摘要
在非线性科学中,非线性波动方程精确解的研究有助于理解孤立子理论的本质属性和代数结构,而且对相应自然现象的合理解释及实际应用将起到重要的作用.
本论文首先借鉴了非线性物理的对称约化思想和线性物理中的分离变量理论,对处理非线性问题的直接代数法和多线性分离变量法进行了研究和推广,对映射变换理论进行了创新,得到了一些新的结果.将基于行波约化的代数方法推广应用到了非线性离散系统和复杂的非线性系统,寻求其精确的行波解和近似解.然后,根据非线性系统的映射变换解和多线性分离变量解,分别讨论了(2+1)维局域激发模式及其相关的非线性动力学行为.围绕一些具有深刻物理背景的非线性波动方程的局域激发模式及其相关非线性特性—分形特征和混沌行为展开了讨论,这些非线性系统源于流体力学、等离子体物理、固体物理、超导物理、凝聚态物理和光学等实际问题.本文研究表明,Charkson-Kruskal的直接约化方法,映射变换方法和多线性分离变量方法蕴藏着内在的有机联系.另外,本文所得结果说明混沌和分形存在于高维非线性系统是相当普遍的现象.现将本文的主要内容概述如下:
第一章,简要回顾了孤立波的发现与研究历史,概述了当前研究孤子解的一些基本方法,其中包括:反散射方法,Darboux变换和B(?)cklund变换,Painlevé分析法和Hirota双线性方法等,最后给出了本论文的主要工作和结构按排.
第二章,首先以Boussinesq方程为例,介绍了寻求非线性波动方程相似约化解的三个基本方法:经典李群法、非经典李群法和CK(Clarkson和Kruskal)直接法.然后以sine-Gordon方程组为例,给出了楼和马最近提出的对CK直接法进行的一种修正.在他们的方法中,并没有对原方程进行低维约化,而是在要求自变量不减少的情况下,直接地得到原方程的李对称群.同时,如果利用多直线孤子解的群变换,可以得到各种多曲线激发.最后,仍基于CK直接相似约化的思想,给出了目标约化的一般理论,即:对一个给定的非线性方程,事先建立一个目标函数,通过假设得到的相似约化方程为常微分方程,将其中的系数分解整理成多个规范系数的比率,而不仅仅只有一个.由此,来得到非线性系统有着更加直接而简便的丰富的精确解,其中包括孤立波解,周期波解和有理函数解.
第三章,利用CK直接相似约化思想,给出了非线性方程的映射变换理论,突破了已有映射理论只能得到系统行波解的约束,并成功地运用到了多个非线性波动方程组中,如:(2+1)维色散长波方程组、(2+1)维广义Broer-Kaup方程组、(1+1)维非线性Schr(?)dinger方程组、修正的(2+1)维色散水波方程组、(2+1维Nizhnik-Novikov-Veselov方程组等,得到了新型的变量分离的精确解.根据所求得的映射解,分析了若干新的或典型的局域激发模式,如:双周期结构孤子,单值与多值复合的半折叠孤子,裂变孤子和聚合孤子及其演化行为特性等.讨论了一些典型孤子所蕴涵的分形特征和混沌动力学行为.研究结果表明混沌、分形存在于高维非线性系统是相当普遍的现象,其根源在于可积系统的初始状态或边界条件具有“不可积”的分形特性或混沌行为.修正了人们长期认为孤波产生于可积非线性系统而混沌、分形只存在于不可积非线性系统的认识局限性.分析表明,所有由多线性分离变量法得到的(2+1)维非线性系统的局域激发,利用映射变换理论也可以找到.映射变换方法不仅突破了原映射理论只能求解非线性系统行波解的约束,还有望进一步推广应用到其它的非线性系统中,由此,丰富和发展了非线性科学的基本理论.
