非线性系统的对称性约化和孤立子研究
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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学科。如生物学、化学、通讯和几乎所有的物理分支,如凝聚态物理、场论、低温物理、流体力学、等离子物理、光学等等,这之中涌现了大量的非线性系统。为此,人们很自然地考虑到:如何求解描述非线性系统的非线性偏微分方程呢?非线性系统的解具有什么样的特性呢?如何对非线性耦合系统进行对称约化,求精确解?如何构造非线性耦合偏微分方程的Lie-B(a丨¨)cklund变换?
通过众多科学家的努力,人们已经建立和发展了很多求解非线性系统的有效方法,特别是针对其中一些被归为可积的非线性系统。常用的方法有反散射方法、达布变换方法、贝克隆变换方法、分离变量法、双线性方法和多线性方法、经典和非经典李群法、CK直接法、形变映射法、Painlevé截断展开法、函数展开法等。
但是随着孤立子理论的发展,许多实际问题不能简化为一维问题,或者简化成一维问题后会失去它的重要特征,或者不能简化为单个方程问题。例如在研究粒子物理中的单极子、瞬子等问题遇到的是高维非线性系统;在研究两根耦合光纤的相互利用时,所遇到的是耦合NLS方程组;又如描述大气和海洋现象的二层流体模型问题时遇到的是耦合KdV演化方程组等等,因此由一维到多维、由单一的孤子演化方程到耦合演化方程组的研究是当前国际研究热点对象之一。本论文主要围绕着非线性耦合系统展开了讨论研究。
对称性研究是自然科学中的最基本方法之一。在可积模型的研究中,由于无穷多对称和守恒律的存在,对称性研究就更为重要。通过许多数学家和物理学家的努力,在连续可积模型中建立了许多强有力的方法。
对于给定的非线性偏微分方程,传统的对称群研究方法通常是局限于寻找点李对称群。在标准的李群理论中,研究无穷小形式就足够了。通过解相应的常微分方程组的初值问题就可以唯一地求得对应的Lie群及Lie代数。然而,这对于非线性偏微分方程来说还是不够的。还有不少的问题有待研究,比如说:(1)即使得到了李代数,相应的求解初值问题来得到有限变换即对称群还是一件十分困难的事。(2)在很多情况下,即使能得到初值问题的解,其显式表达式仍然是繁琐异常的,在实际当中很难得到应用。(3)一般的对称群根本就不是所谓的Lie群,而是更为一般的连续群。显然,这样就使得这些非线性系统的求解难上加难。
众所周知,对于求解非线性系统,特别是求解非线性系统的约化和约化解,存在着三大方法:CK直接法和经典、非经典李群法。前者是从代数的角度来求解非线性偏微分方程的,而后者的求解是基于群论的。在非线性系统的对称性约化研究中传统的看法已相当完善,一些标准类型的约化已被穷尽,为此我们必须给出一些寻求新对称性约化的新思路对非线性耦合系统进行对称约化。通常,增加或者削弱研究过程中的某些限制条件会引发一些新的结果。
本文第二章介绍了求解非线性系统的一些常用方法,并阐述了一些李群在微分方程中应用所涉及的有关基本概念。在第三章中我们首先介绍了一类新的具有实际应用背景的非线性耦合系统。把经典李群法推广应用到非线性耦合系统,然后把这个方法应用到耦合KdV系统,我们得到了该系统的不变群、群不变解、李代数结构等,同时我们还应用这个方法通过求出一个典型的耦合KdV系统的对称,首次用势对称研究了可积耦合非线性方程组,从而得到了此系统的Lie-B(a丨¨)cklund变换。我们还发现用经典李群法和点李对称法所得到的约化是相同的。在第四章中,我们通过增强约束条件,把非经典李群法推广应用到非线性耦合系统,然后把这个方法应用到耦合KdV系统,从而求得更多的新解。在第五章中,我们建立了一种改进了的直接约化法,将修正的直接约化法应用到非线性耦合系统。结果发现,用改进的直接约化法得到的约化包括了经典李群法所得结果,且任一个直接约化法得到的相似约化都可以对应找到一个群论解释。本文的第六章,给出了强色散DGH方程丰富的解,如钟型孤立波解、紧致孤立波解、尖峰孤立波解、双峰孤立波解、奇异孤立波解、指数解及周期波解的一般式等。证明了强色散DGH方程在双Hamilton结构和无穷多对称意义下可积。对强色散DGH方程用WTC法进行了奇异分析。利用齐次平衡法得到了强色散DGH方程的B(a丨¨)cklund变换,结果表明,用Painlevé截断展开法和用齐次平衡法得到方程的B(a丨¨)cklund变换是等价的。同时,我们利用齐次平衡法得到了该系统的对称约化,并且说明了用CK直接法和利用齐次平衡法得到的相似约化是等价的。
在本文的最后我们展望了今后的研究工作。
本文的创新与特色是:
(1)在研究对象上,主要研究一类新的具有实际应用物理背景的非线性耦合系统,这些耦合非线性系统首先是在两层流体体系中导出。他们可以描述许多需要多层流体体系刻画的物理问题,如气象科学,海洋物理学以及大气和海洋相互作用等等。
(2)在思想上,提出了一种改进直接约化法新思路:我们对直接法假设形式进行了修正,提出一个原场(约化前场)可以与两个约化场相联系,而在原方法中,一个原场只可以联系于一个约化场。
(3)在方法上,我们对传统做法中约束条件进行了修改,即不要求约化方程所有项系数成一定比例关系,而是要求部分项系数之间有一定比例关系,而且不同部分之间比例关系只是时空变量函数而不是群不变量函数。
(4)用非点李对称(势对称)研究非线性方程组对称性约化和严格解.用势对称对单个方程研究国际上已有不少先例。Bluman等用势对称方法研究了一些C可积模型(如HBurgers),楼森岳等用势对称方法研究了一些S可积模型(如KdV)。本文首次用势对称研究了S可积耦合非线性方程组。
