AC=BD在Backlund变换中的应用及偏微分方程解的完备性
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摘要
本文根据数学机械化的思想，在导师张鸿庆教授“AC=BD”模式的指导下，以源于物理、力学等领域中的非线性问题所对应的非线性偏微分代数方程(组)为研究对象，研究了它们的一些问题，如精确解、Backlund变换以及解的完备性等。
第一章介绍了数学机械化的思想、应用、历史与发展情况以及非线性演化方程解的构造性方法，同时介绍了一些关于该学科领域的国内外学者所取得的成果。
第二章以“AC=BD”模式为指导，考虑了非线性偏微分方程(组)的精确解的构造。给出了“AC=BD”模式的基本概念及理论初步，C、D算子的构造方法以及C-D可积系统理论，介绍了Reid将线性偏微分方程化为标准型的方法。
第三章运用数学机械化的思想，将“AC=BD”模式应用到Backlund变换中，给出了显式C-D对、隐式C-D对及Backlund变换的基本概念，并分别给出了求自-Backlund变换的算法及例子。
第四章介绍了吴微分特征列及伪带余除法的理论，讨论了线性偏微分代数方程组的解(局部)的完备性问题。
In this dissertation, by applying the ideas of the mathematics mechanization, under the instruction of the AC=BD model of Professor Zhang Hongqing, the nonlinear partial differential algebraic equation or equations (PDAE or PDAEs) related to some nonlinear topics which origin from physics, mechanics et al are studied, including exact solutions, Backlund transformation, the compatibility of the solutions and so on.
Chapter 1 is devoted to investigating the idea, application, history and development of the mathematics mechanization and the construction of the solutions of nonlinear partial differential equation(NPDE). In addition, some achievements on the subject domestic and abroad are presented.
Chapter 2 considers the construction of exact solutions of NPDE(s) under the instruction of the AC=BD model. The basic concept and theory about AC=BD model, the construction of the operators of C and D, the theory of C-D integral system are introduced. In addition, this dissertation introduces the methods of transforming NPDE into standard type advanced by Reid.
Chapter 3 applies AC=BD model to Backlund transformation by the ideas of the mathematics mechanization. The basic concepts of the explicit and implicit C-D pair, Backlund transformation are presented. The arithmetics and examples are also presented.
Chapter 4 introduces Wu differential character set, division with remainder and the compatibility of the solutions (local) of linear PDEs.
第一章介绍了数学机械化的思想、应用、历史与发展情况以及非线性演化方程解的构造性方法，同时介绍了一些关于该学科领域的国内外学者所取得的成果。
第二章以“AC=BD”模式为指导，考虑了非线性偏微分方程(组)的精确解的构造。给出了“AC=BD”模式的基本概念及理论初步，C、D算子的构造方法以及C-D可积系统理论，介绍了Reid将线性偏微分方程化为标准型的方法。
第三章运用数学机械化的思想，将“AC=BD”模式应用到Backlund变换中，给出了显式C-D对、隐式C-D对及Backlund变换的基本概念，并分别给出了求自-Backlund变换的算法及例子。
第四章介绍了吴微分特征列及伪带余除法的理论，讨论了线性偏微分代数方程组的解(局部)的完备性问题。
In this dissertation, by applying the ideas of the mathematics mechanization, under the instruction of the AC=BD model of Professor Zhang Hongqing, the nonlinear partial differential algebraic equation or equations (PDAE or PDAEs) related to some nonlinear topics which origin from physics, mechanics et al are studied, including exact solutions, Backlund transformation, the compatibility of the solutions and so on.
Chapter 1 is devoted to investigating the idea, application, history and development of the mathematics mechanization and the construction of the solutions of nonlinear partial differential equation(NPDE). In addition, some achievements on the subject domestic and abroad are presented.
Chapter 2 considers the construction of exact solutions of NPDE(s) under the instruction of the AC=BD model. The basic concept and theory about AC=BD model, the construction of the operators of C and D, the theory of C-D integral system are introduced. In addition, this dissertation introduces the methods of transforming NPDE into standard type advanced by Reid.
Chapter 3 applies AC=BD model to Backlund transformation by the ideas of the mathematics mechanization. The basic concepts of the explicit and implicit C-D pair, Backlund transformation are presented. The arithmetics and examples are also presented.
Chapter 4 introduces Wu differential character set, division with remainder and the compatibility of the solutions (local) of linear PDEs.
引文
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[2] X.S.Gao, J.Z.Zhang, and S.C.Chou, Geometry Expert(in Chinese), Nine Chapters Pub, Talwan, 1998.
[3] S.C.Chou, Automated reasoning in differential geometry and mechanics using the characteristic set method. I. An improved version of Ritt-Wu's decomposition algorithm. Journal of Automated Reasoning, 1993, 10: 161.
[4] X.S.Gao, An Introduction to Wu's Method of Mechanical Geometry Theorem Proving, IFIP Transaction on, Automated Reasoning, North-Holland, 1993, 13-21.
[5] 关霭雯，吴文俊消元法讲义，北京理工大学，1994．
[6] J.Z.Wu and Z.J.Liu, Proceedings of 1st Asian Symposium on Computer Mathematics. Beijing of China, 1995.