第四章,首先简要介绍了分离变量法在几个方向的发展情况,并以(2+1)维非线性系统为例,给出了楼提出的多线性变量分离法的一般步骤.然后求出了(1+1)维浅水波方程,(2+1)维mBK方程组和另外若干著名的方程组的变量分离解及其推广形式,并讨论了在普适公式下的孤子的结构及其相互作用的行为.一般说来,所得到的多线性分离变量解,可以用来描述系统场量或相应的势函数,进而讨论基于多线性分离变量解引起的系统局域激发及其相关非线性特性.文中报导了一些典型的局域激发模式,如:在所有方向都呈指数衰减的相干局域结构-dromion,在各个方向同时褶皱的多值孤波-折叠子,三维空间中的偶极子型孤子等.
第五章,将基于行波约化的代数方法推广应用到了非线性离散系统和复杂的非线性系统,寻求其精确的行波解和近似解.首先给出了微分-差分系统双曲函数法的一般理论,其推广形式及在非线性离散系统中的应用,然后给出了基于行波约化的形变映射理论及非线性Schr(?)dinger方程基于行波的约化解,最后则引进了Adomian分解法,来研究在一定初始条件下的高阶非线性Schr(?)dinger方程的基于行波的近似解.
第六章,给出了本文的主要结果,提出了一些未来相关研究工作的设想.
In nonlinear science,the study on the exact solution of nonlinear wave equations is helpful in clarifying the underlying algebraic structure of the soliton theory and plays an important role in reasonable explaining of the corresponding natural phenomenon and application.
In this dissertation,with the help of the symmetry reduction theory in nonlinear physics and the variable separation approach in linear physics,the direct albebra method and the multilinear variable separation approach are studied and extended to nonlinear physics successfully, then a new algorithm - the mapping transformation approach is proposed and many kinds of new results are obtained from there.To seekig for the exact travelling wave solution and approximate solution,the algebra method based on travelling wave reduction is extended and applied to some nonlinear discrete systems and complicated equations.Based on the mapping transformation solutions and the multilinear variable separation solutions respectively, abundant localized excitations and related nonlinear dynamical behaviors for some (2+1)-dimensional(two spatial-dimensions and one time dimension) nonlinear models are investigated. The localized excitations and related fractal and chaotic behaviors in nonlinear wave equations are discussed,which are orginated from many natural sciences,such as fluid dynamics,plasma physics,solid physics,superconducting physics,condensed matter physics and optical problems.The research results indicate that one can establish the relationship among the Charkson-Kruskal(CK) direct reduction method,the mapping transformation approach and the multilinear variable separation approach.Meanwhile,fractals and chaos in higher-dimensional nonlinear soliton systems are quite universal phenomena.The main contents are summarized as folows:
In the first chapter,a brief history of finding solitary waves and solitons is outlined and several improtant methods for studying soliton solutions are listed,including the inverse scatterng method,the Darboux transformation and the Backlund transformation,the Painlevéanalysis approach and the Hirota bilinear method.The research arrangement of the dissertation is given in the end of this chapter.
In the second chapter,taking an example of the Boussinesq equation,three powerful methods are presented for finding similarity reduction solutions of a nonlinear wave equa- tion,i.e.,the Lie group method of infinitesimal transformations,the nonclassical Lie group method and the Clarkson and Krnskal direct method.Starting from the Lax expression of the(2+l)-dimensional sine-Gordon system,a modified technic for the CK direct method was put forward by Lou and Ma recently.The symmetry group and then the Lie symmetries and the related algebra can be reobtained via a simple combination of a gauge transformation of the spectral function and the transformations of the space time variables.Meanwhile,applying the group transformation theorem on the multiple straight line soliton solutions,one can obtain various types of multiple curved line excitations.Finally,the objective reduction approach is presented based on the idea of the above CK direct method.The main theory of this method is:For a given nonlinear equation,an objective function is first established, supposing that the obtained similarity reduction equation is an ordinary partial equation, through setting one coefficient as the normalizing coefficient in each part of this equation,we may separate all coefficients into several parts and require that the ratios of these coefficients satisfy the same objective reduction equation.From there,the abundant exact solutions of a given system are derived,including solitary wave solutions,periodic wave solutions and rational function solutions.
