物理学中几个非线性演化方程的研究
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摘要
随着非线性科学的发展,非线性物理也逐渐发展起来。在非线性物理学中,我们常常把复杂的非线性物理系统简化为非线性演化方程来研究,通过对方程的求解来确定物理量之间的定量或定性关系,并可以通过解的图形给出物理量之间关系的直观印象。求解非线性方程并画出解的图形对物理学的发展具有重要意义。本论文对典型的差分-微分方程Toda链方程和两个非线性偏微分方程组(2+1)维Boiti-Leon-Pempinelli方程组、(2+1)维Konopelchenko-Dubrovsky方程组进行了求解,并对解的性质进行了一定的分析。
主要工作如下:
1.介绍了非线性科学的发展概况、一些重要的非线性演化方程和归纳总结了求非线性偏微分方程精确解的一些主要方法。
2.对齐次平衡方法做了简单的介绍,参照求解一维Toda链方程的过程,将此方法应用于(2+1)维Toda链方程,最终得到了(2+1)维Toda链方程的单孤子解、双孤子解、三孤子解和总结归纳出其N孤子解的一般形式,并对一定参数的若干解进行了数值模拟。
3.介绍了双曲正切函数展开法和拓展的Riccati方程映射法,分别利用这两种方法求得了(2+1)维Boiti-Leon-Pempinelli方程组和(2+1)维Konopelchenko-Dubrovsky方程组的若干解析解,最后得出了一定结论。
Nonlinear physics developed gradually with the development of nonlinear science.In nonlinear physics,some simplified nonlinear evolution equations are often needed to describe the complex nonlinear physics symtem.We can make sure the quantificational connection or the qualitative connection of physics quantities by solving the nonlinear evolution equation. Besides of this,we can also give the firsthand impression of the physics quantities by giving the picture of solution of the nonlinear evolution equation.Then,it is very important to the development of physics by solving the nonlinear evolution equation and giving the picture of the solution.In this paper we obtained many solutions of(2+1)-dimensional Toda lattice equation,(2+1)-dimensional Boiti-Leon-Pempinelli equation and(2+1)-dimensional Konopelchenko-Dubrovsky equation and studied characters of equations.There are mainly three sections in this dissertation.
1.Firstly,we introduced the development of nonlinear science and some nonlinear evolution equation.And the methods for solving the exact solution of nonlinear evolution equation are introduced.
2.The Homogeneous method are introduced.According to the method of solving (1+1)-dimension Toda lattice equation,The single-soliton,the double-solitons,the three-solitons and simulation results of(2+1)-dimensional Toda lattice equation have been obtained by using this method.Finally,the general form of the N-solitons solution of the equation is derived.
3.We introduced the hyperboloic function expansion method and the extend Riccati mapping approach detailly.Then those approaches were applied to the(2+1)-dimensional Boiti-Leon-Pempinelli equation and(2+1)-dimensional Konopelchenko-Dubrovsky equation. Finally,some solutions of the(2+1)-dimensional Boiti-Leon-Pempinelli equation and (2+1)-dimensional Konopelchenko-Dubrovsky equation are gained.In addition,there are arbitrary function in these solutions.
主要工作如下:
1.介绍了非线性科学的发展概况、一些重要的非线性演化方程和归纳总结了求非线性偏微分方程精确解的一些主要方法。
2.对齐次平衡方法做了简单的介绍,参照求解一维Toda链方程的过程,将此方法应用于(2+1)维Toda链方程,最终得到了(2+1)维Toda链方程的单孤子解、双孤子解、三孤子解和总结归纳出其N孤子解的一般形式,并对一定参数的若干解进行了数值模拟。
3.介绍了双曲正切函数展开法和拓展的Riccati方程映射法,分别利用这两种方法求得了(2+1)维Boiti-Leon-Pempinelli方程组和(2+1)维Konopelchenko-Dubrovsky方程组的若干解析解,最后得出了一定结论。
Nonlinear physics developed gradually with the development of nonlinear science.In nonlinear physics,some simplified nonlinear evolution equations are often needed to describe the complex nonlinear physics symtem.We can make sure the quantificational connection or the qualitative connection of physics quantities by solving the nonlinear evolution equation. Besides of this,we can also give the firsthand impression of the physics quantities by giving the picture of solution of the nonlinear evolution equation.Then,it is very important to the development of physics by solving the nonlinear evolution equation and giving the picture of the solution.In this paper we obtained many solutions of(2+1)-dimensional Toda lattice equation,(2+1)-dimensional Boiti-Leon-Pempinelli equation and(2+1)-dimensional Konopelchenko-Dubrovsky equation and studied characters of equations.There are mainly three sections in this dissertation.
1.Firstly,we introduced the development of nonlinear science and some nonlinear evolution equation.And the methods for solving the exact solution of nonlinear evolution equation are introduced.
2.The Homogeneous method are introduced.According to the method of solving (1+1)-dimension Toda lattice equation,The single-soliton,the double-solitons,the three-solitons and simulation results of(2+1)-dimensional Toda lattice equation have been obtained by using this method.Finally,the general form of the N-solitons solution of the equation is derived.
3.We introduced the hyperboloic function expansion method and the extend Riccati mapping approach detailly.Then those approaches were applied to the(2+1)-dimensional Boiti-Leon-Pempinelli equation and(2+1)-dimensional Konopelchenko-Dubrovsky equation. Finally,some solutions of the(2+1)-dimensional Boiti-Leon-Pempinelli equation and (2+1)-dimensional Konopelchenko-Dubrovsky equation are gained.In addition,there are arbitrary function in these solutions.
引文
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[72]J H He,X H Wu.Chaos,Solitons and Fractals 2006(30).
[73]J H He,M A Abdou.Chaos,Solitons and Fractals 2006(05).
[74]Li Ya-Zhou,Feng Wei-Gui,Li Kai-Ming and Lin Chang.Chinese Physics.2007,16(9):2510-2513
[75]G W Bluman,S Kumei.J.Math.Phys.1988(29):86.
[76]S Y Lou.Commun.Theor.Phys.1991(15):465.
[77]C T Yan,Phys.Lett.A.1996(224):77.
[78]N A Kudryashov.Phys.Lett.A.1990(147):287.
[79]B Tian and Y T Gao.Phys.Lett.A.1990(212):247.
[80]W Malfliet.Am J Phys.1992,60(7):650.
[81]P A Clarkson,M D Kruskal.J.Math.Phys,1989(30):2201.
[82]刘式适,付遵涛,刘式达,赵强.物理学报,2002,51(1),10.
[83]刘式适,付遵涛,刘式达,赵强.物理学报,2002,51(9),1923.
[84]S K Liu,Z T Fu,S D Liu,Q Zhao.Phy.Lett.A.2001(289):69.
[85]廖世俊,1992年博士论文.(上海交通大学)第1页.
[86]P J Olver.Application of Lie Group to differential equation,New York:Springer,1986
[87]G W Bluman,S Kumei.Symmetries and differential equations,Berlin:Springer,1989.
[88]李志斌,张善卿.数字物理学报,1997(17):81.
[89]吴文俊.几何定理机械证明的基本原理,科学出版社,1984.
[90]吴文俊.中国科学,1977,6:507;数学物理学报,1982,2:125;数学学报,1986,3:204;中国科学,数学专辑(i),1979,94;系统科学与数学,1984,4:207.
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[92]H Shi.Proc.1992 International Workshop Math.Mech.International Acad- emic,1992,79.
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