Kruskal M D. Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. rev. Lett. 1965, 15:240-243.
[37] Wadati M. The modified Korteweg-de Vries equation, J. Phys. Soc. Japan. 1973, 34:1289-1296.
[38] Ablowitz M J, Kaup Z J, Newell A C. et.al. Nonlinear-evolution equations of physical significance. Phys. Rev. Lett. 1973, 31: 125-127.
[39] Wahlquist H D, Estabrook F B. Prolongation structures of nonlinear evolution equations. J. Math. Phys. 1975, 16: 1-7.
[40] Wu K, Guo H Y, Wang S K. Prolongation structures of nonlinear evolution systems in higher dimensions. Commun.Theoret.Phys. 1983, 2:1425-1437.
[41] Guo H Y, Wu K, Wang S K. Prolongation structure, Backlund transformation snd principal homo- geneous Hilbert problem in general relativity, Commun. Theoret. Phys. 1983, 2:883-898.
[42] 屠规彰，非线性方程的逆散射方法，应用数学与计算数学．1979，24(20)：913-917．
[43] 李翊神．一类发展方程和谱的变形，中国科学，A．1982，5：386-390．
[44] 曹策问．孤立子与反散射，郑州大学出版社， 1983，35-89．
[45] 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.
[46] Cunningham E. The principle of relativity in electrodynamics and an extension thereof. Proc. London Math. Soc. 1909, 8: 77-97.
[47] Nother E. Invariante varlationsprobleme. Nachr Koning. Cesell. Wissen. Gottingen, Meth. Phys. kl, 1918, 235-257.
[48] Ovsiannikov L V. Group properties of differential .equations. Novosibirsl, 1962.
[49] Sedov L I. Similarity and dimensional method, 4th ed. New York: Academic Press, 1959.
[50] Venikov V A. Theory of Similarity and Simulation. London: Mac Donald Technical and Scientific, 1969.
[51] Bluman G W, Cole J D. The general similarty solution of the heat equation, J.Math.Mech. 1969, 18:1025-1042.
[52] Levi D, Winternitz P. Group theoretical analysis of a rotating shallow liquid in a rigid container. J. Phys. A, 1989, 22:4743-4767.
[53] Vorob'er E M. Reduction and quotient, equations for differential equations with symmetries. Acta Appl. Math. A. 1991, 23: 1-24.
[54] Olver P J. Evolution equations possessing infinitely many symmetries. J. Math. Phys. 1977, 18:1212- 1215.
[55] Olver P J. On the Hamiltonian structure of evolution equations. Math. Proc. Camb. Phill Soc. 1980, 88:71-88.
[56] Fuchsseiner B, Fokas A S. Symplectic structures, their Backlund transformations and hereditary symmetries. Physica D. 1981, 4:47-66.
[57] Chen H H, et.al. In Advance in_nonlinear wave, ed L.Debnath. 1982, 233.
[58] 李翊神，朱国城．一个谱可变演化方程的对称．科学通报．1986，19：1449-1453；可积方程新的对称，李代数及可变演化方程．中国科学A．1987，30：1243-1250；”C可积”非线性方程的代数性质Burgers方程和Calogero方程．中国科学A．1990，33：513-520．
[59] 田踌，Burgers方程的新的强对称，对称和李代数．中国科学A．1987，31：141—151．
[60] Clarkson P A, Kruskal M D. New similarity reductions of the Boussinesq equation. J. Math. Phys., 1989, 30(10): 2210- 2213.
[61] Clark.son P A. New similarity reductions of the modified Bonssinesq equation. J. Phys.A. 1989, 22(13): 2355- 2367.
[62] Lou S Y. A note on the new similarity reductions of the Bonssinesq equation. Phys. Lett. A. 1990, 151:133- 135.
