对某类变系数非线性发展方程精确解的讨论
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摘要
本文以探究非线性发展方程精确解的孤子理论为依据,在导师孙福伟教授的指导下,研究了在诸多自然科学领域有着广泛应用的低维变系数非线性发展方程与高维变系数非线性耦合方程求解精确解尤其是精确孤立波解的方法。
论文第一章介绍国内外非线性科学中孤立子理论研究的历史与发展,重点简述求解非线性发展方程精确解的几种主要方法及该方法在国内外的研究现状。
论文第二章第一部分详述Painleve分析方法在国内外的发展情况和用它求解非线性发展方程精确解的具体步骤,第二部分将Painleve分析方法用于研究非线性发展方程精确解的前沿领域之一：变系数非线性发展方程,以变系数KdV-Burgers方程为实例,求解变系数KdV-Burgers方程的精确解表达式及Backlund变换,并通过形象的、简单易行的图示描述由此变系数非线性发展方程所确定的精确孤立波解。
论文第三章在理解第二章研究成果的基础上,继续探究非线性发展方程应用的高新技术领域：变系数、高维、耦合系统,继续用Painleve分析方法求解高维变系数非线性耦合方程的精确解,以有背景相互作用的2+1维变系数非线性Schodinger耦合系统为典型,求解出该变系数高维耦合系统的精确解表达式及Backlund变换,分析由此高维变系数耦合方程所确定的精确解,同样以图示方式阐释其中的精确孤立波解,深化用Painleve分析方法求解高维、变系数、耦合非线性系统精确解的研究。
论文第四章总结研究成果,分别指出第二章、第三章用Painleve分析方法处理变系数、高维、耦合等这些非线性发展方程研究中的复杂问题时的物理理论根源,解释据此所采取的数学解决方法的变通与改进,分析在综合以上复杂问题时所能确定出的精确孤立波解的物理意义,延伸求解非线性发展方程精确解中对孤立子理论的探讨,挖掘课题研究中的独立创新及缺点不足,展望非线性科学发展的前景。
Under the guidance of Professor Sun Fuwei, this paper explore the exact solutions of nonlinear evolution equations based on the theory of soliton, It mainly studied a wide range of low-dimensional variable coefficient nonlinear equations and high-dimensional variable Exact solutions of linear coupled equations especially exact solitary wave solutions in many natural sciences applications.
The first chapter introduces the domestic and foreign history and development of non-linear science, soliton theory focusing on brief exact solutions of nonlinear evolution equations in several major ways and relate methods of research status at home and abroad.
The first part of the second chapter details the Painleve analysis of development at home and abroad and specific steps of using it gain exact solutions of nonlinear evolution equations. By the Painleve analysis method, the second part study variable coefficient nonlinear equations which is one frontier of the research of nonlinear evolution equations. For variable coefficient KdV-Burgers equation as an example, the paper get the variable coefficient KdV-Burgers equation and Backlund transformation expression and describe the nonlinear evolution equations by the exact solitary wave solutions determined by variable coefficients through simple image.
In the basis of understanding research results in chapter II, the third chapter continues to explore the application of nonlinear evolution equations in high-tech areas:variable coefficient, high dimension, coupled system. With the Painleve analysis method, the paper has again studied high-dimensional nonlinear coupled coefficient equation. For 2+1 dimensional variable coefficient nonlinear Schodinger system as typical of the variable coefficient problems, it gains the exact solutions of expression and the Backlund transformation of high-dimensional coupled system and makes analysis of the exact solution that determined by this high-dimensional variable coupled equations, In the same way of the second chapter, it explains exact solitary wave solutions with images, which deepen with the Painleve analysis method to solve exact solutions of high-dimensional, variable coefficients, nonlinear coupled systems research..
The forth chapter summarizes the research conclusions pointed out the complex issues of physical theory causes that explain accordingly adopted Painleve analysis approach with variable coefficient, high dimension, nonlinear evolution coupled equations in chapterⅡ, ChapterⅢ.The paper identifies the modifications and improvements in the above analysis and the physical meaning of exact solitary wave solutions of complex problems, extending the exact solutions of nonlinear evolution equations on the theory of solitons, At last, it mines independent innovation and weaknesses during the research and looks forward to the prospects for the development of nonlinear science.
