摘要
本文研究内容主要涉及孤立子理论中精确求解非线性发展方程的Backlund变换法,Painlevé截断展开法,CK直接约化法等几个方面。引言中主要介绍了孤立子概念的产生、孤立波的发展及其意义和本文的主要工作。第二章介绍了Painlevé分析法及其新进展。第三章利用Painlevé分析法得到了修正KdV方程的递推算子及其共振点。第四章应用Painlevé截断展开法求解了几个非线性发展方程。第五章在修改文献[57]几处笔误的基础上应用CK直接约化法求解了一类描述方向上存在可变剪切流动的长波变系数Boussinesq方程的相似解,这种解不同于用Painlevé截断展开法求出的解。
This dissertation mainly studies the methods in soliton theory for finding exact solutions of nonlinear evolution equations, such as the Backlund transformation method, truncated Painlevéexpansion method, the CK direct method and so on.
The first chapter introduces the concept of solitons, its developments and meanings with our main work.
In the second chapter, the Painlevéproperty and its new development are introduced.
In the third chapter, the recursion operator and the total resonances of the modified KdV equation are obtained by the use of Painlevéanalysis.
In the forth chapter, exact travelling wave solutions to several nonlinear evolution equations are obtained by truncated Painlevéexpansion method .
In the fifth chapter, the CK direct method is extended to reduce a variable–coefficient Boussinesq equation and some new similarity solutions are found. A few mistakes in [57] are also corrected.
引文
[1] Russell J S. Report on Waves [M]. Rep.14th. Meet. Brit. Assoc. Adv. Sic. York, London: John Murray 1844, 311-390.
[2] Korteweg D J, de Vries G. On the change of from long advancing in a rectangular canal and on a new type of long stationary waves [J]. phil. Mag., 1895, 39: 422-443.
[3] Fermi A, Pasta J, Ulam S. Studies of nonlinear problems, I, Los Alamos Report, LA 1940, 1955.
[4] Zabusky N J,Kruskal M D .Interaction of solutions in a collisions plasma and the recurrence of initial states [J]. Phys. Rev. Lett., 1965, 15: 240-243.
[5] Ablowitz M J, Clarkson P A. Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Notes Series 149, Cambridge University Press, 1991.
[6] Gu C H, Guo B L, Cao C W, et al. Soliton Theory and its Aapplications. New York, Sprinnger-Verlag, 1995.
[7] Infeld E, Rowlands G. Nonlinear Waves, Solitons and Chaos.2nd Edition. Cambridge, Endland:Cambridge University Press, 2000.
[8] 楼森岳,唐晓艳.非线性数学物理方法.科学出版社,2006.
[9] 陈登远.孤子引论.科学出版社,2006.
[10] 李羽神.孤子与可积系统.上海科技教育出版社,1999.
[11] 屠规彰.一族新的可积系统及其 Hamilton 结构.中国科学 A,1988,12:1243-1252.
[12] 张玉峰,张鸿庆.一族 Liouvill 可积系统及其约束流的 Lax 表示、Darboux 变换.应用数学和力学,2002,23(11):23-30.
[13] 徐桂琼.非线性演化方程的精确解与可积性及符号计算研究.华东师范大学博士学位论文,2004.
[14] Painlevé P. Sur les équations différentielles d, ordre suéprieur dont l’intégerale n’admet qu,un nombre fini determinations, C.R.Acad.Sc. Paris, 1888.
[15] Gambier B. Sur les équations différentielles dont l’intégerale general est uniforme. Paris: C.R. Acad. Sc., 1096, 142: 266-269, 1403-1406, 1497-1500:1906, 143: 741-743.
[16] Fuchs R. Sur quelques équations différentielles linéaires du second ordre. C.R.Acad. Sc.Paris, 1905, 141:555-558.
[17] Painlevé P. Sur les équations différentielles du second ordre et d’ordre suéprieur dont l’intégerale générale est uniforme. Acad Math., 1902, 25:1-85.
[18] 何育赞,袁文俊,李叶舟. 潘勒韦方程解析理论讲义.科学出版社,2005.
[19] Ablowitz M J,Kaup D J, Newell A C, Segur H. The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math. 1974, 53:249-315.
