对称约化在非线性演化方程的初值问题和群分类中的应用
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摘要
非线性科学被深入研究并广泛应用到了各个自然科学领域中.如化学,数学,生物学,物理,经济学等等.同时涌现了大量的非线性系统.在研究过程中我们遇到各种各样的非线性偏微分方程.其解法以及解的性质等问题是非线性科学的重要组成部分.人们已经建立和发展了很多求解非线性偏微分方程的有效方法.其中对称群理论是我们研究非线性偏微分方程精确解的有效方法之一.
随着对非线性理论的研究,出现许多非线性偏微分方程依赖于一个小参数,通常称为扰动(近似)偏微分方程,需要寻求它们的近似解.以非扰动方程的理论框架为基础,人们对扰动方程进行了相关的分析,得到了许多有效的结论.一些以对称理论为基础的扰动方法相继产生.
第一章,介绍了研究工作的背景,现状,基本理论和本文的主要工作.
第二章,利用广义条件对称方法讨论了三阶演化方程ut=F(u,ux)Uxxx+G(u,ux,uxx),F(u,ux)≠0和四阶演化方程
ut=-F(u)uxxxx-G(u,ux)uxxx+H(u,ux,uxx),F(u)≠0初值问题的分类和对称约化,并得到其解.
第三章,将广义条件对称方法进行推广,并成功地用于非线性偏微分方程组ut=f(u,v)uxx+9(u,v),f(u,v)≠0 vt=p(u,v)vxx+g(u,v),p(u,v)≠0初值问题的分类和对称约化.
第四章,我们将该方法成功地推广到了扰动方程的情形,利用近似广义条件对称方法解决了扰动KdV-Burger方程ut=guxxx+(?)G(u)uxx+F(u,ux),g≠0和扰动的反应扩散型方程ut=(A(u)ux)x+(?)B(u)ux+Q(u)的初值问题的分类和对称约化.
第五章,讨论了波动方程utt=f(x,u,ux)uxx+g(x,u,ux)的初等群分类问题.
在本文的最后给出主要的研究结论和今后的研究工作.
本文的创新与特色是：
(1)在研究对象上,主要研究了具有实际应用物理背景的非线性偏微分方程(组),从单个方程到方程组的情形,以及从非扰动方程到扰动方程的情形.
(2)在方法上,我们将该方法推广到了方程组初值问题的对称约化上.提出了一种解决扰动方程初值问题的方法,利用近似广义条件对称方法解决扰动方程初值问题.
(3)在内容上,我们解决了方程组和扰动方程初值问题的对称约化,并得到约化后方程的解,为进一步研究这些方程提供了重要的信息.同时,我们研究了一类波动方程的初等群分类问题.
Nonlinear science is substantially studied and widely applied in natural sciences such as chemistry, mathematics, biology, physics, economy and so on. There are many nonlinear systems. In the research process, we encounter a wide variety of nonlinear partial differential equations (PDEs). The method to solve the equations and the properties of their solutions are important parts of nonlinear science. Up to now, many methods have been established and developed to solve the nonlinear PDEs. It is well known that symmetry group theory is one of the effective ways in studying exact solutions of nonlinear PDEs.
As the study of the nonlinear systems developing, it is found that many PDEs in application depend on a small parameter, which are usually called perturbed (approximate) PDEs, we need to seek their approximate solutions. Based on the theories frame of the unperturbed equations, some related anal-ysis have been done on the perturbed equations and many efficient conclusions have been achieved. Some symmetry perturbation methods based on the Lie theory have been established.
In the first chapter, we study the background, current situation and basic notation and theory of our research work, at the same time introduce the main work of this paper.
In the second chapter, classification and symmetry reduction of initial value problems for third-order evolution equation ut= F(u, ux)uxxx+G(u, ux, uxx), F(u, ux)≠0 and fourth-order evolution equation ut= -F(u)uxxxx-G(u, ux)uxxx+H(u, ux, uxx), F(u)≠0 are discussed by generalized conditional symmetry method,and the solutions are obtained.
In the third chapter,classification and symmetry reduction of initial value problem for nonlinear systems of PDEs ut=f(u,v)uxx+g(u,v),f(u,v)≠0 vt=p(u,v)vxx+q(u,v),p(u,v)≠0 is successfully presented by a developing generalized conditional symmetry method.