论文第一章介绍国内外非线性科学中孤立子理论研究的历史与发展,重点简述求解非线性发展方程精确解的几种主要方法及该方法在国内外的研究现状。
论文第二章第一部分详述Painleve分析方法在国内外的发展情况和用它求解非线性发展方程精确解的具体步骤,第二部分将Painleve分析方法用于研究非线性发展方程精确解的前沿领域之一：变系数非线性发展方程,以变系数KdV-Burgers方程为实例,求解变系数KdV-Burgers方程的精确解表达式及Backlund变换,并通过形象的、简单易行的图示描述由此变系数非线性发展方程所确定的精确孤立波解。
论文第三章在理解第二章研究成果的基础上,继续探究非线性发展方程应用的高新技术领域：变系数、高维、耦合系统,继续用Painleve分析方法求解高维变系数非线性耦合方程的精确解,以有背景相互作用的2+1维变系数非线性Schodinger耦合系统为典型,求解出该变系数高维耦合系统的精确解表达式及Backlund变换,分析由此高维变系数耦合方程所确定的精确解,同样以图示方式阐释其中的精确孤立波解,深化用Painleve分析方法求解高维、变系数、耦合非线性系统精确解的研究。
论文第四章总结研究成果,分别指出第二章、第三章用Painleve分析方法处理变系数、高维、耦合等这些非线性发展方程研究中的复杂问题时的物理理论根源,解释据此所采取的数学解决方法的变通与改进,分析在综合以上复杂问题时所能确定出的精确孤立波解的物理意义,延伸求解非线性发展方程精确解中对孤立子理论的探讨,挖掘课题研究中的独立创新及缺点不足,展望非线性科学发展的前景。
Under the guidance of Professor Sun Fuwei, this paper explore the exact solutions of nonlinear evolution equations based on the theory of soliton, It mainly studied a wide range of low-dimensional variable coefficient nonlinear equations and high-dimensional variable Exact solutions of linear coupled equations especially exact solitary wave solutions in many natural sciences applications.
The first chapter introduces the domestic and foreign history and development of non-linear science, soliton theory focusing on brief exact solutions of nonlinear evolution equations in several major ways and relate methods of research status at home and abroad.
The first part of the second chapter details the Painleve analysis of development at home and abroad and specific steps of using it gain exact solutions of nonlinear evolution equations. By the Painleve analysis method, the second part study variable coefficient nonlinear equations which is one frontier of the research of nonlinear evolution equations. For variable coefficient KdV-Burgers equation as an example, the paper get the variable coefficient KdV-Burgers equation and Backlund transformation expression and describe the nonlinear evolution equations by the exact solitary wave solutions determined by variable coefficients through simple image.
In the basis of understanding research results in chapter II, the third chapter continues to explore the application of nonlinear evolution equations in high-tech areas:variable coefficient, high dimension, coupled system. With the Painleve analysis method, the paper has again studied high-dimensional nonlinear coupled coefficient equation. For 2+1 dimensional variable coefficient nonlinear Schodinger system as typical of the variable coefficient problems, it gains the exact solutions of expression and the Backlund transformation of high-dimensional coupled system and makes analysis of the exact solution that determined by this high-dimensional variable coupled equations, In the same way of the second chapter, it explains exact solitary wave solutions with images, which deepen with the Painleve analysis method to solve exact solutions of high-dimensional, variable coefficients, nonlinear coupled systems research..
The forth chapter summarizes the research conclusions pointed out the complex issues of physical theory causes that explain accordingly adopted Painleve analysis approach with variable coefficient, high dimension, nonlinear evolution coupled equations in chapterⅡ, ChapterⅢ.The paper identifies the modifications and improvements in the above analysis and the physical meaning of exact solitary wave solutions of complex problems, extending the exact solutions of nonlinear evolution equations on the theory of solitons, At last, it mines independent innovation and weaknesses during the research and looks forward to the prospects for the development of nonlinear science.
