AC=BD及其在非线性方程中的应用
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摘要
本文研究的主要内容为：以构造性的变换和符号计算为工具，来研究非线性波和可积系统中的一些问题：精确解、Liouville可积的发展方程族及其N-Hamilton结构、约束流、Lax表示、r-矩阵、对合系统及对合解。
第二章考虑非线性偏微分方程的精确解的构造：首先给出了C-D对和C-D可积系统的基本理论，然后研究了它们具体的应用：在张鸿庆教授的AC=BD的模式下，利用闫振亚博士的Riccati方程展开法求解Modified Improved Boussinesq方程的精确解，得到了方程的26个解，包括新的孤波解和周期解，其中包含文献[65]中所给出的解。
第三章研究了Lax可积的新的方程族和Liouville可积的N-Hamilton结构方面的问题：利用屠格式计算一个等谱问题，得到了一族Lax可积的广义热传导方程族，并且构造了它的Liouville可积的N-Hamilton结构，给出了此发展方程族的Bargman约束流的r-矩阵，一个新的对合系统和解的对合表示。
第四章研究了Maxwell方程组用外微分形式和余微分形式表示的形式，了解了Maxwell方程组和电荷守恒定律可以互相推得，尤其是由电荷守恒定律可以推得Maxwell方程组，这对于未知的物理定律的发现有着深远的影响。
The major contents in this paper include:with the aid of many types of constructive transformations and symbolic computation, some topics in nonlinear waves and integrable system are studied, including exact solutions,Liouville integrable hie-rarchy,and its N-Hamilton structure, constraint flow, Lax representation, r-matrix,in-volutive system and involutive solutions.
Chapter 2 is devoted to investigating exact solutions of nonlinear wave equations: Firstly, the basic theories of C-D pair and C-D integrable system are presented. Secondly, we choose some examples to illustrate them .Based on Prof. Zhang hong-qing' AC=BD theory, using one of Dr.Yan zhenya' transformations based on one Riccati equation, twenty-six families of exact solutions of Modified Improved Bou-sinnesq equation are found, including new solitary solution and periodic solution, except solutions given by literature[65].
Chapter 3 concentrates on new Lax integrable hierarchies of equations and Liouville integrable N-Hamilton structures. A spectral problem is studied by using the Tu's scheme , its Lax integrable hierarchies of equations and Liouville integrable Hamilton structures are obtained , r-matrix, new involutive system and involutive solutions of Bargman constraint flow of this hierarchy are found.
In chapter 4,Maxwell equations presented in the form of exterior and codiff-erential are studied , and Maxwell equations and charge conservation law can be derived by each other. In particular , the derivation of Maxwell equations from charge conservation law is very important for the discovery of unknown physical law.
第二章考虑非线性偏微分方程的精确解的构造：首先给出了C-D对和C-D可积系统的基本理论，然后研究了它们具体的应用：在张鸿庆教授的AC=BD的模式下，利用闫振亚博士的Riccati方程展开法求解Modified Improved Boussinesq方程的精确解，得到了方程的26个解，包括新的孤波解和周期解，其中包含文献[65]中所给出的解。
第三章研究了Lax可积的新的方程族和Liouville可积的N-Hamilton结构方面的问题：利用屠格式计算一个等谱问题，得到了一族Lax可积的广义热传导方程族，并且构造了它的Liouville可积的N-Hamilton结构，给出了此发展方程族的Bargman约束流的r-矩阵，一个新的对合系统和解的对合表示。
第四章研究了Maxwell方程组用外微分形式和余微分形式表示的形式，了解了Maxwell方程组和电荷守恒定律可以互相推得，尤其是由电荷守恒定律可以推得Maxwell方程组，这对于未知的物理定律的发现有着深远的影响。
The major contents in this paper include:with the aid of many types of constructive transformations and symbolic computation, some topics in nonlinear waves and integrable system are studied, including exact solutions,Liouville integrable hie-rarchy,and its N-Hamilton structure, constraint flow, Lax representation, r-matrix,in-volutive system and involutive solutions.
Chapter 2 is devoted to investigating exact solutions of nonlinear wave equations: Firstly, the basic theories of C-D pair and C-D integrable system are presented. Secondly, we choose some examples to illustrate them .Based on Prof. Zhang hong-qing' AC=BD theory, using one of Dr.Yan zhenya' transformations based on one Riccati equation, twenty-six families of exact solutions of Modified Improved Bou-sinnesq equation are found, including new solitary solution and periodic solution, except solutions given by literature[65].