[63] Lou S Y. Twelve sets of symmetries of the Caudrey-Dodd-Gibbon- Sawada-Sawada-Kotera equation; Generalized Boussinesq equation and KdV equation-Painlevé properties, Backlund transformations and Lax pairs. Sci.China A. 1991, 34(9):1098-1108.
[64] Zhdanov R Z, Fushchych W I. Conditional symmetry and new classical solutions of the Yang-Mills equations. J. Phys. A: Math. Gen. 1995,28:6253-6264.
[65] Ablowitz M J. et al., A connection between nonlinear evolution equations and ordinary differential equations of P-type. J. Phys. 1980, 21(4): 715-721.
[66] Weiss J. The Painlevé property for partial differential equations. Ⅱ. Backlund transformation, Lax pairs, and the Schwarzian derivative. J. Math. Phys. 1983, 24: 1405-1413.
[67] Jimbo M. et al., Painlevé test for the self-dual Yang-Mills equation. Phys. Lett. A. 1982, 92(2):59-60.
[68] Fordy A. Picketing A, Analysing negative resonances in the Painlevé test. Phys. Lett. A. 1991, 160 (4):347-354.
[69] 曾云波．与A_l相联系的半Toda方程的Lax对与Backlund变换．数学学报．1992，35(4)：454—459．
[70] 曾云波．递推算子与Painlevé性质．数学年刊，1991，12A(1)：78—88．
[71] Hirota R. Exact solution of the KdV equation for multiple collisions of solutions, Phys. Rev. Lett. 1971, 27: 1192-1194.
[72] Hu X B, Tam H W. New integral differential -difference systems: Lax pairs, bilinear forms and solition solutions. 2000, 46: 99-105.
[73] Hu X B, Rational solutions of Integral equations via nonlinear superposition formulae. J. Phys. A. 1997, 30:8225-8240.
[74] Hu X B, et. al., Lax pairs and B acklund transformations for a coupled Ramani equation and its related system. Apll. Math. Lett. 2000, 13:45-48.
[75] Boiti M, et. al. On the spectral transform of a Korterweg-de Vries equation in two spatial dimensions. Inv. Probl. 1986, 2:271-280.
[76] Lou S Y, Generalized dromion solutions of the (2+1)-dimensiona] KdV equation. J. Phys. A. 1995, 28: 7227-7232.
[77] Zhang J F. Abundant dromion-like structures to the (2+1)-dimensional KdV equation. Chin. Phys. 2000, 9:1-4.
[78] Lou S Y, Ruan H Y.Revisitation of the localized excitations of the (2+1)-dimensional KdVequation. J. Phys. A, 2001, 34:305-316.
[79] Rosenau P, Hyman M. Compactons: Solitions with finite wavelength, Phys. Rev. Lett. 1993, 70:564- 567.
[80] Lou S Y, Wang Q X. Painleve integrability fo two sets of nonlinear evolution equations with nonlinear dispersions. Phys. Lett. A, 1999, 262:344-349.
[81] Wazwaz A M. Exact special solutions with solitary patterns for the nonlinear dispersive K(m,n) equations. Chaos, Solitons and Fractals. 2002, 13:161-170;
[82] Olver P J, Rosenau P. TRi-Hamiltonian duality between solitions and solitary-wave solutions having compact support. Phys. Rev. E. 1996, 53:1990-1906.
[83] Grammaticos B, Ramani A, Hietarinta J. Multilinear operators: the natural extension of Hirota's bilinear formalism. Phys. Lett. A. 1994, 190:65-70.
[84] Liu Q P, Hu X B, Zhang M X. Supersymmetric modified Korteweg-de Vries equation: bilinear approach. Nonlinearity. 2005, 18(4):1597-1603.
[85] Liu Q P, Hu X B. Bilinearization of N = 1 supersymmetric Korteweg-de Vries equation revisited. J. Phys. A. 2005, 38(28):6371-6378.
[86] Wang M L. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. Lett. A. 1996, 216(1-5): 67-75.