[20] Ablowitz M J,Ramani A,Segur H. A connection between evolution equations and ordinary differential of P–type,Ⅰ,Ⅱ, J. Math. Phys 1980, 21:715-721;1006 -1015.
[21] Weiss J, Tabor M and Carneval G .The Painlevé property of partial differential equations, J.Math. Phys.1983, 24:522-526.
[22] Weiss J. The Painlevé property of partial differentialⅡ:equations Backlund transformation, Lax pairs, and the Schwarzian derivative [J]. J. Math. Phys.1983, 24: 1405-1413 .
[23] Conte R. Invariant Painlevé analysis of partial differential equations [J]. Phys. Lett. A, 1989, 140: 383-390.
[24] Lou S Y. Extended Painlevé expansion, nonstandard truncation and special reductions of nonlinear evolution equations [J]. Z. Naturforsch. 1998, 53a :251-258.
[25] Perring J K, Skyrmc T H R. A model uniform field equation [J]. Nucl. Phys. 1962, 31: 550-553.
[26] Conte R, Fordy A P, Pickering A. A perturbative Painlevé approach to nonlinear differential equations [J]. Phys. D .1983, 69: 33-58.
[27] Fordy A P and Picking A. Analyzing negative resonances in the Painlevé test [J]. Phys. Lett . A., 1991,160:347-354.
[28] Musette M and Conte R. Algorithmic method for deriving Lax pairs from the invariant Painlevé analysis of nonlinear partial differential equations [J] .J. Math .Phys., 1991, 32,:1450-1457.
[29] 朱思明,施齐焉.现代数学与力学(MMM-VⅡ),上海大学出版社,1997,482.
[30] Xie F D, Chen Y. An algorithmic method in Painlevé analysis of PDE, Comput. Phys. Commun, 2003, 154:197-202.
[31] Hereman W, Angenet S. The Painlevé test for nonlinear ordinary and partial differential equations, Macsyma Newsletter, 1989, 6:11.
[32] Hereman W. Algorithmic integrability tests of nonlinear differential and lattice equations, Comput. Phys. commun., 1998, 115:428.
[33] Xu G Q, Li Z B. A Maple package for the Painlevé test of nonlinear partial differential equations, Chin. Phys. Lett., 2003, 20: 975.
[34] Lou S Y, Chen C L, Tang X Y.(2+1)-dimensional (M+N)-component AKNS system : Painlevé integrability, infinitely many symmetries similarity reductions and exact solutions, J. Math. Phys., 2002, 43:4078-4109.
[35] Zhang S L, Wu B, Lou S Y. Painlevé analysis and special solutions of generalized Broer-Kaup equations. Phys. Lett. A, 2002, 300:40.
[36] Lou S Y, Wu Q X .Painlevé integrability of two sets of nonlinear evolution equations with nonlinear dispersions. Phys. Lett. A, 1999, 262:344.
[37] Newell A C, Tabor M, Zeng Yunbo. A unified approach to Painlevé expansions. Physica D,1987, 29: 1-68.
[38] 曾云波.递推算子和 Painlevé性质.数学年刊,1991,12A:78—88.
[39] 朱左侬.若干非线性偏微分方程的 Painlevé分析和 Backlund 变换.东南大学学报,1994, 24(2):132—136.
[40] Gardner C S, Greene J M, Kruskal M D, Mirura R M. Korteweg-deVries equation and generalizationⅥ. Method for exact solutions. Comm. Pure. Appl. Math., 1974,27:97-133.
[41] Gardner C S, Greene J M, Kruskal M D, Mirura R M. Method for solving Korteweg-deVries equation[J]. Phys.Rev.Lett., 1967,19:1095-1097.
[42] Matveev V B, Salle M A. Darboux Transformations and Solitons. Springer-Verleg, Berlin, 1991.
[43] Mirura R M. Backlund Transformations. Springer-Verleg, Berlin, 1978.
[44] Rogers C, Schief W K. Backlund and Doarbox transformations geometry and modern applications in soliton theory. Cambridge University Press, Cambridge Textes in Applied Mathematics, 2002.
[45] 谷超豪,胡和生,周子翔.孤子理论中的达布变换及其几何应用.上海科技出版社,2001.