引文
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[2]莫嘉琪.一个厄尔尼诺大气物理非线性模型.数学物理学报.2007.27A(5)：931-934
[3]桂占吉,康岚.预测胎儿体重的非线性模型.数学的实践与认识.Vol.32.No.42002.7.537-540
[4]楼森岳,唐晓艳.非线性数学物理方法[M].北京：科学出版社,2006
[5]Russell. J. S. Report on waves. Rep.14th. meet. Brit. Assoc. Adv. Sci. York, London, John Murray,1844:311
[6]Wang M. L. Solitary Wave solulation
[7]杨伯君.高等数学物理方法.北京：北京邮电大学出版社.2003.
[8]曾听.一类非线性偏微分方程若干求精确解方法的研究.[硕士学位论文],2005.3
[9]Zabusky N J, Kruskal M D. Phys. Rev. Lett.1965,14:240
[10]Hasgawa A, TappePF. App.Phys. Lett,1973,23:142
[11]Kivshar Y S. Physics Reports,1998,298:81
[12]Gordon J. P. Opt. Lett,1983,8:596:1986,11:665
[13]Agrawal G. P. Nonlinear Fiber Optics, second edition, SanDiego:Academic Press,1995
[14]王明亮.非线性发展方程与孤立子.兰州大学出版社.1990
[15]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.上海科学技术出版社,2001
[16]Miura M R. Backlund Transformation. Berlin, Springer-Verlay,1978
[17]Hirota R. The direct method in soliton theory. Edited and translated by Nagai A, Nimmo J. Gilson C.Cambridge Trctc in Mathematics No.155,2004
[18]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
[19]Picking A. [J]. Phys. A:Math Gen,1993.26:4395; J. Math. Phys.,1996,37: 1894
[20]Fan EG. Phys. Lett.A:Math. Gen.,2001,34:513
[21]Bluman G W, Kumei S. Symmtries and Differential Equation. Appl. Math. Sci.81.Springer, Berlin,1989
[22]Weiss J. The Painleve property for partial differential equation II. Backlund transformation Lax pairs and Schwar:derivative[J]. Math. Phys,1983,24:1405
[23]Gardner C S, Greene J M. Kruskal.M D, Miura P M. Phys. Rev. Lett. 1976.19:1095
[24]冯国鑫,王卿,钱贤民.2+1维Broer-Kaup方程推广的Painleve非标准展开和精确解[J].绍兴文理学院学报,2003,23(10)：16-20
[25]Weiss J. Tabor M.Carnevale G. [J]. Math. Phys,1983,24:522
[26]Conte R. Invariant Painleve analysis of partical differentical
equation[J]. Phys. Lett. A,1989,140:383
[27]曾云波.与A相联系的半Toda方程的Lax对与Backlund变换.数学学报.1992.35：454
[28]曾云波,递推算子与Painleve性质.数学年刊.1991.12A：778
[29]Gao. YT, Tian. B. New families of exact solutions to the integrable dispersive long wave equations in2+1-dimensional spaces. J. Phys. A, 1986,29:2895
[30]Fan EG, Zhang HQ. New exact solutions to a System of coupled KdV equation. Phys. Lett. A.1998.245:389-392
[31]刘式适,刘式达.物理学中的非线性方程.北京：北京大学出版社.2000
[32]毛杰健,杨建荣.非线性-Kuramoto方程新的行波解[J].兰州理工大学学报,2006,32(2)：150-153
[33]JIANG Lei, CHEN Xi. Periodic Solutions to KdV-Burgers-Kuramoto Equation. commun[J]. Theor.Phys,2006,45(5):815-818
[34]谢元喜,朱署华.KdV-Burgers方程和KdV-Burgers-方程的精确解[J].安徽大学学报(自然科学版),2007,31(6)：44-47
[35]Winternity P, Gayeau J P. Allowed Transformations and Symmetry classes of variable coefficient Korte weg-de Vires equations [J]. Phys Lett,1992 A167:246-250
[36]武祥,郭鹏.一类非线性粘弹性杆波动方程的求解.科技信息：2008,第二期：203
[37]张辉群.F-展开法的扩展及应用.应用数学.2005,18(4)：629-633
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