(5)在内容上,得到了非线性耦合系统更多的条件相似约化解并给出了群论解释;另一方面我们得到了强色散DGH方程的一些新的孤立子解。
The nonlinear science is substantially studied and widely applied in natural sciences such as biology,chemistry,communication and almost all the physical branches like condensed matters,field theory,low temperature physics,hydrodynamics,plasma physics, optics and so on,where a large amount of nonlinear systems emerge.Therefore,a lot of questions are naturally asked:How to solve these nonlinear partial differential equations which describe the nonlinear systems? What kind of characters do the solutions of these nonlinear systems possess? How to symmetry reduction and solve for a coupled nonlinear system? How to construct B(a|¨)cklund transformation for a partial differential equation?
Through many efforts of scientists,many methods have been established and developed to solve the nonlinear systems especially those integrable ones.For instance, the Inverse scattering transformation,Darboux transformation,B(a|¨)cklund transformation, Functional variable separation approach,Bilinear method and Multilinear method, Classical and non-classical Lie group approaches,Clarkson-Kruskal's direct method,Deformation mapping method,Truncated Painlevéexpansion,Function expansion method and so on.
As the development of soliton theory,some actual problems cannot be simplified to either one-dimention or mono-equation,like high-dimention non-linear system should be considered when studying on monopole in particle physics,or coupling NLS equation group should be concerned when dealing with interaction between coupling optical fiber, or coupling KdV equation group in two-layer fluid model when describing both atmosphere phenomenon and sea phenomenon,etc.Otherwise,its important characteristics will lost. Therefore,study on from mono-dimention to muti-dimention,from mono-solition equation to coupling equation group is a hot object of nowaday's international research.This dissertation is carried out around the nonlinear coupled systems.
Symmetry method is one of the most fundamental methods in natural science.In the study of the integrable models,symmetries play an important role because there exist infinitely many symmetries and conservation laws.Through many efforts of mathematicians and physicists,a lot of powerful approaches have been established in the continuous integrable systems.