[46] Chen C L, Tang X Y ,Lou S Y. Solutions of (2+1)-dimensional dispersive long wave equation [J]. Phys. Rev. E, 2002, 66:603-605.
[47] 楼森岳.推广的 Painlevé 展开及 KdV 方程的非标准截断解.物理学报,1998, 47(12):1937—1945.
[48] 冯国鑫、王卿、钱贤民.2+1 维 Broer--Kaup 方程推广的 Painlevé 非标准截断展开和精确解.绍兴文理学院,2003, 23(10):16—20.
[49] Gu Chaohao, Guo Bailing, Li Yishen,et al. Soliton Theory and its Application [M]. Hangzhou: Zhejiang Pulish House of Science and Technology, 1990.
[50] 刘式达,傅遵涛,刘式适等.Jacobi 椭圆函数展开法及其在求解非线性波动方程中的应用[J]. 物理学报,2001,50(11):2068- 2073.
[51] Clarkson P A, Kruskal M D. New similarity reductions of Boussinesq equation. J. Math.Phys., 1989, 30:2201-2213.
[52] Clarkson P A, et al. Symmetry reductions of a generalized cylindrical Schroginger equation. J. Phys. A , 1993, 26:133-150.
[53] Wang Mingliang, Wang Yueming, Zhou Yubin. An auto- Backlund transformation and ageneralized KdV equation with variable coefficients and their applications. Phys. A, 2002,303: 45-51.
[54] 张金良,王跃明,王明亮.(1+1)维 KdV 型方程的变速孤波解.工程数学学报,2004, 21(5):810—812.
[55] 斯仁道尔吉,孙炯.两个非线性发展方程自 Backlund 变换及精确形波解.内蒙古师范大学学报, 2002, 31(2):95—99.
[56] Wang Ming-liang, Li Xiang-zheng, Nie Hui. Solitary wave for nonlinear evolution equations . 应用数学,2006,19(2):460-468 .
[57] 郭玉翠,徐淑奖.描述顺流方向可变剪切流动的一类变系数 Boussinesq 方程的 Painlevé 分析和相似约化. 应用数学学报,2006, 29(6):1054 -1062.
[58] Hodyss D, Nathan T R. Phys. Rev. Lett., 2004,93:074502 -1-4.
[59] Levi D, Winternitz P. Painlevé Transcendents: Their Asymtotics and Applicatons. New York: Plenum Press, 1992, NATO Adv. Sci .Inst . Ser B (Phys.) V278.
[60] 刘式适, 刘式达. 物理学中的非线性方程.北京大学出版社,2000.
[61] 秦惠曾,商妮.第四类 Painlevé方程的渐进性态分析.山东理工大学学报,2004, 2(18):12-15.
[62] Wadati M, et al., Simple derivation of Backlund transformations from Riccati form of inverse method, Prog Theor. Phys., 1975, 53: 418-425 .
[63] 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(11):1957—1961.
[64] 斯仁道尔吉.组合 KDV 方程的新的相似约化.内蒙古师范大学学报,1996, 2(2):1-6.
[65] Zhang Yufeng and Zhang Hongqing, An extension of the direct method and similarity reductions of a generalized Burgers equation with an arbitrary derivative function, Chinese Physics, 2002,11(4):319-322.
[66] 王烈衍.扰动非线性薛定谔方程的 Painlevé分析和精确孤立波解.宁波大学学报, 1999, 12(2):37—41.
[67] 斯仁道尔吉.李方程组的 Painlevé性质及其相似约化.内蒙古师范大学学报,1998, 27(4):253—259.
[68] 刘启铭.系数依赖于时间的 KdV --MKdV –Burgers 方程的 Painlevé性质.应用数学学报,1990, 13(3):267—271.
[69] Ma Wenxiu. Exact solutions to Tu system through painlevé analysis.复旦大学学报,1994,33(3):319—326.
[70] Liu Jianbin, Yang Kongqing. The extended F-expansion method and solutions of nonlinear PDEs.Chaos, Solitons and Fractals, 2004,22:111—121.
[71] 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(11):1957—1961.
[72] 范恩贵,张鸿庆.非线性孤子方程的齐次平衡法.物理学报,1998,47(3):353—361.