[7] H.Shi, On solving the Yang-Baxter equations. Proceeding of the 1992, International workshop on Math. Mech, 28-37.(International Academic Publishers)
[8] 石赫，机械化数学引论，湖南教育出版社，1998．
[9] X.D.Sun, K.Wang, K.Wu, Classification of six-vertex-type solutions of the colored Yang-Baxter equation. J.Math.Phys. 1995, 10:6043.
[10] X.D.Sun, K.Wang, K.Wu, Solutions of Ya.ng-Baxter equation with color parameters.中国科学 A, 1995, 38:1105.
[11] 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．
[12] Z.B.Li, Analytic solitary wave solutions for the non-linear schrSdinger equation coupled to the KdV equation. Proc. of the 3rd Asian Symposition on computer mathematics(ASCM), Lanzhou University Press, 1998, 153.
[13] 朱思铭，施齐焉，符号计算与吴-消元法在Painlevé检验分析中的应用．现代数学和力学(MMM-Ⅶ)，上海大学出版社，1997，482-485．
[14] 范恩贵，孤立子和可积系统：精确解．相似约化．Hamilton结构·零曲率表示·吴方法和计算机代数的应用，大连理工大学博士论文，1998．
[15] 闫振亚，非线性波与可积系统，大连理工大学博士论文，2002．
[16] 陈勇，孤立于理论中的若干问题的研究及机械化实现，大连理工大学博士论文，2003．
[17] 朝鲁，吴-微分特征列法及其在微分方程对称和力学中的应用，大连理工大学博士论文，1997．
[18] 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，1978，18(3)：23-47．
[19] 张鸿庆，Maxwell方程的多余阶次与恰当解．应用数学和力学，1981，2(3)：321-331．
[20] 张鸿庆，算子系统通解的代数构造．力学特刊，1981：152-161．
[21] 张鸿庆，王震宇，胡海昌解的完备性和逼近性．科学通报，1985，30(5)：342-344．(10)：667．
[22] 张鸿庆，线性算子方程组一般解的代数构造．大连工学院学报，1979，3：1-16．
[23] 张鸿庆，冯红，非齐次线性算子方程组一般解的代数构造．大连理工大学学报，1994，34(3)：249-255．
[24] 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(2)：135-139．
[25] 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．
[26] 张鸿庆，吴方向，构造板壳问题基本解的一般方法．科学通报，1993，38(7)：671-672．
[27] 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．
[28] 张鸿庆，力学的代数化，机械化，辛化与几何化．现代数学和力学，Ⅶ(MMM-Ⅶ)，1997：20-25．
[29] Zhang Hongqing, Proceedings of 5th Asian Symposirm on Computer Mathematics, World Scientific, Singapore, 2001:210.
[30] H.Q.Zhang and Y.F.Chen, The compatible conditions of system. Proceeding of the 3rd ACM, Lanzhou University Press, 1998:147-154.
[31] H.Q.Zhang, Proceeding of the 3rd ASCM, World Scientific Press, 2001:254.
[32] 陈玉福，微分特征列法理论研究及应用，大连理工大学博士论文，1999．
[33] 张玉峰，孤立子方程求解与可积系统，大连理工大学博士论文，2002．
[34] C.Rogers, and W.R.Shadwick, Bcklund transformations and their application, academic Press, New York, 1982.
[35] 郭柏灵，庞小峰，孤立子，北京：科学出版社，1987．
[36] 谷超豪等．孤立子理论与应用，杭州：浙江科技出版社，1990．
[37] H.D.Wahlquist and F.B.Estabrook, Bcklund transformations for solitons of the Korteweg-de Vriu equation. Phys,Rev,Lett, 1973, 31:1386-1390.
[38] J.Weiss, M.Tabor and G.Carnvale, The Painlevé property for partial differential equations. J.Math.Phys., 1983, 24:522.
[39] J.Weiss, Bcklund transformation, Lax pairs, and Schwarzian derivative. J.Math.Phys, 1983, 24:1405.
[40] G.Darboux, Compts Rendus Hebdomadaires des Seances de l'Academie des Sciences, Pairs, 1882, 94:1456.
[41] 谷超豪等，孤立于理论中的Darboux变换及其几何应用，上海：上海科技出版社，1999．
[42] M.Wadati et al., Simple derivation of Bcklund transformation from Pdccati form of inverse method. Prog.Theor.Phys, 1975, 53:418.
[43] C.H.Gu, A unified explicit form of Bcklund transformations for generalized hierarchies of KdV equations. Lett. Math. Phys., 1986, 31:325.
[44] C.H.Gu and Z.X.Zhou, On the Darboux matrices of Bcklund transformations for AKNS systems. Lett. Math.Phys., 1987, 13:179.
[45] V.B.Matveev and E.K.Sklyanin, On Bcklund transformations for many-body systems. J.Phys.A, 1998, 31:2241.
[46] E.K.Sklyanin, Canonicity of Bcklund transformation: r-matrix approach. I. L. D. Faddeev's Seminar on Mathematical Physics. Amer.Math.Soc.Transt.Ser., 2000, 2:201.
[47] Y.B.Zeng, et al., Canonical explicit Bcldund transformations with spectrality for constrained flows of soliton hierarchies. Phys.A, 2002, 303:321.
[48] M.J.Ablowitz et al., A connection between nonlinear evolution equations and ordinary differential equations of P-type. J.Math.Phys., 1980, 21:715.
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