In the traditional studies of the symmetry group of a given nonlinear PDE,one usually restrict himself to find the Lie point symmetry group.In the standard Lie group theory, it is in principle enough to study its infinitesimal form,Lie algebra,because the related Lie group can be uniquely obtained by solving an initial problem related to an ordinary differential equation(ODE) system.However,for a given nonlinear PDE,there are still some serious problems.For instance,(ⅰ) once the Lie algebra is obtained,it is still very difficult to solve the related initial problem to give out the finite transformations,the symmetry group.(ⅱ) In many cases,even if the initial problem can be obtained,the final expressions are very complicated and it is not convenient in the real applications.(ⅲ) In some other cases,the general symmetry groups of the nonlinear systems are not Lie groups at all and there may be some types of more general continuous groups.Obviously, it is very difficult or even impossible to solve all these problem for any nonlinear systems.
As is well known,to solve nonlinear systems especially to obtain reduction solutions, there are three fundamental methods,CK's direct method,and classical/nonclassical Lie group approach.The former solves PDEs algebraically and the latter is based on group theory.It is believed that traditional reduction methods are quite perfect and some standard types of reduction solutions are exhausted.So,one has to find some novel ideas to get some new types of symmetry reductions.
In chapter 2,many methods have been briefly narrated to solve the nonlinear systems, and basis concept have been introduced for applications of Lip groups to differential equations.In chapter 3,a kind nonlinear systems which has the actuai applied physics background have been introduced.we extend the classical Lie group approaches to coupled nonlinear partial differential system,and then apple it to coupled KdV nonlinear partial differential system,we have obtained symmetry groups,Lie algebras of local Lie groups group-invariant solutions and so on.The especial symmetry be obtained to a classical coupled KdV nonlinear partial differential system by use the classical Lie group approaches,and introducing some potentials for a coupled KdV equation,then we obtain Lie-B(a|¨)cklund transformation for the coupled system.Using the classical Lie group approaches to find the symmetry reductions of nonlinear evolution equations is equal to Lie point symmetry method for the reduction of equation.In chapter 4,we extend the nonclassical Lie group approaches to coupled nonhnear partial differential system,and then apple it to coupled KdV nonlinear partial differential system by strengthening some constraints,that will lead to new results.In chapter 5,we discuss the the modified Clarkson and Kruskal's direct method,and the modified CK direct method is used to reduce the coupled nonlinear partial differential system.The result tells us that the results obtained by the CK direct method contain those obtained by the classical Lie group approach and the results of the nonclassical Lie approach include those of the direct method,and the diect method can have a corresponding group explanation.In chapter 6, abundant exact solutions are obtained such as the bell shaped soliton,the compacton,the peakon,Double solitary wave solution with peakon,singular solitary wave solution,the exponential solution,the periodic travelling wave solution and so on.it is proved that the strong dispersive DGH equation is integrable under the meaning that it possesses Bi-Hamiltonian structure and infinitely many symmetries.The singularity analysis of the strong dispersive DGH equation is performed by using the WTC method.By homogenous balance(HB) method,B(a|¨)cklund transformation for the strong dispersive DGH equation is established.The result tells us that using the truncated Painlevéexpansion method to find the B(a|¨)cklund transformation of nonlinear evolution equations is equal to by homogenous balance(HB) method.The symmetry reduction of this equation is deduced by the HB method.The result tells us that using the homogeneous balance method to find the symmetry reductions of nonlinear evolution equations is equal to CK direct method for solving similarity solutions.
This dissertation ends with summary and research prospects.
The innovations and features of this dissertation are as follows:
(1)New integrability model:we mostly study some new types of coupled KdV equation systems with some arbitrary parameters from a two-layer fluid model.They can describe many physical problem for multiple-layered fluid system,such as the atmospheric blockings, the interactions between the atmosphere and ocean,the oceanic circulations and even hurricanes or typhoons.
(2)New ideas:Put forward a new idea of using modified direct method,we have modified the suppositional form of the direct method.It is proposed that one original field can be related to two reduced fields while in the old approach,one original field can only be related to one reduced field.
(3)New method:We are enlightened to modify the constraints for the traditional direct method,namely,require parts of the coefficients have one common ratio instead of all and the ratios among different parts should be only a function with respect to space time variables.
(4) After introducing some potentials for a coupled KdV equation which is derived from the two layer fluid model,the nonlocal Lie B(a|¨)cklund transformation is obtained. Using the Lie B(a|¨)klund transformation theorem to the trivial zero solution,the single soliton solution is also found.
