一类非线性偏微分方程若干求精确解方法的研究
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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”理论的指导下,研究在流体
力学、空气动力学、等离子体物理、生物物理和化学物理等现代科学技术中引出的非线性偏微
分方程的若干求精确解的方法。
第一章介绍了数学机械化的思想与应用的情况;回顾了孤立子研究的历史与发展以及非线
性偏微分方程精确解的若干构造性方法,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。
第二章在“AC=BD”统一理论框架下考虑非线性偏微分方程(组)精确解的构造。给出
了“AC=BD”理论的基本思想和应用,通过具体的变换给出了构造C-D对的算法。
第三章主要介绍了我们推广的一种直接求解方法--广义代数方法。以(2+1) 维色散
长波方程为例,说明了广义代数方法具体的应用。推广后的方法可以获得非线性偏微分方程
(组)的更多类型的精确解(孤波解、类孤波解、周期解、类周期解、有理解)。
第四章考虑非线性偏微分方程的Painleve性质和Backlund变换。介绍了Painleve奇性分
析的一般原理,利用WTC方法证明了(2+1) 维Boussinesq方程具有Painleve性质,并经截
断展开原理获得了方程的Backlund变换;对Backlund变换作了简单介绍,通过对(2+1)
维Boussinesq方程的种子解作适当的未知函数替换,进一步发展了Backlund变换,并得到了方
程形式丰富的精确解(类孤子解,有理解)。
In this dissertation, by applying the ideas of the mathematics mechanization, under the instruction of the AC=BD theory of Professor Zhang Hongqing, considers some methods seeking exact solutions for the nonlinear partial differential equation(s) arising from the fields of fluid mechanics, aerodynamics, plasma physics, biophysics and chemical physics.Chapter 1 of this dissertation is devoted to investigating the theory and application of mathematics mechanization; Reviewing the history and development of the soliton theory and the construction of the Nonlinear partial differential equation. In addition, some achievements on the subject domestic and abroad are presented.Chapter 2 concerns the construction of exact solutions of nonlinear partial differential equation(s) under the uniform frame work of AC=BD theory. The basic theory and application about AC=BD model and the construction of the operators of C and D are introduced.Chapter 3 is devoted mainly to generalized algebraic method, which is a direct method. the generalized algebraic method is shown to solve (2+l)-dimensional dispersive long wave equations, and which can obtain abundant exact solution(including solitary solutions, soliton-like solutions, periodic solutions, periodic-like solutions and rational solutions) of nonlinear partial differential equation(s).Chapter 4 mainly deals with the Painleve property and Backlund transformation of nonlinear partial differential equation. The general theory of Painleve singular analysis is discussed, (2+l)-dimensional Boussinesq equation's Painleve property is shown to pass the Painleve test by using WTC method, and its Backlund transformation is obtained through Painleve truncating expansion; Backlund transformation is discussed simply and is further expanded, (2+1)-dimensional Boussinesq equation's abundant exact solution(including soliton-like solutions, rational solutions) is obtained by replacing the seed solution with unknown function.
