非线性演化系统的精确解求解及其自动实现研究
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摘要
非线性演化方程通常指的是描述随时间演化的物理现象的一类数学模型,它是非线性系统科学的孤立子理论研究中最前沿的课题之一。近年来,随着非线性科学的快速发展,非线性演化方程的精确孤立波解已在物理学、生物学、图形学、通讯技术等自然科学和工程技术的各个领域得到了广泛的研究。
本文通过对非线性演化方程的精确孤立波解的求解方法的学习和研究,借鉴专家学者的理论思想,对tanh-sech方法从算法基础上做了相应的改进,并以非线性演化方程(组)为研究对象,借助符号计算系统Maple这一有力工具研究了多种不同维度的非线性演化方程(组)。为了使大家能更好的利用改进的tanh-sech方法研究非线性演化方程(组),我们还编写不同维度下非线性演化方程(组)的自动求解软件包,使大家在今后的研究中更得心应手。
本文主要从以下几个方面展开研究：
第一章阐述了孤立波的发现与提出,同时描述了孤立波从受到科学界的一致质疑到在理论上证实它的存在性并得到大家的认可的曲折过程,最后列举了人们在研究过程中发现的几中孤立波解形式。
第二章通过对各种典型的精确孤立波解解法的介绍,向大家阐明了非线性演化方程(组)精确解求解方法的基本思想和方法原理,从而使大家对非线性演化方程(组)的精确孤立波解的求解方法有了深刻的理解。
第三章对改进的tanh-sech方法的基本原理做了详细的介绍,并指出改进tanh-sech方法的必要性和有效性,最后借助符号计算系统Maple对(2+1)维KP方程、(2+1)维Ito方程、(2+1)维KD方程组和(3+1)维Burgers方程组分别做了研究,结果证明了改进的tanh-sech结果证明了改进的tanh-sech方法在求解高维非线性演化方程(组)精确孤立波解时的正确性和有效性。
第四章详细介绍自动求解软件包ASTNS的运行原理以及在Maple下的调用方法,并通过符号计算系统Maple调用ASTNS软件包对高维非线性演化方程(组)进行实例演示与分析,使ASTNS软件包在求解非线性演化方程(组)精确孤立波解的整个过程一目了然。
第五章总结全文并对精确孤立波解的求解研究工作进行展望。
Nonlinear evolution equation is a mathematical model for describing physical phenomenon which developing with time and is an important frontier field in the study of nonlinear physical, especially in the study of soliton theory. In recent years, nonlinear science has developed deeply and the solitary wave solutions of nonlinear evolution equation are widely studied in natural science and engineering technology, such as physics, biology, graphics, communication and so on.
In my thesis, based on the study and research on the method of obtaining exact solitary wave solutions of nonlinear evolution equations and absorbed insight from the ideas of experts and scholars, we make a modification of the tanh-sech method and give an implementation of it in symbolic computation system Maple. We investigate several multidimensional nonlinear evolution equations with it. In order to make people to use this method better in studying nonlinear evolution equations, we present an automatized programme for solving and studying all kinds of multidimensional nonlinear evolution equations.
In this thesis, we focus on the following four parts:
In Section 1, we introduce the history about how to discover and how to propose the solitary wave, also we narrate the tortuous process of it from being oppugned by scientists to prove its existence in theoretical aspect and then how to get everyone's approve. Finally, we list several forms of solitary wave solutions that were found in the course of investigating solitary wave.
In Section 2, we illustrate the idea and the principle of the method of constructing exact solutions for nonlinear evolution equations. Then a variety of typical methods of constructing exact solitary wave solutions are introduced, so that everyone has a profound understanding on this method.
In Section 3, we introduce the principle of the improved tanh-sech method and show its necessity and effectiveness. Finally, we investigate several multidimensional nonlinear evolution equations by our symbolic computation Maple package, such as (2+1) KP equations, (2+1) Ito equations, (2+1) KD equations and (3+1) Burgers equations. The results demonstrate the improved tanh-sech method being effectiveness as a tool for solving solitary wave solutions of high-dimensional nonlinear evolution equations.