Chapter 3 concentrates on new Lax integrable hierarchies of equations and Liouville integrable N-Hamilton structures. A spectral problem is studied by using the Tu's scheme , its Lax integrable hierarchies of equations and Liouville integrable Hamilton structures are obtained , r-matrix, new involutive system and involutive solutions of Bargman constraint flow of this hierarchy are found.
In chapter 4,Maxwell equations presented in the form of exterior and codiff-erential are studied , and Maxwell equations and charge conservation law can be derived by each other. In particular , the derivation of Maxwell equations from charge conservation law is very important for the discovery of unknown physical law.
引文
[1]谷超豪等，孤立子理论与应用，杭州：浙江科技出版社，1990．
[2]J. K. Perring and T. H. R. Skyrme, A model unified field eqnation, Nucl. Phys., 1962,31(4), 550-554.
[3]M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transformation, SIAM: Philadelphia,1981~
[4]M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering,Cambridge: Cambridge University Press, 1991.
[5]A. C. Newell, Soliton in mathematics and physics, SIAM: Philadelphia, 1985.
[6]V. B. Matveev and M. A. Salle, Darboux transformation and Solitons, Berlin:Springer, 1991.
[7]郭柏灵，庞小峰，孤立子，北京：科学出版社，1987．
[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]V. A. Venikov, Theory of Similarity and simulation, London:Mac Donald Technical and Scientific, 1969.
[10]H. D. Wahlquist and F. B. Estabrook, Maximally slicing a black hole, Phys. Rev. D., 1973, 7(10),2814-2817.
[11]J. weiss, M. Tabor and G. Carnvale, The painleve property for partical differential equations, J.Math. Phys., 1983, 24(3), 522-526.
[12]J. Weiss, The painleve property for partical differential equations. Ⅱ :Backlund transformation, Lax pairs, and the Schwarzian derivativer, J. Math. Phys., 1983, 24 (6), 1405-1413.
[13]M. Wadati et al., Simple derivation of Backlund transformation from Riccati form of inverse method,Prog. Theor. Phys., 1975, 53(6), 652-656.
[14]R. M. Miura, Method for solving Korteweg - deVries equation, Phys. Rev, Lett., 1%7, 19(19), 1095-1097.
[15]M. J. Ablowitz et al., The evolution of multi-phase models for nonlinear dispersive waves, stud~ Appl.Math., 1970,49(3), 225-238.
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[17]X. B. Hu, Rational solitons of integrable equations via nonlinear superposition formulae, J. Phys.1997, 30(23), 8225-8240.
[18]X, B. Hu et al., Lax Pairs and Bcklund Transfbrmations for a Coupled Ramani Equation and its Related System, Apl. Math. Lett., 2000, 13 (6), 45-48.
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[23]H. Q. Zhang and Engui Fan, Applications of Mechanical Methods to Partial Differential Equations,Mathematics Mechanization and Application , London :Academic Press, 2000, 409-432.
[24]Chen, YuFu, Zhang hongqing, A generalization of recursion operators of differential equations.Appl. Math. Mech, 1999, 20(11) , 1230-1236.
[25]陈玉福，大连理工大学博士论文，1999．
[26]Zhang hongqing, Chen yufu, A reduction method for overdetermined systems of equations. (Chinese)J, Dalian Univ. Technol. , 1999, 39(2) , 137—143.
[27]Li Biao, Chen Yong, Zhang hongqing ,Explicit exact solutions for new general two-dimensional KdV-type and two-dimensional KdV-Burgers-type equations with nonlinear terms of any order, J. Phys. A, 2002, 35(39),825:3—8265.
[28]M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A, 19996, 213(5), 279-287.
[29]Y. T. Gao and B. Tian, new families of exact solutions to the integrable dispersive long wave equations in (2+1)dimensional spaces, J. Phys. A, 1996, 29(11), 2895-2903.
[30]E. G. Fan and H. Q. Zhang, A note on the homogeneous balance method, Phys. Lett. A, 1998, 246(5), 403-406.
[31]E. G. Fan, Integrable system of derivative nonlinear schrodinger type and their multi-Hamiltonian structure, J. Phys. Math. Gen., 2001, 34(3), 513-519.
[32]C .T .Yea, A simple transformation for nonlinear waves ,Phys. Lett. A, 1996, 224(1), 77-84.