力学、空气动力学、等离子体物理、生物物理和化学物理等现代科学技术中引出的非线性偏微
分方程的若干求精确解的方法。
第一章介绍了数学机械化的思想与应用的情况;回顾了孤立子研究的历史与发展以及非线
性偏微分方程精确解的若干构造性方法,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。
第二章在“AC=BD”统一理论框架下考虑非线性偏微分方程(组)精确解的构造。给出
了“AC=BD”理论的基本思想和应用,通过具体的变换给出了构造C-D对的算法。
第三章主要介绍了我们推广的一种直接求解方法--广义代数方法。以(2+1) 维色散
长波方程为例,说明了广义代数方法具体的应用。推广后的方法可以获得非线性偏微分方程
(组)的更多类型的精确解(孤波解、类孤波解、周期解、类周期解、有理解)。
第四章考虑非线性偏微分方程的Painleve性质和Backlund变换。介绍了Painleve奇性分
析的一般原理,利用WTC方法证明了(2+1) 维Boussinesq方程具有Painleve性质,并经截
断展开原理获得了方程的Backlund变换;对Backlund变换作了简单介绍,通过对(2+1)
维Boussinesq方程的种子解作适当的未知函数替换,进一步发展了Backlund变换,并得到了方
程形式丰富的精确解(类孤子解,有理解)。
In this dissertation, by applying the ideas of the mathematics mechanization, under the instruction of the AC=BD theory of Professor Zhang Hongqing, considers some methods seeking exact solutions for the nonlinear partial differential equation(s) arising from the fields of fluid mechanics, aerodynamics, plasma physics, biophysics and chemical physics.Chapter 1 of this dissertation is devoted to investigating the theory and application of mathematics mechanization; Reviewing the history and development of the soliton theory and the construction of the Nonlinear partial differential equation. In addition, some achievements on the subject domestic and abroad are presented.Chapter 2 concerns the construction of exact solutions of nonlinear partial differential equation(s) under the uniform frame work of AC=BD theory. The basic theory and application about AC=BD model and the construction of the operators of C and D are introduced.Chapter 3 is devoted mainly to generalized algebraic method, which is a direct method. the generalized algebraic method is shown to solve (2+l)-dimensional dispersive long wave equations, and which can obtain abundant exact solution(including solitary solutions, soliton-like solutions, periodic solutions, periodic-like solutions and rational solutions) of nonlinear partial differential equation(s).Chapter 4 mainly deals with the Painleve property and Backlund transformation of nonlinear partial differential equation. The general theory of Painleve singular analysis is discussed, (2+l)-dimensional Boussinesq equation's Painleve property is shown to pass the Painleve test by using WTC method, and its Backlund transformation is obtained through Painleve truncating expansion; Backlund transformation is discussed simply and is further expanded, (2+1)-dimensional Boussinesq equation's abundant exact solution(including soliton-like solutions, rational solutions) is obtained by replacing the seed solution with unknown function.
引文
[1] 吴文俊.吴文俊论数学机械化.山东:山东教育出版社,1996.
[2] Wang.ML. Solitary wave solutions for variant Boussinesq equations.Phys.Lett.A, 1996, 212: 353.
[3] 石赫.机械化数学引论.湖南:湖南教育出版社,1998.
[4] Fan.EG, Dai.HH. A direct approach with computerized symbolic computation for finding a series of traveling waves to nonlinear equations. Comput Phys Commun, 2003, 153: 17-30.
[5] Fan. EG, Hon.YC. A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves. Chaos,Solitons and Fractals, 2003, 15: 559-566.
[6] Fan.EG. Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics. Chaos,Solitons and Fractals, 2003, 16: 819-839.
[7] Fan.EG. Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems. Physics Letters A, 2002, 300: 243-249.
[8] 王明亮,李志斌,周宇斌.齐次平衡原理及其应用.兰州大学学报(自然科学版),1999,35(3) :8.
[9] 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.
[10] Wang ML, Li.ZB. Proc.Of the 1994 Beijing Symposium on nonlinear evolution equations and infinite dimensional dynamics systems. Zhongshan University Press, 1995, 181-185.
[11] Li ZB, Wang ML. Travelling wave solutions to the two-dimensional KdV-Burgers equaiton. J.Phys.A, 1993, 26: 6027-6031.
[12] 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:(博士学位论文).大连:大连理工大学,1998.
[13] 朝鲁.微分方程(组)对称向量的吴-微分特征列算法及其应用.数学物理学报,1999,19(3) :326-332.
[14] 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.
[15] 朱思铭,施齐焉.符号计算与吴-消元法在Painleve检验分析中的应用.现代数学和力学(MMM-VII), 上海:上海大学出版社,1997:482-485.
[16] 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.
[17] Xie.FD, Chen.Y. An algorithmic method in Painleve analysis of PDE. Compu.Phys.Commun, 2003, 154: 197-204.
[18] 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.
[19] 吕卓生.计算微分方程对称与精确解的机械化算法及实现:(博士学位论文).大连:大连理工大学, 2003.