In the fourth chapter,we use the approximate generalized conditional symmetry method to research the classification and symmetry reduction of initial value problem for perturbed KdV-Burger equation ut=guxxx+(?)G(u)uxx+F(u,ux),g≠0 and perturbed diffusion-reaction type equation ut=(A(u)ux)x+(?)B(u)ux+Q(u) and develop the method from unperturbed case to perturbed one.
In the fifth chapter,we discuss the preliminary group classification for one wave equation utt=f(x,u,ux)uxx+g(x,u,ux).
This dissertation end with summary and research prospects.
The innovations and features of this dissertation are as follows:
(1)In research objection,we discuss the nonlinear PDE or systems of non-linear PDE which have physical background,such as from one single equation to system of equation and from unperturbed equation to perturbed equation.
(2)In the method, we successfully develop the method to initial value problem for system of equation and put forward the method to deal with the initial value problem for perturbed equation.
(3)In content, we research the classification and symmetry reduction of initial value problem for higher-order evolution, systems of PDE and per-turbed equation, and obtain the analytical solutions of initial value problem which offer important information for further study these equations. At the same time, we study the preliminary group classification for one class of wave equation.
随着对非线性理论的研究,出现许多非线性偏微分方程依赖于一个小参数,通常称为扰动(近似)偏微分方程,需要寻求它们的近似解.以非扰动方程的理论框架为基础,人们对扰动方程进行了相关的分析,得到了许多有效的结论.一些以对称理论为基础的扰动方法相继产生.
第一章,介绍了研究工作的背景,现状,基本理论和本文的主要工作.
第二章,利用广义条件对称方法讨论了三阶演化方程ut=F(u,ux)Uxxx+G(u,ux,uxx),F(u,ux)≠0和四阶演化方程
ut=-F(u)uxxxx-G(u,ux)uxxx+H(u,ux,uxx),F(u)≠0初值问题的分类和对称约化,并得到其解.
第三章,将广义条件对称方法进行推广,并成功地用于非线性偏微分方程组ut=f(u,v)uxx+9(u,v),f(u,v)≠0 vt=p(u,v)vxx+g(u,v),p(u,v)≠0初值问题的分类和对称约化.
第四章,我们将该方法成功地推广到了扰动方程的情形,利用近似广义条件对称方法解决了扰动KdV-Burger方程ut=guxxx+(?)G(u)uxx+F(u,ux),g≠0和扰动的反应扩散型方程ut=(A(u)ux)x+(?)B(u)ux+Q(u)的初值问题的分类和对称约化.
第五章,讨论了波动方程utt=f(x,u,ux)uxx+g(x,u,ux)的初等群分类问题.
在本文的最后给出主要的研究结论和今后的研究工作.
本文的创新与特色是：
(1)在研究对象上,主要研究了具有实际应用物理背景的非线性偏微分方程(组),从单个方程到方程组的情形,以及从非扰动方程到扰动方程的情形.
(2)在方法上,我们将该方法推广到了方程组初值问题的对称约化上.提出了一种解决扰动方程初值问题的方法,利用近似广义条件对称方法解决扰动方程初值问题.
(3)在内容上,我们解决了方程组和扰动方程初值问题的对称约化,并得到约化后方程的解,为进一步研究这些方程提供了重要的信息.同时,我们研究了一类波动方程的初等群分类问题.
Nonlinear science is substantially studied and widely applied in natural sciences such as chemistry, mathematics, biology, physics, economy and so on. There are many nonlinear systems. In the research process, we encounter a wide variety of nonlinear partial differential equations (PDEs). The method to solve the equations and the properties of their solutions are important parts of nonlinear science. Up to now, many methods have been established and developed to solve the nonlinear PDEs. It is well known that symmetry group theory is one of the effective ways in studying exact solutions of nonlinear PDEs.
As the study of the nonlinear systems developing, it is found that many PDEs in application depend on a small parameter, which are usually called perturbed (approximate) PDEs, we need to seek their approximate solutions. Based on the theories frame of the unperturbed equations, some related anal-ysis have been done on the perturbed equations and many efficient conclusions have been achieved. Some symmetry perturbation methods based on the Lie theory have been established.
In the first chapter, we study the background, current situation and basic notation and theory of our research work, at the same time introduce the main work of this paper.