[87] Wang M L, Wang Y M, Zhou Y B. An auto-Backlund transformation and exact solutions to a generalized KdV equation with variable coefficients and their applications. Phys. Lett: A. 2002, 303(1): 45-51.
[88] Gao Y T, Tian B. New families of exact solutions to the integrable dispersive long wave equaitons in 2+1-dimensional spaces. J. Phys. A, 1996, 29(11): 2895-2903.
[89] Fan E G, Zhang H Q. A note on the homogeneous balance method. Phys. Lett. A. 1998, 246: 403-406.
[90] Fan E G, Zhang H Q. New exact solutions to a system of coupled KdV equations. Phys. Lett. A. 1998, 245: 389-392.
[91] Tian B; Gao Y T. Variable-coefficientbalancing-actmethodandvariable-coe?cientKdVequation from- fluiddynamicsandplasmaphysics. Eur. Phys. J. B. 2001,22:351-360.
[92] 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.
[93] 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.
[94] 吕卓生．大连大连大连理工大学博士学位论文，2004．
[95] Barnett M P. Symbolic calculation inchemistry: Selected examples. Int. J. uantumChem. 2004, 100: 80-104.
[96] Lou S Y, Ruan H Y, Huang G X. Exact solitary wave sina convecting fluid. J. Phys. A, 1991, 24: 587-590.
[97] Conte R, Musette M. Link between solitary wave sand projective Riccatiequations. J. Phys. A. Math. Gen. 1992,25:5609-5623.
[98] 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.
[99] 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.
[100] Fan E G. Soliton solutions for ageneralized Hirota-Satsuma coupled KdV equation and acoupled mKdV equation. Phys. Lett. A. 2001, 282: 18-22.
[101] Yan C T. A simple transformation for nonlinear waves. Phys. Lett. A, 1996, 224: 77-84.
[102] Yah Z Y, Zhang H Q. New explicit and exact travelling wave solutions for asystem of variant Boussinesq equations in mathematical physics. Phys. Lett. A. 1999, 252: 291-296.
[103] Rosenau P, Hyman M. Compactons: Solitons with finite wavelength. Phys. Rev. Lett. 1993,70: 564-567.
[104] 张鸿庆，弹性力学方程组一般解的统一理论，大连工学院学报， 1978，3：23-47．
[105] 张鸿庆，杨光．变系数偏微分方程组一般解的构造．应用数学和力学， 1991，12：135—139．
[106] 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(4)：315—322．
[107] 张鸿庆，Maxwell方程的多余阶次与恰当解．应用数学与力学．1981，2(3)：321-331．
[108] Lou S Y, Ni G J. The relations among a special type of solutions in some (D+1) dimensional nonlinear equations. J. Math. Phys. 1989, 30: 1614-1620.
[109] Zakharov V E, Shabat A B. A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. Funct. Anal. Appl. 1974, 8: 226-235.
[110] Dolye P W. Separation of variables for scalar evolution equations in one space dimension. J. Phys. A. 1996, 29: 7581-7596.
[111] 李翊神等．非线性科学选讲．中国科学技术大学出版社．1994．
[112] 吴文俊．吴文俊论数学机械化．山东：山东教育出版社．1996．
[113] 高小山，数学机械化进展综述数学进展，2001 30(5)385-401．
[114] 吴文俊．几何定理机器证明的基本原理．北京：科学出版社， 1984．
[115] 陈玉福，高小山．微分多项式系统的对合特征集．中国科学A．33(2003)97-113．
[116] Wu W J. On the decision problem and the mechanization of theorem-proving in elementary geom- etry. Scintia Sinica, 1978,21:159-172.
[117] Wu W T. On zeros of algebraic equations: an application of Ritt principle. Kexue Tongbao, 1986, 31: 1-5.
[118] Ritt J F. Differential Algebra. New York: American mathematical Society, 1950.
[119] Wu W.J. On the foundation of algebraic differential geometry. Systems Sci. Math. Sci. 1989,2(4): 289-312.