In the third chapter,a new mapping transformation for a nonlinear equation is proposed using the CK direct similarity reduction theory.The approach breaks through the traditional idea,which only the travelling wave solution of a nonlinear equation can be obtained,and is applied to some nonlinear wave equations,such as the(2+1)-dimensional dispersive long wave system,the(2+1)-dimensional generalized Broer-Kaup system,the(1+1)-dimensional nonlinear Schrodinger system,the(2+1)-dimensional modified dispersive water-wave system, and the(2+1)-dimensional Nizhnik-Novikov-Veselov system.Based on the new variable separation solutions,some new or typical localized coherent excitations and their evolution properties are revealed.By introduing suitable arbitrary functions,considerably novel localized structures are constructed,such as doubly periodic patterns from the Jacobi elliptic functions,semifolded localized structures including multi-valued and single-valued solitons, and certain localized excitations with fission and fusion behaviors.Some typical localized excitations with fractal property and chaotic behavior are also discussed.Why the localized excitations possess such kinds of chaotic behavior and fractal property? If one considers the boundary or initial condition of the chaotic and fractal solutions obtained here,one can find straightforwardly that the initial or boundary condition possesses chaotic and fractal properties.These chaotic and fractal properties of the localized excitations for an integrable model essentially come from certain "nonintegrable" chaotic and fractal boundary or initial condition.From these theoretical results,one may interpret that chaos and fractals in higherdimensional integrable physical models would be a quite universal phenomenon.Therefore, all the localized excitations based on the multilinear variable separation solution can be rederived by the mapping transformation solution.Meanwhile,we have established a simple relation between the multilinear variable separation solutions and the mapping transformation solutions,which are essentially equivalent by taking certain variable transformation.The mapping transformation approach not only outbreaks its original limitation merely searching for traveling wave solutions to nonlinear systems,but also can be extended to many other nonlinear dynamical systems,which also means that the mapping approach has been richened and developed to the basic theory of nonlinearity.
In the fouth chapter,several studying aspects of the variable separation method are first introduced briefly,and then the general process for a(2+1)-dimensional nonlinear system of the Lou's multilinear variable separation approach is given.As a result,the variable separation solutions and their generalizations of the(1+1)-dimensional shallow water wave equation, the(2+1)-dimensional mBK equations and some other famous equations are solved.Some localized excitations and their interaction behaviours are revealed under a quite "universal" variable separation formula by selecting appropriate initial and/or boundary conditions. Based on the plots and theoretical analysis,we explored some typical localized excitations. Dromions are localized solutions decaying exponentially in all directions,which can be driven not only by straight line solitons but also driven by curved line solitons and can be located not only at the cross points of the lines but also at the closed points of the curves.Foldons are a class of multi-valued solitary waves,which can be folded in all directions.The fractal solitons and chaotic solitons reveal fractal characteristics and chaotic dynamic behaviors in solitary waves,respectively.The dipole-type dromion and the paraboloid-type camber soliton solution are plotted in their equipotential form or projective density form for the(3+1)-dimensional Burgers equation.
In the fifth chapter,to seekig for the exact travelling wave solution and approximate solution for some nonlinear discrete systems and complicated equations,the algebra method based on travelling wave reduction is extended and applied to them.The tanh function approach is devised for the nonlinear differential-difference equations,its two generalized applications are presented.The deformation mapping theory based on travelling wave reduction and the reduction solutions of the nonlinear Schrodinger equation are given.At last,the Adomian decomposition method is implemented for solving a hlgher-order nonlinear Schrodinger equation in atmospheric dynamics with the initial condition to obtain the approximate solutions based on the travelling wave transformation.
Finally,some main and important results as well as future research topics are outlined in the last chapter.
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