In the second chapter, classification and symmetry reduction of initial value problems for third-order evolution equation ut= F(u, ux)uxxx+G(u, ux, uxx), F(u, ux)≠0 and fourth-order evolution equation ut= -F(u)uxxxx-G(u, ux)uxxx+H(u, ux, uxx), F(u)≠0 are discussed by generalized conditional symmetry method,and the solutions are obtained.
In the third chapter,classification and symmetry reduction of initial value problem for nonlinear systems of PDEs ut=f(u,v)uxx+g(u,v),f(u,v)≠0 vt=p(u,v)vxx+q(u,v),p(u,v)≠0 is successfully presented by a developing generalized conditional symmetry method.
In the fourth chapter,we use the approximate generalized conditional symmetry method to research the classification and symmetry reduction of initial value problem for perturbed KdV-Burger equation ut=guxxx+(?)G(u)uxx+F(u,ux),g≠0 and perturbed diffusion-reaction type equation ut=(A(u)ux)x+(?)B(u)ux+Q(u) and develop the method from unperturbed case to perturbed one.
In the fifth chapter,we discuss the preliminary group classification for one wave equation utt=f(x,u,ux)uxx+g(x,u,ux).
This dissertation end with summary and research prospects.
The innovations and features of this dissertation are as follows:
(1)In research objection,we discuss the nonlinear PDE or systems of non-linear PDE which have physical background,such as from one single equation to system of equation and from unperturbed equation to perturbed equation.
(2)In the method, we successfully develop the method to initial value problem for system of equation and put forward the method to deal with the initial value problem for perturbed equation.
(3)In content, we research the classification and symmetry reduction of initial value problem for higher-order evolution, systems of PDE and per-turbed equation, and obtain the analytical solutions of initial value problem which offer important information for further study these equations. At the same time, we study the preliminary group classification for one class of wave equation.
引文
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[2]Olver P. J.. Application of Lie Groups to Differential Equations [M]. New York:Springer,1993
[3]Bluman G. W., Kumei S.. Symmetries and Differential Equations [M]. New York:Springer,1989
[4]Ibragimov H. N.. Transformation Groups Applied to Mathematical Physics [M]. Dordrecht:Reidel,1985
[5]Bluman G. W., Anco S. C.. Symmetry and Integration Methods for Dif-ferential Equations [M]. New York:Springer,2002
[6]布卢曼G.W.,安科S.C.著,闫振亚译.微分方程的对称与积分方法[M].北京：科学出版社,2009
[7]Ovsiannikov L. V.. Group Analysis of Differential Equations [M]. New York:Academic,1982
[8]Noether E.. Invariant variations problem [J]. Nachr. Konig. Gesell. Wis-sell. Gottinge, Math. Phys.,1917,1:235-257
[9]Bluman G. W., Cole J. D.. The general similarity solution of the heat equation[J]. J.Math. Mech.,1969,18:1025-1042
[10]Fokas A. S., Liu Q. M.. Generalized conditional symmetries and exact solutions of non-integrable equations [J]. Theor.Math.Phys.,1994,99: 263-277
[11]Fokas A. S., Liu Q. M.. Nonlinear interaction of traveling waves of non-integrable equations [J]. Phys. Rev. Lett.,1994,72:3293-3296
[12]Zhdanov R. Z.. Conditional Lie - Backlund symmetry and reduction of evolution equations [J]. J.Phys.A:Math.Gen.,1995,28:3841-3850
[13]Bluman G. W., Kumei S., Reid G. J.. New classes of symmetries for partial differential equation [J]. J. Math. Phys.,1988,29:806-811
[14]Bluman G. W., Kumei S.. On invariance properties of the wave equation [J]. J. Math. Phys.,1987,28:307-318
[15]李翊神,朱国城.一个谱可变演化方程的对称[J].科学通报,1986,19：1449-1456
[16]田畴.李群及其在微分方程中的应用[M].北京：科学出版社,2001
[17]Qu C. Z.. Group classification and generalized conditional symmetry re-duction of the nonlinear diffusion-convection equations with a nonlinear source [J]. Stud. Appl. Math.,1997,99:107-136
[18]Qu C. Z.. Reductions and exact solutions of some nonlinear partial differ-ential equations under four types of generalized conditional symmetries [J]. J. Aust. Math. Soci. B.,1999,41:1-40
[19]Qu C. Z.. Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method [J]. IMA J. Appl. Math.,1999, 62:283-302
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