[120] Chou S C, Gao X S, Zhang J Z. Machine proofs in geometry. Automated production of readable proofs for geometry theorems, Singapore, 1994.
[121] Gao X S, Zhang J Z. and Chou S C. Geometry Expert(in Chinese), Nine Chaoters Pub. Taiwan, 1998.
[122] Wang D M, Gao X S. Geometry theorems proved mechanically using Wu's method. MM-Research Preprints, 1987, 2.
[123] Wang D M, Hu S. Mechanical proving system for constructible theorems in elementry gepmetry. J. Sys. Sci. Math. Scis, 1995, 7.
[124] Lin D D, Liu Z J. Some results in theorem proving in finite geometry. Proc of IWMMM. Inter Academic Pub. 1992.
[125] Wu J Z, Liu Z J. Proceedings of 1st Asian Symposium on Computer Mathematics, Beijing of China, 1995.
[126] Wu J Z, Liu Z J. Science in China (Series E), 1996, 39(6): 608-619.
[127] 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.
[128] Kaput D, Wan H K. Refutational proofs of geometry theorems via characteristic sets. Proc of ISSAC-90, Tokyo, Japan, 1990, 277-284.
[129] Ko H. Geometry theorem proving by decomposition of quasi-algebraic sets: an application of the Ritt-Wu principle. Artif. Intell. 1988, 37: 95-122.
[130] Li H B, Cheng M T. Proving theorems in elementary geometry with Cliffird algebraic method. Chinese Math Progress, 1997, 26(4):357-371.
[131] Li H B, Cheng M T. Clifford algebraic reduction method for mechanical theorem proving in differ- ential geometry. J. Automated Reasoning, 1998, 21(1):1-21.
[132] 石赫，机械化数学引论，湖南教育出版社， 1998．
[133] Sun X D, Wang K, Wu K. Wu algorithm and solutions of Yang-Baxter equation for eight-vertex model. Comm. Theoret. Phys. 1995, 23(1): 125-128.
[134] 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.
[135] Li Z B, Shi H. Exact solutions for Belousov-Zhabotinskiǐreaction-diffusion system. Appl. Math. J. Chinese Univ. Set. 1996, 11B(1): 1-6.
[136] Li Z B. Proc. Of Asian Symposition on computer Mathematics, Lanzhou University, 1998: 153.
[137] Li Z B, Liu Y P. RATH: A Maple package for finding travelling solitary wave solutions to nonlinear evolutions. Comput. Phys. Comm. 2002, 148(2): 256-266.
[138] Xie F D, Chen Y. An algorithmic method in Painlevé analysis of PDE. Compu. Phys. Commun. 2003, 154: 197-204.
[139] 郑学东．一类非线性偏微分方程组的机械化求解与P-性质的机械化检验： (硕士学位论文)．大连：大连理工大学，2003．
[140] Cohen A M, Davenport J H, Heck J P. An overview of computer algebra. Computer Algebra for Industry. 1994,16:1-25.
[141] 斯特列可夫CII著，何文蛟译．振动理论引论(上册)北京：高等教育出版社， 1956．
[142] Sturm C. Sur les equations differentielles linearies on second order. J. Math. Pure, 1886, 1106-186.
[143] Cesari L. Asymptotic behavior and stability problem in ordinary differential equations. Berlin, Springer 1963.
[144] willett D. Classification of second orders linear differential equations with respect to oscillation, Advance. Math, 1969, 3: 594-623.
[145] Barrett J H. Oscillation theory of ordinary linear differential equations. Advances in Math. 1969, 3: 415-509.
[146] Wong J S W. On second order nonlinear oscillation, Funkcial. Ekvac, 1968, 11(3): 207-234.
[147] Kartsatos A G. Recent results on oscillation of solutions of forced and perturbed nonlinear dif- ferential equations of even order, J. R. Graef. Ed. stability of Dynamical Systems Theory and Applications. 1977, Vol.28, Charpter 3 Marcel. Dekker, N.Y.