[20] Yan.ZY. New families of nontravelling wave solutions to a new (3+1) -dimensional potential-YTSF equation. Phys.Lett.A, 2003, 218: 78-83.
[21] 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.
[22] Weiss.J, Tabor.M, Carnvale.G. The Painleve property for partial differential equations. J. Math. Phys, 1983, 24: 522.
[23] Weiss.J. The Painleve property for partial differential equations. Ⅱ. Backlund transformation, Lax pairs, and the Schwarzian derivative. J. Math. Phys, 1983, 24: 1405.
[24] Gu.CH. A unified explicit form of Backlund transformations for generalized hierarchies of KdV equations, Lett. Math. Phys, 1986, 31: 325.
[2
[25] Gu.CH, Zhou.ZX. On the Darboux matrices of Backlund transformations for AKNS systems. Let. Math. Phys, 1987, 13: 179.
[26] 谷超豪等.孤立子理论与应用.杭州:浙江科技出版社,1990.
[27] Fan EG. Two new applications of the homogeneous balance method. Phys.Lett.A, 2000, 265: 353-357.
[28] 范恩贵,张鸿庆.二类变式Boussinesq的对称性约化和精确解.数学物理学报,1999,19(4) .
[29] Fan EG, Zhang HQ. A note on the homogeneous balance method. Phys.Lett.A, 1998, 246: 403-406.
[30] Fan EG, Zhang HQ. New exact solutions to a system of coupled KdV equations. Phys.Lett.A, 1998, 245: 389-392.
[31] 张本样,孙博文.社会科学非线性方法论.哈尔滨:哈尔滨出版社,1997.
[32] Hu.XB. Rational solutions of integrable equations via nonlinear superposition formulae. J.Phys.A, 1997, 30: 8225.
[33] 曾云波.与Al相联系的半Toda方程的Lax对与Backlund变换.数学学报,1992,35:454;
[34] 曾云波.递推算子与Painleve性质.数学年刊,1991,12A:778.
[35] 郭柏灵,庞小峰.孤立子.北京:科学出版社,1987.
[36] 李翊神.孤子与可积系统.上海:上海科技出版社,1999.
[37] Wang.ML. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. Lett. A, 1996, 216: 67.
[38] Wang.ML, Wang.YM, Zhou.YB. An auto-Backlund transformation and exact solutions to a generalized KdV equation with variable coefficients and their applications. Phys. Lett. A, 2002, 303: 45.
[39] Li.ZB. Exact solutions for two nonlinear equations. Adv. in Math., 1997, 26: 129.
[40] Gao.YT, Tian.B. New families of exact solutions to the integrable dispersive long wave equaitons in 2+1-dimensional spaces. J.Phys.A, 1986, 29: 2895.
[41] 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.
[42] 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.
[43] 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.
[44] 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.
[45] Zhang HQ. Proceedings of 5th Asian Symposium on Computer Mathematics, World Scientific, Singapore, 2001, pp210.
[46] 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.
[47] 闫振亚,张鸿庆.具有三个任意函数的变系数Kdv-Mkdv方程的精确类孤子解.物理学报,1999,48: 1957.
[48] Ablowitz.MJ, Clarkson.PA. Solitons,nonlinear evolution equations and inverse scattering. Cambrideg: Cambridge University. Press, 1991.
[49] Zheng XD, Chen.Y, Zhang.HQ. Generalized extended tanh-function method and its application to (1+1) -dimensional dispersive long wave equation. Physics Letters A, 2003, 311: 145157.
[50] 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.
[51] 谢福鼎.Wu-Ritt消元法在偏微分代数方程中的应用:(博士学位论文).大连:大连理工大学,1999.
[52] 贾屹峰.偏微分方程和偏微分方程组的变化解法及约化:(硕士学位论文).大连:大连理工大学,2001.
[53] Zhang HQ, Chen.YF. Proceeding of the 3rd ACM. Lanzhou University Press, 1998, 147.