In Section 4, we illustrate the operation principle and the loading method of the automated solving package ASTNS, also we analyses several high-dimensional nonlinear evolution equations by the ASTNS in Maple. It shows that the ASTNS is indeed effective and clear in the study of solitary wave solutions.
Section 5 is a summary and short review of this work.
本文通过对非线性演化方程的精确孤立波解的求解方法的学习和研究,借鉴专家学者的理论思想,对tanh-sech方法从算法基础上做了相应的改进,并以非线性演化方程(组)为研究对象,借助符号计算系统Maple这一有力工具研究了多种不同维度的非线性演化方程(组)。为了使大家能更好的利用改进的tanh-sech方法研究非线性演化方程(组),我们还编写不同维度下非线性演化方程(组)的自动求解软件包,使大家在今后的研究中更得心应手。
本文主要从以下几个方面展开研究：
第一章阐述了孤立波的发现与提出,同时描述了孤立波从受到科学界的一致质疑到在理论上证实它的存在性并得到大家的认可的曲折过程,最后列举了人们在研究过程中发现的几中孤立波解形式。
第二章通过对各种典型的精确孤立波解解法的介绍,向大家阐明了非线性演化方程(组)精确解求解方法的基本思想和方法原理,从而使大家对非线性演化方程(组)的精确孤立波解的求解方法有了深刻的理解。
第三章对改进的tanh-sech方法的基本原理做了详细的介绍,并指出改进tanh-sech方法的必要性和有效性,最后借助符号计算系统Maple对(2+1)维KP方程、(2+1)维Ito方程、(2+1)维KD方程组和(3+1)维Burgers方程组分别做了研究,结果证明了改进的tanh-sech结果证明了改进的tanh-sech方法在求解高维非线性演化方程(组)精确孤立波解时的正确性和有效性。
第四章详细介绍自动求解软件包ASTNS的运行原理以及在Maple下的调用方法,并通过符号计算系统Maple调用ASTNS软件包对高维非线性演化方程(组)进行实例演示与分析,使ASTNS软件包在求解非线性演化方程(组)精确孤立波解的整个过程一目了然。
第五章总结全文并对精确孤立波解的求解研究工作进行展望。
Nonlinear evolution equation is a mathematical model for describing physical phenomenon which developing with time and is an important frontier field in the study of nonlinear physical, especially in the study of soliton theory. In recent years, nonlinear science has developed deeply and the solitary wave solutions of nonlinear evolution equation are widely studied in natural science and engineering technology, such as physics, biology, graphics, communication and so on.
In my thesis, based on the study and research on the method of obtaining exact solitary wave solutions of nonlinear evolution equations and absorbed insight from the ideas of experts and scholars, we make a modification of the tanh-sech method and give an implementation of it in symbolic computation system Maple. We investigate several multidimensional nonlinear evolution equations with it. In order to make people to use this method better in studying nonlinear evolution equations, we present an automatized programme for solving and studying all kinds of multidimensional nonlinear evolution equations.
In this thesis, we focus on the following four parts:
In Section 1, we introduce the history about how to discover and how to propose the solitary wave, also we narrate the tortuous process of it from being oppugned by scientists to prove its existence in theoretical aspect and then how to get everyone's approve. Finally, we list several forms of solitary wave solutions that were found in the course of investigating solitary wave.
In Section 2, we illustrate the idea and the principle of the method of constructing exact solutions for nonlinear evolution equations. Then a variety of typical methods of constructing exact solitary wave solutions are introduced, so that everyone has a profound understanding on this method.
In Section 3, we introduce the principle of the improved tanh-sech method and show its necessity and effectiveness. Finally, we investigate several multidimensional nonlinear evolution equations by our symbolic computation Maple package, such as (2+1) KP equations, (2+1) Ito equations, (2+1) KD equations and (3+1) Burgers equations. The results demonstrate the improved tanh-sech method being effectiveness as a tool for solving solitary wave solutions of high-dimensional nonlinear evolution equations.
In Section 4, we illustrate the operation principle and the loading method of the automated solving package ASTNS, also we analyses several high-dimensional nonlinear evolution equations by the ASTNS in Maple. It shows that the ASTNS is indeed effective and clear in the study of solitary wave solutions.
Section 5 is a summary and short review of this work.
引文
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[2]陈登远.孤子引论.北京：科学出版社[M].2006.