(5)New results:Get conditional similarity reduction solutions of coupled nonlinear systems and give out their corresponding group explanations;obtain the new soliton of the strong dispersive DGH equation.
通过众多科学家的努力,人们已经建立和发展了很多求解非线性系统的有效方法,特别是针对其中一些被归为可积的非线性系统。常用的方法有反散射方法、达布变换方法、贝克隆变换方法、分离变量法、双线性方法和多线性方法、经典和非经典李群法、CK直接法、形变映射法、Painlevé截断展开法、函数展开法等。
但是随着孤立子理论的发展,许多实际问题不能简化为一维问题,或者简化成一维问题后会失去它的重要特征,或者不能简化为单个方程问题。例如在研究粒子物理中的单极子、瞬子等问题遇到的是高维非线性系统;在研究两根耦合光纤的相互利用时,所遇到的是耦合NLS方程组;又如描述大气和海洋现象的二层流体模型问题时遇到的是耦合KdV演化方程组等等,因此由一维到多维、由单一的孤子演化方程到耦合演化方程组的研究是当前国际研究热点对象之一。本论文主要围绕着非线性耦合系统展开了讨论研究。
对称性研究是自然科学中的最基本方法之一。在可积模型的研究中,由于无穷多对称和守恒律的存在,对称性研究就更为重要。通过许多数学家和物理学家的努力,在连续可积模型中建立了许多强有力的方法。
对于给定的非线性偏微分方程,传统的对称群研究方法通常是局限于寻找点李对称群。在标准的李群理论中,研究无穷小形式就足够了。通过解相应的常微分方程组的初值问题就可以唯一地求得对应的Lie群及Lie代数。然而,这对于非线性偏微分方程来说还是不够的。还有不少的问题有待研究,比如说:(1)即使得到了李代数,相应的求解初值问题来得到有限变换即对称群还是一件十分困难的事。(2)在很多情况下,即使能得到初值问题的解,其显式表达式仍然是繁琐异常的,在实际当中很难得到应用。(3)一般的对称群根本就不是所谓的Lie群,而是更为一般的连续群。显然,这样就使得这些非线性系统的求解难上加难。
众所周知,对于求解非线性系统,特别是求解非线性系统的约化和约化解,存在着三大方法:CK直接法和经典、非经典李群法。前者是从代数的角度来求解非线性偏微分方程的,而后者的求解是基于群论的。在非线性系统的对称性约化研究中传统的看法已相当完善,一些标准类型的约化已被穷尽,为此我们必须给出一些寻求新对称性约化的新思路对非线性耦合系统进行对称约化。通常,增加或者削弱研究过程中的某些限制条件会引发一些新的结果。
本文第二章介绍了求解非线性系统的一些常用方法,并阐述了一些李群在微分方程中应用所涉及的有关基本概念。在第三章中我们首先介绍了一类新的具有实际应用背景的非线性耦合系统。把经典李群法推广应用到非线性耦合系统,然后把这个方法应用到耦合KdV系统,我们得到了该系统的不变群、群不变解、李代数结构等,同时我们还应用这个方法通过求出一个典型的耦合KdV系统的对称,首次用势对称研究了可积耦合非线性方程组,从而得到了此系统的Lie-B(a丨¨)cklund变换。我们还发现用经典李群法和点李对称法所得到的约化是相同的。在第四章中,我们通过增强约束条件,把非经典李群法推广应用到非线性耦合系统,然后把这个方法应用到耦合KdV系统,从而求得更多的新解。在第五章中,我们建立了一种改进了的直接约化法,将修正的直接约化法应用到非线性耦合系统。结果发现,用改进的直接约化法得到的约化包括了经典李群法所得结果,且任一个直接约化法得到的相似约化都可以对应找到一个群论解释。本文的第六章,给出了强色散DGH方程丰富的解,如钟型孤立波解、紧致孤立波解、尖峰孤立波解、双峰孤立波解、奇异孤立波解、指数解及周期波解的一般式等。证明了强色散DGH方程在双Hamilton结构和无穷多对称意义下可积。对强色散DGH方程用WTC法进行了奇异分析。利用齐次平衡法得到了强色散DGH方程的B(a丨¨)cklund变换,结果表明,用Painlevé截断展开法和用齐次平衡法得到方程的B(a丨¨)cklund变换是等价的。同时,我们利用齐次平衡法得到了该系统的对称约化,并且说明了用CK直接法和利用齐次平衡法得到的相似约化是等价的。
在本文的最后我们展望了今后的研究工作。
本文的创新与特色是:
(1)在研究对象上,主要研究一类新的具有实际应用物理背景的非线性耦合系统,这些耦合非线性系统首先是在两层流体体系中导出。他们可以描述许多需要多层流体体系刻画的物理问题,如气象科学,海洋物理学以及大气和海洋相互作用等等。
(2)在思想上,提出了一种改进直接约化法新思路:我们对直接法假设形式进行了修正,提出一个原场(约化前场)可以与两个约化场相联系,而在原方法中,一个原场只可以联系于一个约化场。
(3)在方法上,我们对传统做法中约束条件进行了修改,即不要求约化方程所有项系数成一定比例关系,而是要求部分项系数之间有一定比例关系,而且不同部分之间比例关系只是时空变量函数而不是群不变量函数。
(4)用非点李对称(势对称)研究非线性方程组对称性约化和严格解.用势对称对单个方程研究国际上已有不少先例。Bluman等用势对称方法研究了一些C可积模型(如HBurgers),楼森岳等用势对称方法研究了一些S可积模型(如KdV)。本文首次用势对称研究了S可积耦合非线性方程组。
(5)在内容上,得到了非线性耦合系统更多的条件相似约化解并给出了群论解释;另一方面我们得到了强色散DGH方程的一些新的孤立子解。
The nonlinear science is substantially studied and widely applied in natural sciences such as biology,chemistry,communication and almost all the physical branches like condensed matters,field theory,low temperature physics,hydrodynamics,plasma physics, optics and so on,where a large amount of nonlinear systems emerge.Therefore,a lot of questions are naturally asked:How to solve these nonlinear partial differential equations which describe the nonlinear systems? What kind of characters do the solutions of these nonlinear systems possess? How to symmetry reduction and solve for a coupled nonlinear system? How to construct B(a|¨)cklund transformation for a partial differential equation?