[73] 刘启铭.变系数 Kdv 方程的 painlevé分析.工程数学学报,1989, 6(1):103—105.
[74] 程艺,王存启,田畴,李翊神.Burgers 方程族的对称、 Backlund 变换和 Painlevé性质之间的联系.应用数学学报,1991, 14(2):180—184.
[75] 张全举.一类变系数 KP 方程的 Painlevé 分析和容许变换.西安建筑科技大学,1998, 20 (2):186—188.
[76] 魏光美、许晓革.一类变系数 KdV 方程的 Painlevé分析和自 Backlund 变换,数学的实践与认识. 2006, 36(6):308-312.
[77] 田涌波.方程族的 Painlevé相容组的求解公式.数学年刊,2000, 21A(3):271—276.
[78] Zeng Yubo. On the use of Painlevé property for obtaining Miura type transformation. Acta Mathematical. 1991,7(2):150—164.
[79] 包霞.非线性发展方程求解方法研究.内蒙古师范大学硕士论文,2004.
[80] 叶彩儿.几个非线性发展组的精确解与 Painlevé 分析.浙江大学硕士论文,2003.
[81] 李彪.孤子理论中若干求解方法的研究及应用.大连理工大学博士学位论文,2004.
[82] 王烈衍,楼森岳,耦和 KdV 方程的 Painlevé分析和非标准截断解. 吉首大学学报,1999, 20(3):30-35.
[83] 田涌波.KdV 方程和 Kupersmidt 方程的递推求解公式.数学进展,1999,28(2):151-156.
[84] 范恩贵.齐次平衡法、Weiss-Tabor-Carnevale 法及 Clarkson-Kruskal 约化法之间的联系.物理学报,2000,49(08):1409-1412.
[85] Fokas A S,Ablowitz M J. Linearrization of the KdV and Painlevé Ⅱ equation . Phys. Rev. Lett. , 1981, 47,:1096-1100.
[86] 斯仁道尔吉,孙炯.二维色散长波方程组的精确解.高校应用数学学报,2002,17A(3):308- 312.
[87] 斯仁道尔吉,孙炯.齐次平衡法的一个新应用.内蒙古师范大学学报,2001,30(4):287 - 290.
[88] 杜先云.变系数(2+1)维色散长波方程组的精确解.聊城大学学报,2005,18(2):17 - 20.
[89] 谢福鼎,闫振亚,张鸿庆.源于 Fermi-Pasta-Ulam 问题的非线性发展方程的相似约化,应用数学和力学.2002,23(4):347-352.
[90] 闫振亚,张鸿庆.(2+1)维非线性色散长波方程组的相似约化和解析解.数学物理学报,2001,21A(3):384–390.
[91] 殷久利,田立新,桂贵龙. 广义 Camassa-Holm 方程的对称性相似约化和精确解.江苏大学学报,2005,26(4):312 – 315.
[92] 闫振亚,张鸿庆.具有阻尼项的非线性波动方程的相似约化.物理学报,2000,49(11): 2113 – 2117.
[93] 郑春龙,张解放.(2+1)维 Camassa-Holm 方程的相似约化和解析解.物理学报,2002,51(11):2426 – 2430.
[94] 范恩贵,张鸿庆.二类变式 Boussinesq 方程的对称约化和精确解.数学物理学报,1999,19 (4):373–378.
[95] E.卡姆克著,张鸿林译.常微分方程手册.科学出版社,1977.
[96] 高本庆.椭圆函数及其应用.国防工业出版社.1991.
[97] Liu Guang-ting. The linearizing of some nonlinear evolution equations and their new exact solutions. 内蒙古师范大学学报, 2007,36(1):1-5.
[98] 刘官厅.准晶弹性的复变方法与非线性方程的显示解.内蒙古人民出版社.2005,5.
[99] 套格图桑,斯仁道尔吉.(2+1)维色散长波方程组和组合 KdV-Burgers 方程的新的精确孤立波解.工程数学学报,2006,23(5):943-946.
[100] 套格图桑,斯仁道尔吉.mBBM 方程和 KdV 方程新的 Jaccobi 椭圆函数周期解. 内蒙古师范大学学报,2006,35(3): 278-281.