[148] Swanson C A. Comparison and oscillation theory of linear differential Equations. New York Acad. Press, 1968.
[149] 张炳根．二阶常微分方程解的有介性．数学学报．1964，1：128—136．
[150] 燕居让．常微分方程振动理论．太原：山西教育出版社．1992．
[151] Ladde G S, Lakshimikantham V, Zhang B G. Oscillation Theory of Differential Equations with Deviating Arguments. New York: Marcel Dekker, 1987.
[152] Bainov D D, Mishev D P. Oscillation Theory for Neutral Differential Equations with Delay. New York: Adam Hilger, 1991.
[153] Gyori I, Ladas G. Oscillation Theory of Delay Differential Equations. Oxford: Clarendon Press, 1991.
[154] Agarwal R P, Grace S R, O'Regan D. Oscillation Theory for Difference and Differential Equa- tions. Dordrecht: Kluwer Academic Pubhshers, 2000.
[155] Wu J h. Theory and Applications of Partial Functional Differential Equations. New York: Springer- Verlag, 1996.
[156] Allegretto W, Erbe L. Oscillation criteria for matrix differential inequalities. Canad. Math. Bull. 1973, 16: 5-10.
[157] Atkinson F V, Kaper H G, Kwong M K. An oscillation for linear second order differential systems. Proc. Amer. Math. Sec. 1985, 94: 91-96.
[158] Butler G J, Erbe L H. Oscillation results for second order differential systems. J. Math. AnaL. 1986, 17:19-29
[159] Butler G J, Erbe L H. Oscillation results for self-adjoint differential systems. J. Math. Anal. Appl. 1986, 115: 470-481.
[160] Butler G J, Erbe L H, Mingarelli A B. Riccati techniques and variational principles in oscil- lation theory for linear systems. Trans. Amer. Math. Soc. 1987, 303: 263-282.
[161] 郑召文，孟凡伟，俞元洪．二阶线性矩阵微分系统的振动性．数学学报．1998，41(6)：1231—1238．
[162] 庄容坤．二阶矩阵微分系统的区间准则．数学学报．2001，44(6)：1037-1044．
[163] Zhang H Q. Proceedings of 5th Asian Symposium on Computer Mathematics, World Scientific, Singapore, 2001.
[164] Yan Z Y, Zhang H Q. Explict and exact solutions for the generalized rection Duffing equation, Commun. in Nonlinear and Numerical Simulation. 1999, 4:146.
[165] Yah Z Y, A new sine-Gordon equation expansion algorithm to investigate some special nonlinear differential equations. Chaos, Solitons and Fractals. 2005, 23:767-775.
[166] Hirota R. The direct method in soliton theory, translated from Japanese and .edited by Atsushi Nagai Jon Nimmo and sClaire Gilson. Cambridge University press, 2002.
[167] Parkes E J, Duffy, B R. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput. Phys. Commun, 1996, 98: 288-300.
[168] Hereman W. Exact solitary wave solutions of coupled nonlinear evolution equations using MAC- SYMA. Comput. Phys. Commun. 1991, 65: 143-150.
[169] Yah Z Y, Zhang H Q. New explicit and exact travelling wave solutions for system of variant Boussinessq equations in mathematical physics. Phys. Lett. A. 996,224:77.
[170] Ma W X. et. al. Explicit and exact solutions to a Komog0rov-Petrovskii-Piskunov equation. Int. J. Non-linear Mech. 1996, 31: 329.
[171] Chert H T, Zhang H Q. New mutliple solition-like solutions to the generalized (2+1)-dimensional KP equation. Applied Mathmatics and Computation. 2004, 157: 765-773.
[172] Fu Z T, Liu S K, Liu S D. Lame Function and Its Application to Some Nonlinear Evolution Equations. Commun. Theor. Phys. (Beijing, China) 2003, 40: 53-56.