Through many efforts of scientists,many methods have been established and developed to solve the nonlinear systems especially those integrable ones.For instance, the Inverse scattering transformation,Darboux transformation,B(a|¨)cklund transformation, Functional variable separation approach,Bilinear method and Multilinear method, Classical and non-classical Lie group approaches,Clarkson-Kruskal's direct method,Deformation mapping method,Truncated Painlevéexpansion,Function expansion method and so on.
As the development of soliton theory,some actual problems cannot be simplified to either one-dimention or mono-equation,like high-dimention non-linear system should be considered when studying on monopole in particle physics,or coupling NLS equation group should be concerned when dealing with interaction between coupling optical fiber, or coupling KdV equation group in two-layer fluid model when describing both atmosphere phenomenon and sea phenomenon,etc.Otherwise,its important characteristics will lost. Therefore,study on from mono-dimention to muti-dimention,from mono-solition equation to coupling equation group is a hot object of nowaday's international research.This dissertation is carried out around the nonlinear coupled systems.
Symmetry method is one of the most fundamental methods in natural science.In the study of the integrable models,symmetries play an important role because there exist infinitely many symmetries and conservation laws.Through many efforts of mathematicians and physicists,a lot of powerful approaches have been established in the continuous integrable systems.
In the traditional studies of the symmetry group of a given nonlinear PDE,one usually restrict himself to find the Lie point symmetry group.In the standard Lie group theory, it is in principle enough to study its infinitesimal form,Lie algebra,because the related Lie group can be uniquely obtained by solving an initial problem related to an ordinary differential equation(ODE) system.However,for a given nonlinear PDE,there are still some serious problems.For instance,(ⅰ) once the Lie algebra is obtained,it is still very difficult to solve the related initial problem to give out the finite transformations,the symmetry group.(ⅱ) In many cases,even if the initial problem can be obtained,the final expressions are very complicated and it is not convenient in the real applications.(ⅲ) In some other cases,the general symmetry groups of the nonlinear systems are not Lie groups at all and there may be some types of more general continuous groups.Obviously, it is very difficult or even impossible to solve all these problem for any nonlinear systems.