[33]Z. Y. Yan and H. Q. Zhang, New explicit and exact traveling waves solutions for a system of variant Boussinesq equations in mathematical physics ,Phys. Lett. A, 1999,252(6), 291-296.
[34]P. Rosenau et al, Compactions :solitons with finite wave length, Phys. Rev. Lett., 1993,70(5), 564-567.
[35]范恩贵，大连理工大学博士论文，1998．
[36]V. I. Arnold, Mathematical method of classical mechanics, Berlin: Springer, 1978.
[37]G. Z. Tu, The trace identity, a powerful tool for construting the Hamiltonian structure of integrable systems, J. Math. Phys., 1989, 30(2), 330-338;On Liouville integrablility of zero-curvature equations and the Yang hierarchy, J. Phys. A, 1989, 22(13), 2375-2392.
[38]马文秀，一个新的Liouville可积的广义Hamilton方程族及其约化，数学年刊，1992，13A(1)，115-123．
[39]C. W. Cao, A classical integrable system and the involutive solution of the Kdv equation, Acta Math.Sci. New Ser., 7(3), 1991, 216-223; Nonlinearization of the Lax system for AKNS hierarchy, Sci. China A,1990, 33 (5) , 528-536.
[40]C. W. Cao and X. G. Geng, A nonconfical generator of involutive systems and three associated soltion hierarchies, J. Math. Phys., 1991,32(9), 2323-2328.
[41]Y. B. Zeng and Y. S. Li, The constraints of potential and the finite-dimensional integrable systems,J. Math. Phys., 1989,30(8), 1679-1689.
[42]Y. B. Zeng, Phys. Lett. A, An approach to the deduction of the finite-dimensional integrability from the infinite-dimensional integrability, 1991, 160 (6) , 541-547 ;The highter-order constraints and integrable systems associates with the Boussinesq equation, Chin. Sci. Bull., 37(23), 1992, 1937-1942.
[43]斯仁道尔吉，由伴随坐标得到的Dirac族的可积约束流，数学学报，1999，42(5)，845-852．
[44]J. C. Eilkerck et al. Linear r~-matrix algebra for system separable in parabolic coordinates, Phys.Lett. A, 1993, 180(3), 208-214.
[45]L. D. Faddeev, L. A. Takhtajan, Hamiltonian method in the theory of solitons, Berlin:$pringer-Verlag, 1987.
[46]Y. B. Zeng, Separation of variables for the constrained flows, J. Math. Phys., 1997, 38(1), 321-329.
[47]Y. B. Zeng and R. L. Lin, Families of dynamical r~-matrices and Jacobi inverse problem for nonlinear evolution equations, J. Math. Phys., 1998, 39(11), 5964-5983.
[48]闫振亚，大连理工大学博士论文，2002．
[49]吴文俊，初等几何定理判定问题与机械化证明，中国科学，1977，6A，507-516；走向几何的机械化，数学物理学报，1982，2(2)，125-135．
[50]X. D. Sun, S. K. Wang, K. Wu, Wu algorithm and eight-vertex solutions of colored Yang-Baxer equation,Commun. Theor. Phys. , 26(2), 1996, 213-218. Wu algorithm and solutions of Yang-Baxer equation, 1995,23(1),125-128.
[51]X. D. Sun, S. K .Wang, K. Wu, Classification of six-vertex-type of the colored Yang-Baxer quation, J .Math. Phys., 1995,36(10), 6043-6063.
[52]朱思铭，施齐焉，现代数学和力学，上海大学出版社，1997：482．
[53]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．
[54]Z. B. Li, Exact solitary wave solutions of nonlinear evolution equations, Mathematics Mechanization and Application , London:Academic Press, 2000, 389-408.
[55]Z. B. Li and H. Shi, Exact solutions for Belousor-Zhabotinskii Reaction-diffusion system, Appl.Math- JCU, 1996, 11B, 1-6.
[56]Yao, Ruo-xia, Li Zhi-Bin, Exact analytical solutions to a set of coupled nonlinear differential equations using symbolic computation, Lecture Notes Ser. Comput., 9 World Sci. Publishing, River Edge, NJ,2001, 201-210.
[57]朝鲁，大连理工大学博士论文，1997．
[58]H. Q. Zhang, C-D integrable system and computer aided solver for differential equations, Lecture Notes Ser. Comput.,9 World Sci. Publishing River Edge, NJ, 2001, 221-226.