[3]陈陆同,梁昌洪.孤立子理论及其应用-光孤子理论孤子通讯[M].西安：西安电子科技大学出版社.1997.
[4]刘式适.物理学中的非线性方程[M].北京：北京大学出版社.2001.
[5]李政道.粒子物理和场论简引[M].北京：科学出版社.1984.
[6]Russell J S. Report on waves, Report of the 14th meeting of the Birtish association for the advancement of science [M], John Murray, London,1944,311-390.
[7]D. J. Korteweg and G. de Vires, On the change of form of long waves advancing in a stationary waves [J], Phil. Mag.39:(1985) 422-443.
[8]D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves [J], Phil. Mag,39: (1895)442-443.
[9]C.S. Gardner and G.M. Morikawa, Similarity in the asymptotic behavior of collision free hydromagnetic waves and water waves [J]. Courant Inst. Math. Sci. Report, (1960).
[10]Zabusky N J and Kruskal M D. interactions of solitions in a collisionless plasma and the recurrence of initial states [J]. Phys. Rev. Leet.15:(1965) 240-243.
[11]Kruskal M D. The Korteweg-devries equation and related evolution equtions[J]. Applied Mathematcis.68:(1974) 61-83.
[12]楼森岳,唐晓燕.非线性数学物理方法[M].北京：科学出版社.2006.
[13]陈同兴.非线性物理概论[M].北京：中国科学技术出版社.1984.
[14]王跃明,王明亮.两个非线性偏微分方程的分离变量法解[J].商洛工学院学报.21：(2000)88-90.
[15]R.Hirota. Exact N-soliton solutions of the wave equation of long waves in shallow-water and in nonlinear lattices [J]. Math.Phys.14:(1973) 810
[16]F.Cariello and M.Tabor. Similarity reductions from extended Painleve expansions for nonintegrable evolution equations [J]. Physica D 53:(1991) 59-70.
[17]S.K.Liu,Z.T.Fu,S.D.Liu and Q.Zhao. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations [J]. Phys.Lett. A 289:(2001) 69-74.
[18]Z.Y. Yan. Abundant families of Jacobi elliptic function solutions of the (2+ l)-dimensional integrable Davey-Stewartson-type equation via a new method [J]. Chaos Solitons Fractals 18:(2003) 299-309.
[19]M.L. Wang and Y.B. Zhou. The periodic wave equations for the KleinC GordonC Schordinger equations [J]. Phys. Lett. A 318 (2003) 84.
[20]M.L. Wang and X.Z. Li. expansion and periodic wave solutions for the generalized Zakharov equations [J]. Phys. Lett. A.343:(2005) 48.
[21]A.M. Wazwaz. Distinct variants of the KdV equation with compact and noncompact structures [J]. Appl. Math. Comput.150:(2004) 365-377.
[22]A.M. Wazwaz. The tanh and the sine-cosine methods for a reliable treatment of the modified equal width equation and its variants [J].Comput. Math. Appl.47:(2004) 583.
[23]J.H. He and X.H. Wu. Exp-function method for nonlinear wave equations [J]. Chaos Solitons Fractals 30:(2006) 700-708.
[24]C. S. Gardner, J. M.Greene, M. D. Kruskal and R. M. Miura, Method for solving the Koreweg-devries equation [J]. Phys. Rev. Lett.19:(1967) 1095-1097.
[25]C. S. Gardner, J. M.Greene and M. D. Kruskal. Korteweg-devries equation and generalization VI. Methods for exact solution [J]. Comm. Pure Appl. Math.27: (1974)97-133.
[26]P. D. Lax. Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math [J].21:(1968) 467-490.
[27]M. J. Ablowity, D. J. Kaup, A. C. Newell and H. Segur. Method for Solving the Sine-Gordon Equation [J]. Phys. Rev. Lett.30:(1973) 1262-1264.
[28]杨伯君.高等数学物理方法[M].北京：北京邮电大学出版社.(2003).
[29]Ryogo Hirota and Junkichi Satsuma. Nonlinear Evolution Equations Generated from the Backlund Transformation for the Boussinesq Equation [J]. Prog. Theor. Phys.57:(1977) 797-807.
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