[173] Arai A. Exactly solvable supersymmetric quantum mechanics. J. Math. Anal. Appl. 1991, 158: 63-79.
[174] Arai A. Exact solutions of multi-component nonlinear Schrodinger and Klein- Gordon equations in two-dimensional space-time. J.Phys. A. 2001, 34: 4281-4288.
[175] Liu X Q, Jiang S, The SeC_q - tanh_q method and its applications. Phys. Lett. A. 2002, 298: 253-258.
[176] Wang M L, Zhou Y B. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Physics. lett. A. 1996, 216:67-75.
[177] Zhang J F, Multiple solition solutions for the approrimate equations of long wave water. Acta Physics Sinica, 1998, 47: 1416(in Chinese).
[178] Yah Z Y. Zhang H Q. Explicit exact solutions for nonlinear approrimate equations with long waves in shallow water. Acta Physica sinica. 1999, 48: 1962(in Chinese).
[179] Liu S K. et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave .equations. Phys. Lett. A. 2001, 289: 69-74.
[180] Liu S K, Fu Z T. et. al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 2001, 289(1-2):69-74.
[181] Fan E G, Hon B Y. Double periodic solutions with Jacobi elliptic functions for two generalized Hirota-Satsuma coupled KdV systems. Phys. Lett. A. 2002, 292: 335-337.
[182] Fan E G, Zhang J. Applications of the Jacobi elliptic function method to special-type nonlinear equations. Phys. Lett. A. 2002, 305: 383-392.
[183] Yah Z Y. Abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional integrable Davey-Stewartson-type equation via a new method. Chaos Solitons Practals. 2003, 18(2):299-309.
[184] Yan Z Y. Extended Jacobian elliptic function algorithm with symbolic computation to construct new doubly-periodic solutions of nonlinear differential equations. Comput. Phys. Comm. 2002, 148(1):30- 42.
[185] Wang M L, Zhou Y B. The periodic wave solutions for the Klein-Gordon- Schrǒinger equations. Physics Letter A. 2003, 318: 84-92.
[186] Zhou Y B, Wang M L, Wang Y M. Periodic wave solutions to a coupled KdV equations with variable coefficients. Physics Letter A. 2003, 308: 31-36.
[187] Chen Y, Wang Q, Li B. Jacobi Elliptic Function Rational Expansion Method with Symbolic Compu- tation to Construct New Doubly-periodic Solutions of Nonlinear Evolution Equations. Z. Naturforsch 2004, 59a: 529-536.
[188] Wang Q, Chen Y, Zhang H Q. A new Jacobi elliptic function rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation. Chaos Solitons Fractals 2005,23 (2):477-483.
[189] Chen H T, Zhang H Q, New double periodic and multiple soliton solutions of the generalized (2+1)-dimentional Boussineq equations, Chaos, Solitions and Fractals. 2004, 20(4): 765-776.
[190] Chen H T, Zhang H Q, New double periodic solutions for a kind of nonlinear wave equation, Progress of Theoretical Physics. 2003, 109(5): 709-716.
[191] Chandrasekharan K. Elliptic Functions, Berlin, Springer, 1978.
[192] Patrich D V. Elliptic Function and Elliptic Curves, Cambridge, At the University Press, 1973.
[193] Wang Z X, Xia X J. Special Functions, Singapone:World Scientific, 1989.
[194] Ito J M. extension of nonlinear evolution equation of the KdV(mKdv) type to higher orders. J. Phys. Soc. Jpn. 1980, 49: 771.
[195] Tam etal. Two Integral couple nonlinear system, J. Phys. Soc. Jpn. 1999, 68: 369-379.
[196] Karasu K etal. Integrallity of generalized Ito system: The Painleve-test. J. phys. soc, Jpn. 2001, 70: 1165-1166.