[59]张鸿庆，电动力学基本方程的多余阶次与恰当解，大连理工大学学报，1981，20(1)，11-18．
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[63]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，1994，34，249-255．
[64]张解放，长水波近似方程的多孤子解，物理学报，1998，47(9)，1417-1420．
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[67]G. Z. Tu, An extension of a theorem on gradient of conserved densitities of integrable systems ,Northeastern Math. J., 1990, 6(1), 26-32.
[68]张玉峰，大连理工大学博士论文，2002．
[69]现代分析讲义，张鸿庆，1998．
[70]余扬政，冯承天，物理学中的几何方法，高等教育出版社，1998．
[71]姜东梅，张鸿庆．Modified Improved Bousinnesq方程的显式精确解，甘肃工业大学学报，2002，28(3)，104-106．
[72]李翊神，孤子与可积系统，上海：上海科技出版社，1999．
[2]J. K. Perring and T. H. R. Skyrme, A model unified field eqnation, Nucl. Phys., 1962,31(4), 550-554.
[3]M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transformation, SIAM: Philadelphia,1981~
[4]M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering,Cambridge: Cambridge University Press, 1991.
[5]A. C. Newell, Soliton in mathematics and physics, SIAM: Philadelphia, 1985.
[6]V. B. Matveev and M. A. Salle, Darboux transformation and Solitons, Berlin:Springer, 1991.
[7]郭柏灵，庞小峰，孤立子，北京：科学出版社，1987．
[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]V. A. Venikov, Theory of Similarity and simulation, London:Mac Donald Technical and Scientific, 1969.
[10]H. D. Wahlquist and F. B. Estabrook, Maximally slicing a black hole, Phys. Rev. D., 1973, 7(10),2814-2817.
[11]J. weiss, M. Tabor and G. Carnvale, The painleve property for partical differential equations, J.Math. Phys., 1983, 24(3), 522-526.
[12]J. Weiss, The painleve property for partical differential equations. Ⅱ :Backlund transformation, Lax pairs, and the Schwarzian derivativer, J. Math. Phys., 1983, 24 (6), 1405-1413.
[13]M. Wadati et al., Simple derivation of Backlund transformation from Riccati form of inverse method,Prog. Theor. Phys., 1975, 53(6), 652-656.
[14]R. M. Miura, Method for solving Korteweg - deVries equation, Phys. Rev, Lett., 1%7, 19(19), 1095-1097.
[15]M. J. Ablowitz et al., The evolution of multi-phase models for nonlinear dispersive waves, stud~ Appl.Math., 1970,49(3), 225-238.
[16]R. Hirota ,Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys.Rev. Lett., 1971,27(18), 1192-1194;
[17]X. B. Hu, Rational solitons of integrable equations via nonlinear superposition formulae, J. Phys.1997, 30(23), 8225-8240.
[18]X, B. Hu et al., Lax Pairs and Bcklund Transfbrmations for a Coupled Ramani Equation and its Related System, Apl. Math. Lett., 2000, 13 (6), 45-48.
[19]张鸿庆，弹性力学方程组的一般解的统一理论，大连理工大学学报，1978，18，23-27．线性算子方程组一般解的代数结构，力学学报，特刊，1981，152-161．
[20]张鸿庆，王震宇，胡海昌解的完备性与逼近性，科学通报，1985，30(5)，332-334．
[21]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(6)，700-707．构造壳体问题基本解的一般方法，科学通报，1993，38(7)，671-72．
[22]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．
[23]H. Q. Zhang and Engui Fan, Applications of Mechanical Methods to Partial Differential Equations,Mathematics Mechanization and Application , London :Academic Press, 2000, 409-432.
[24]Chen, YuFu, Zhang hongqing, A generalization of recursion operators of differential equations.Appl. Math. Mech, 1999, 20(11) , 1230-1236.
[25]陈玉福，大连理工大学博士论文，1999．
[26]Zhang hongqing, Chen yufu, A reduction method for overdetermined systems of equations. (Chinese)J, Dalian Univ. Technol. , 1999, 39(2) , 137—143.
[27]Li Biao, Chen Yong, Zhang hongqing ,Explicit exact solutions for new general two-dimensional KdV-type and two-dimensional KdV-Burgers-type equations with nonlinear terms of any order, J. Phys. A, 2002, 35(39),825:3—8265.