As is well known,to solve nonlinear systems especially to obtain reduction solutions, there are three fundamental methods,CK's direct method,and classical/nonclassical Lie group approach.The former solves PDEs algebraically and the latter is based on group theory.It is believed that traditional reduction methods are quite perfect and some standard types of reduction solutions are exhausted.So,one has to find some novel ideas to get some new types of symmetry reductions.
In chapter 2,many methods have been briefly narrated to solve the nonlinear systems, and basis concept have been introduced for applications of Lip groups to differential equations.In chapter 3,a kind nonlinear systems which has the actuai applied physics background have been introduced.we extend the classical Lie group approaches to coupled nonlinear partial differential system,and then apple it to coupled KdV nonlinear partial differential system,we have obtained symmetry groups,Lie algebras of local Lie groups group-invariant solutions and so on.The especial symmetry be obtained to a classical coupled KdV nonlinear partial differential system by use the classical Lie group approaches,and introducing some potentials for a coupled KdV equation,then we obtain Lie-B(a|¨)cklund transformation for the coupled system.Using the classical Lie group approaches to find the symmetry reductions of nonlinear evolution equations is equal to Lie point symmetry method for the reduction of equation.In chapter 4,we extend the nonclassical Lie group approaches to coupled nonhnear partial differential system,and then apple it to coupled KdV nonlinear partial differential system by strengthening some constraints,that will lead to new results.In chapter 5,we discuss the the modified Clarkson and Kruskal's direct method,and the modified CK direct method is used to reduce the coupled nonlinear partial differential system.The result tells us that the results obtained by the CK direct method contain those obtained by the classical Lie group approach and the results of the nonclassical Lie approach include those of the direct method,and the diect method can have a corresponding group explanation.In chapter 6, abundant exact solutions are obtained such as the bell shaped soliton,the compacton,the peakon,Double solitary wave solution with peakon,singular solitary wave solution,the exponential solution,the periodic travelling wave solution and so on.it is proved that the strong dispersive DGH equation is integrable under the meaning that it possesses Bi-Hamiltonian structure and infinitely many symmetries.The singularity analysis of the strong dispersive DGH equation is performed by using the WTC method.By homogenous balance(HB) method,B(a|¨)cklund transformation for the strong dispersive DGH equation is established.The result tells us that using the truncated Painlevéexpansion method to find the B(a|¨)cklund transformation of nonlinear evolution equations is equal to by homogenous balance(HB) method.The symmetry reduction of this equation is deduced by the HB method.The result tells us that using the homogeneous balance method to find the symmetry reductions of nonlinear evolution equations is equal to CK direct method for solving similarity solutions.
This dissertation ends with summary and research prospects.
The innovations and features of this dissertation are as follows:
(1)New integrability model:we mostly study some new types of coupled KdV equation systems with some arbitrary parameters from a two-layer fluid model.They can describe many physical problem for multiple-layered fluid system,such as the atmospheric blockings, the interactions between the atmosphere and ocean,the oceanic circulations and even hurricanes or typhoons.
(2)New ideas:Put forward a new idea of using modified direct method,we have modified the suppositional form of the direct method.It is proposed that one original field can be related to two reduced fields while in the old approach,one original field can only be related to one reduced field.
(3)New method:We are enlightened to modify the constraints for the traditional direct method,namely,require parts of the coefficients have one common ratio instead of all and the ratios among different parts should be only a function with respect to space time variables.
(4) After introducing some potentials for a coupled KdV equation which is derived from the two layer fluid model,the nonlocal Lie B(a|¨)cklund transformation is obtained. Using the Lie B(a|¨)klund transformation theorem to the trivial zero solution,the single soliton solution is also found.
(5)New results:Get conditional similarity reduction solutions of coupled nonlinear systems and give out their corresponding group explanations;obtain the new soliton of the strong dispersive DGH equation.
引文
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