[197] Gottwalld G A. The Zakharov-Kuznetsov equation as a two-dimenlonal model for nonlinear Rossby wave. (arXiv: nlin) 2003, 0312009
[198] Fu Z T, Liu S K, Liu S D. Multiple structures of two-dimensional nonlinear Rossby wave. Chaos, Solitons and Fractals. 2005, 24: 383-390.
[199] Yao R X, Li Z B. New exact solutions for three nonlinear evolution equations. Phys. Lett. A. 2002, 297:196-204.
[200] Fan E G. An algebraic method for finding a series of exact solutions to integral and nonintegral nonlinear evolution equations. J. Phys. A: Math. Gen. 2003, 36: 7009-7026.
[201] Lou S Y, Lu J Z. Special solutions from the variable separation approach: the Davey-Stewartson equation, J. Phys. A. 1996, 29: 4016-4029.
[202] Fan E G, Zhang J. Applications of the Jacobi elliptic function method to special-type nonlinear equations. Physics Letters A. 2002, 305: 383-392.
[203] Winter A. A criteria of oscillation stability. Quart. Appl. Math. 1949, 7: 115-117.
[204] Hartman P . On nonoscillatory linear differential equations of second order, Amer. J. Math. 1952, 74: 389-400.
[205] Coles W J. An oscillation criterion for second order diferential equation[J] Proc. Amer. Math. Soc. 1968, 19:175-194
[206] Willett D. On the oscillatory behavior of the solutions of second order li. 二 diferential equations. Ann. Polon. Math. 1969, 21: 175-19.
[207] Leighton W. The detection of the oscillation of solution of a second order linear diferential equation. Duke Math. J. 1950, 17: 57-61.
[208] Kamenev I V. Integral criteria for oscillation of linear differential equations of second order. Mat. zametki. 1978, 23:249-251.
[209] Philos C G. Oscillation theorems for linear differential equations of second order. Arch. Math. (Basel) , 1989, 53: 482-492.
[210] Yan J. Oscillation theorems for second order linear diferential equations with.damping[J]. Proc. Amer. Math. Soc. 1986, 98: 276-282.
[211] Li H J. Oscillation criteria for second order linear di.erential equations. J. Math. Anal. Appl. 1995, 194: 217-234.
[212] Rogovchenko Y V. Note on 'Oscillation criteria for second order linear di.erential equations'. J. Math. Anal. Appl. 1996, 203: 560-563.
[213] Kirane M, Rogovchenko Y V. On oscillation of nonlinear second order differential equation with damping term. Applied Mathematics and Computation. 2001, 117: 177-192.
[214] Grace S R. Oscillation theorems for nonlinear differential equations of second order. J. Math. Anal. Appl. 1992, 171: 220-241.
[215] Grace S R, Lalli B S. Integral averaging technique for the oscillation of second order nonlinear differential equations. J. Math. Anal. Appl. 1990, 149: 277-311.
[216] Rogovchenko Y V. Oscillation theroems for second order differential equations with damping. Non- linear Anal. 2000, 41: 1005-1028.
[217] Li W T, Agarwal R P. Intercal oscillation criteria for second order nonlinear differential equations with damping. Computers and Math. Appl. 2000, 40: 217-230.
[218] Rogovchenko S P, Rogovchenko Y V. Oscillation of second order differential equations with damping. Dynam. Contin. Discrete Impuls. Systems Ser. A. 2003, 10: 447-461.
[219] Mustafa O G, Rogovchenko S P, Rogovchenko Y. V. On oscillation of nonlinear second-order dif- feretial equations with damping term. J. Math. Anal. Appl. 2004, 298: 604-620.
[220] Wong J S W. On Kamenev-Type oscillation theorems, for second-order differential equations with damping. J. Math. Anal. Appl. 2001, 258: 244-257.
[221] Kirane M, Rogovchenko Y V. Oscillation results for a second order damped differential equation with nonmonotonous nonlinearity, J. Math. Anal. Appl. 2000, 250: 118-138.