[28]M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A, 19996, 213(5), 279-287.
[29]Y. T. Gao and B. Tian, new families of exact solutions to the integrable dispersive long wave equations in (2+1)dimensional spaces, J. Phys. A, 1996, 29(11), 2895-2903.
[30]E. G. Fan and H. Q. Zhang, A note on the homogeneous balance method, Phys. Lett. A, 1998, 246(5), 403-406.
[31]E. G. Fan, Integrable system of derivative nonlinear schrodinger type and their multi-Hamiltonian structure, J. Phys. Math. Gen., 2001, 34(3), 513-519.
[32]C .T .Yea, A simple transformation for nonlinear waves ,Phys. Lett. A, 1996, 224(1), 77-84.
[33]Z. Y. Yan and H. Q. Zhang, New explicit and exact traveling waves solutions for a system of variant Boussinesq equations in mathematical physics ,Phys. Lett. A, 1999,252(6), 291-296.
[34]P. Rosenau et al, Compactions :solitons with finite wave length, Phys. Rev. Lett., 1993,70(5), 564-567.
[35]范恩贵，大连理工大学博士论文，1998．
[36]V. I. Arnold, Mathematical method of classical mechanics, Berlin: Springer, 1978.
[37]G. Z. Tu, The trace identity, a powerful tool for construting the Hamiltonian structure of integrable systems, J. Math. Phys., 1989, 30(2), 330-338;On Liouville integrablility of zero-curvature equations and the Yang hierarchy, J. Phys. A, 1989, 22(13), 2375-2392.
[38]马文秀，一个新的Liouville可积的广义Hamilton方程族及其约化，数学年刊，1992，13A(1)，115-123．
[39]C. W. Cao, A classical integrable system and the involutive solution of the Kdv equation, Acta Math.Sci. New Ser., 7(3), 1991, 216-223; Nonlinearization of the Lax system for AKNS hierarchy, Sci. China A,1990, 33 (5) , 528-536.
[40]C. W. Cao and X. G. Geng, A nonconfical generator of involutive systems and three associated soltion hierarchies, J. Math. Phys., 1991,32(9), 2323-2328.
[41]Y. B. Zeng and Y. S. Li, The constraints of potential and the finite-dimensional integrable systems,J. Math. Phys., 1989,30(8), 1679-1689.
[42]Y. B. Zeng, Phys. Lett. A, An approach to the deduction of the finite-dimensional integrability from the infinite-dimensional integrability, 1991, 160 (6) , 541-547 ;The highter-order constraints and integrable systems associates with the Boussinesq equation, Chin. Sci. Bull., 37(23), 1992, 1937-1942.
[43]斯仁道尔吉，由伴随坐标得到的Dirac族的可积约束流，数学学报，1999，42(5)，845-852．
[44]J. C. Eilkerck et al. Linear r~-matrix algebra for system separable in parabolic coordinates, Phys.Lett. A, 1993, 180(3), 208-214.
[45]L. D. Faddeev, L. A. Takhtajan, Hamiltonian method in the theory of solitons, Berlin:$pringer-Verlag, 1987.
[46]Y. B. Zeng, Separation of variables for the constrained flows, J. Math. Phys., 1997, 38(1), 321-329.
[47]Y. B. Zeng and R. L. Lin, Families of dynamical r~-matrices and Jacobi inverse problem for nonlinear evolution equations, J. Math. Phys., 1998, 39(11), 5964-5983.
[48]闫振亚，大连理工大学博士论文，2002．
[49]吴文俊，初等几何定理判定问题与机械化证明，中国科学，1977，6A，507-516；走向几何的机械化，数学物理学报，1982，2(2)，125-135．
[50]X. D. Sun, S. K. Wang, K. Wu, Wu algorithm and eight-vertex solutions of colored Yang-Baxer equation,Commun. Theor. Phys. , 26(2), 1996, 213-218. Wu algorithm and solutions of Yang-Baxer equation, 1995,23(1),125-128.
[51]X. D. Sun, S. K .Wang, K. Wu, Classification of six-vertex-type of the colored Yang-Baxer quation, J .Math. Phys., 1995,36(10), 6043-6063.
[52]朱思铭，施齐焉，现代数学和力学，上海大学出版社，1997：482．
[53]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．
[54]Z. B. Li, Exact solitary wave solutions of nonlinear evolution equations, Mathematics Mechanization and Application , London:Academic Press, 2000, 389-408.
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