[54] Yan.ZY, Zhang.HQ. Explicit and exact solutions for the generalized reaction Duffing equation. Commun, in Nonlinear Science and Numerical Simulation, 1999, 4: 146.
[55] 曾昕,张鸿庆.(2+1) 维色散长波方程的新的类孤子解.物理学报,2005,54(2) .
[56] Thangavel.A, Ambigapathy.U, Kuppusamy.P. The Generalisation of Integrable (2+1) -Dimensional Dispersive Long-Wave Equations. J.Phys.Soc.Jpn, 1997, 66: 1288-1290.
[57] Lou.SY. Symmetries and algebras of the integrable dispersive long wave equations in (2+1) -dimensional spaces. J.Phys.A:Math.Gen, 1994, 27: 3235-3243.
[58] Yan.ZY, Zhang.HQ. Multiple soliton-like and periodic-like solutions to the generalization of integrable (2+1) -dimensional dispersive long-wave equations. J.Phys.Soc.Jpn, 2002, 71: 437-442.
[59] Allen.MA, Rowlands.G. On the transverse instabilities of solitary waves. Phys.Lett.A, 1997, 235: 145.
[60] Senthilvelan.M. On the transverse instabilities of solitary waves. Comput.Math.Appl, 2001, 123: 381.
[61] Chen.Y, Yan.ZY, Zhang.HQ. New explicit solitary wave solutions for (2+1) -dimensional Boussinesq equation and (3+1) -dimensional KP equation Phys.Lett.A, 2003, 307: 107-113.
[62] 刘式适,刘式达.物理学中的非线性方程.北京:北京大学物理学丛书,2000.
[63] 曾昕,张鸿庆.(2+1) 维Boussinesq方程的Backlund变换与精确解.物理学报,2005,54(4) .
[2] Wang.ML. Solitary wave solutions for variant Boussinesq equations.Phys.Lett.A, 1996, 212: 353.
[3] 石赫.机械化数学引论.湖南:湖南教育出版社,1998.
[4] Fan.EG, Dai.HH. A direct approach with computerized symbolic computation for finding a series of traveling waves to nonlinear equations. Comput Phys Commun, 2003, 153: 17-30.
[5] Fan. EG, Hon.YC. A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves. Chaos,Solitons and Fractals, 2003, 15: 559-566.
[6] Fan.EG. Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics. Chaos,Solitons and Fractals, 2003, 16: 819-839.
[7] Fan.EG. Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems. Physics Letters A, 2002, 300: 243-249.
[8] 王明亮,李志斌,周宇斌.齐次平衡原理及其应用.兰州大学学报(自然科学版),1999,35(3) :8.
[9] 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.
[10] Wang ML, Li.ZB. Proc.Of the 1994 Beijing Symposium on nonlinear evolution equations and infinite dimensional dynamics systems. Zhongshan University Press, 1995, 181-185.
[11] Li ZB, Wang ML. Travelling wave solutions to the two-dimensional KdV-Burgers equaiton. J.Phys.A, 1993, 26: 6027-6031.
[12] 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:(博士学位论文).大连:大连理工大学,1998.
[13] 朝鲁.微分方程(组)对称向量的吴-微分特征列算法及其应用.数学物理学报,1999,19(3) :326-332.
[14] 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.
[15] 朱思铭,施齐焉.符号计算与吴-消元法在Painleve检验分析中的应用.现代数学和力学(MMM-VII), 上海:上海大学出版社,1997:482-485.
[16] 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.
[17] Xie.FD, Chen.Y. An algorithmic method in Painleve analysis of PDE. Compu.Phys.Commun, 2003, 154: 197-204.
[18] 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.
[19] 吕卓生.计算微分方程对称与精确解的机械化算法及实现:(博士学位论文).大连:大连理工大学, 2003.
[20] Yan.ZY. New families of nontravelling wave solutions to a new (3+1) -dimensional potential-YTSF equation. Phys.Lett.A, 2003, 218: 78-83.
[21] 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.