[222] Grace S R. Oscillation theorems for second order nonlinear differential equations with damping. Math. Nachr. 1989, 141: 117-127.
[223] Rogovchenko S P, Rogovchenko Y V. Oscillation theorems for differenttial equations with a non- linear damping term. J. Math. Anal. Appl. 2003, 279:121-134.
[224] Tiryaki A, Zafer A. Oscillation of second-order nonlinear differential equations with nonlinear damp- ing. Mathmatical and computer Modelling. 2004, 39:197-208.
[225] Wang Q R. Oscillation criteria for nonlinear second order damped differential equations. Acta Math. Hungar. 2004, 102(1-2): 117-139.
[226] Ayanlar B, Tiryaki A. Oscillation theorems for nonlinear second order differential equations with damping. Acta. Math. Hungar. 2000, 89(1-2): 1-13.
[227] Wong J S W. Second order nonlinear forced oscillations. SIMA J. Math. Anal. 1988, 19(3): 667-675.
[228] Kartsatos A G. Maintenance of oscillations under the effect of a periodic forcing term. Proc. Amer. Math. Soc. 1972, 33: 377-383.
[229] EI-Sayed M A. An oscillation criteria for a forced second-order, linear differential eqaution. Proc. Amer. Math. Soc. 1993, 118(3): 813-817.
[230] Leighton W. Comparison theorems for linear differential equations of second order. Proc. Amer. Math. Soc. 1962, 13: 603-610.
[231] Wong J S W. Oscillation criteria for a forced second-order linear differential eqaution. J. Math. Anal. Appl. 1999, 231: 235-240.
[232] Jaros J, Kusano T. A Picone type identity for second order half-linear differential equation. Acta. Math. Univ. Comeniance. 1999, 68(1): 137-151.
[233] Li W T, Cheng S S. An oscillation criterion for nonhomogeneous half-linear differential equations. Appl. Math. Lett. 2002, 15: 259-263.
[234] Qakmak D, Tiryaki A. Oscillation criteria for certain forced second-order nonlinear differential equations. Appl. Math. Lett. 2004, 17: 275-279.
[235] 李万同．二阶半线性常微分方程的区间振动准则．数学学报．2002，45(3)：509—516．
[236] Nasr A H. Necessary and sufficient conditions for the oscillation of foced nonlinear second order differential eqautions with delayed argument. J. Math. Anal. Appl. 1997, 212: 51-59.
[237] Rogovechenko Y V. Oscillation criteria for certain nonlinear differential equation. J. Math. Anal. Appl. 1998, 229:399-416.
[238] Grace S R, Lalli B S. An oscillation criteria for certain second order strongly sublinear differential equation. J. Math. Anal. Appl. 1987, 123:584-588.
[239] Hamedani G G, Krenz G S. Oscillation criteria for certain second order differential equation. J. Math. Anal. Appl. 1990, 149:271-276.
[240] Wong J S W, Burton T A. Some properties of solutions of u″+ a(t)f(u)g(u′) = 0. Monatsh.Math. 1965, 69: 364-374.
[241] Erbe L. Oscillation criteria for second order nonlinear differential equation. Canad. Math. Bull. 1973, 16:49-56.
[242] Kong Q. Interval criteria for oscillation of second order linear ordinar y differential equations. J. Math. Anal. Appl. 1999, 229: 258-270.
[243] Li W T, Agarwal R P. Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations, J. Math. Anal. Appl. 2000, 245:171-188.
[244] Li W T, Agarwal R P. Interval oscillation criteria for second order linear order nonlinear differential equation with damping. Comput. Math. Appl. 2000, 40: 217-230.
[245] Tang Y, Yang Q. Oscillation of even order nonlinear functional differential equations with dampin. Acta Math Hungar. 2004, 102(3):223-238.
[246] Erbe L H. Oscillation criteria for second order nonlinear delay equation. Can. Math. Bull. 1972, 16: 49-59.