[22] Weiss.J, Tabor.M, Carnvale.G. The Painleve property for partial differential equations. J. Math. Phys, 1983, 24: 522.
[23] Weiss.J. The Painleve property for partial differential equations. Ⅱ. Backlund transformation, Lax pairs, and the Schwarzian derivative. J. Math. Phys, 1983, 24: 1405.
[24] Gu.CH. A unified explicit form of Backlund transformations for generalized hierarchies of KdV equations, Lett. Math. Phys, 1986, 31: 325.
[2
[25] Gu.CH, Zhou.ZX. On the Darboux matrices of Backlund transformations for AKNS systems. Let. Math. Phys, 1987, 13: 179.
[26] 谷超豪等.孤立子理论与应用.杭州:浙江科技出版社,1990.
[27] Fan EG. Two new applications of the homogeneous balance method. Phys.Lett.A, 2000, 265: 353-357.
[28] 范恩贵,张鸿庆.二类变式Boussinesq的对称性约化和精确解.数学物理学报,1999,19(4) .
[29] Fan EG, Zhang HQ. A note on the homogeneous balance method. Phys.Lett.A, 1998, 246: 403-406.
[30] Fan EG, Zhang HQ. New exact solutions to a system of coupled KdV equations. Phys.Lett.A, 1998, 245: 389-392.
[31] 张本样,孙博文.社会科学非线性方法论.哈尔滨:哈尔滨出版社,1997.
[32] Hu.XB. Rational solutions of integrable equations via nonlinear superposition formulae. J.Phys.A, 1997, 30: 8225.
[33] 曾云波.与Al相联系的半Toda方程的Lax对与Backlund变换.数学学报,1992,35:454;
[34] 曾云波.递推算子与Painleve性质.数学年刊,1991,12A:778.
[35] 郭柏灵,庞小峰.孤立子.北京:科学出版社,1987.
[36] 李翊神.孤子与可积系统.上海:上海科技出版社,1999.
[37] Wang.ML. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. Lett. A, 1996, 216: 67.
[38] Wang.ML, Wang.YM, Zhou.YB. An auto-Backlund transformation and exact solutions to a generalized KdV equation with variable coefficients and their applications. Phys. Lett. A, 2002, 303: 45.
[39] Li.ZB. Exact solutions for two nonlinear equations. Adv. in Math., 1997, 26: 129.
[40] Gao.YT, Tian.B. New families of exact solutions to the integrable dispersive long wave equaitons in 2+1-dimensional spaces. J.Phys.A, 1986, 29: 2895.
[41] 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.
[42] 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.
[43] 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.
[44] 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.
[45] Zhang HQ. Proceedings of 5th Asian Symposium on Computer Mathematics, World Scientific, Singapore, 2001, pp210.
[46] 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.
[47] 闫振亚,张鸿庆.具有三个任意函数的变系数Kdv-Mkdv方程的精确类孤子解.物理学报,1999,48: 1957.
[48] Ablowitz.MJ, Clarkson.PA. Solitons,nonlinear evolution equations and inverse scattering. Cambrideg: Cambridge University. Press, 1991.
[49] Zheng XD, Chen.Y, Zhang.HQ. Generalized extended tanh-function method and its application to (1+1) -dimensional dispersive long wave equation. Physics Letters A, 2003, 311: 145157.
[50] 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.
[51] 谢福鼎.Wu-Ritt消元法在偏微分代数方程中的应用:(博士学位论文).大连:大连理工大学,1999.
[52] 贾屹峰.偏微分方程和偏微分方程组的变化解法及约化:(硕士学位论文).大连:大连理工大学,2001.
[53] Zhang HQ, Chen.YF. Proceeding of the 3rd ACM. Lanzhou University Press, 1998, 147.
[54] Yan.ZY, Zhang.HQ. Explicit and exact solutions for the generalized reaction Duffing equation. Commun, in Nonlinear Science and Numerical Simulation, 1999, 4: 146.
[55] 曾昕,张鸿庆.(2+1) 维色散长波方程的新的类孤子解.物理学报,2005,54(2) .
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