【作者】 张祥芝；

【导师】 张玉峰；

【作者基本信息】 中国矿业大学 ， 应用数学， 2020， 博士

【摘要】 本文研究了非线性科学中几类可积系统的生成及相关性质.主要有以下几个方面的工作:利用Loop代数及屠格式方法生成了等谱和非等谱的可积族,并得到其中一个方程族的守恒律;利用屠格式生成了(1+1)-维、(2+1)-维离散可积族及其扩展可积族;利用R-矩阵生成了Toda晶格系统及其扩展离散系统;用对称约化的方法得到了一类广义的浅水波方程,并进一步得到了它的Lax对、对称、不变解和序列解及其对应的自伴系统和守恒律;最后研究了时间分数阶Burgers系统的相似解和数值解,给出了数值模拟及误差估计.第一章分别介绍了非线性科学及可积系统的研究背景和发展现状,数学物理中重要的可积系统的生成方法,可积系统的求解方法,分数阶偏微分方程的研究背景和发展现状,最后阐明了本文的主要工作.第二章利用屠格式生成几类连续的和离散的可积系统.第一小节我们得到了等谱和非等谱的Lax对,并利用屠格式生成了等谱和非等谱的可积族.第二小节我们利用屠格式得到(1+1)-维可积族及其哈密顿结构,另外生成了(2+1)-维离散可积族.而且还利用势函数得到一个新的差分-微分方程.接着我们求出这些方程族的哈密顿结构、遗传算子及对称.另外,还建立了等谱方程族的B¨aclund变换.对等谱方程族约化之后,我们得到了新的长水波方程族,并利用李群方法求出了它的相似解、非相似解和非线性自伴随.最后,我们利用变量平衡法分析了长水波方程族的无穷守恒律.第三章利用R-矩阵方法推导出在统计物理和量子物理等学科具有广泛应用的Toda晶格系统.首先使用R-矩阵构造了一个新的离散可积系统生成公式,得到了扩展的Toda晶格及其Lax对.接着我们再次利用这个公式,得到相应的(2+1)-维Toda晶格系统及它的扩展离散系统,并且求出了它们的Lax对.最后,我们得到了(1+1)-维广义Toda晶格系统和一个新的(2+1)-维晶格系统的无穷守恒律.第四章我们将一类广义的长水波系统约化为标准水波系统,并进一步得到了广义浅水波的Lax对、对称、不变解和序列解.另外,我们还研究了长水波系统对应的自伴系统和守恒律.第五章了讨论了时间分数阶Burgers系统的相似解.利用Lie点对称,将分数阶偏微分方程转化为Riemann-Liouville型的分数阶常微分方程,从而得到了方程的相似解和数值解.另外利用尺度变换,将分数阶偏微分方程转化为Caputo意义下的分数阶常微分方程,我们发现它的解可以用β函数表示.最后我们还得到这种近似方程的数值解.第六章总结了本文工作并对未来进一步的研究工作进行了展望.本论文有图3幅,表3个,参考文献169篇.更多还原

【Abstract】 This dissertation investigates generation and related properties of several kinds of integrable systems in non-linear science.The main work is as follows: Loop algebra and Tu-Scheme methods are used to generate isospectral and non-isospectral integrable equations hierarchy and a conservation law of one of the equations;(1 +1)-dimensional,(2 + 1)-dimensional discrete integrable hierarchy and their extended integrable hierarchy are generated using the Tu-scheme;the R-matrix method is applied to construct Toda lattice systems and Extended discrete systems;a class of generalized shallow water wave equations are obtained by means of symmetric reduction,and their Lax pairs,symmetric,invariant solutions,and sequential solutions and their corresponding self-adjoint systems and conservation laws are obtained;similarity solutions and numerical solutions of time-fractional Burgers system are given.Numerical simulation and error estimation are given.In Chapter 1,an introduction is devoted to review the research background and current situation of nonlinear science and integrable systems,the important integrable system generation methods in mathematical physics,several methods for seeking exact solutions of the integrable system,the research background and current situation of fractional partial differential equations.Finally,the main work of this dissertation is clarified.In Chapter 2,the Tu-scheme method is used to generate several kinds of continuous and discretitye integrable systems.In the first section,we obtain isospectral and non-isospectral Lax pairs,and generate isospectral and non-isospectral integrable hierarchy by the Tu-scheme.In the second part of this chapter,we still use the Tu-scheme to obtain the(1 + 1)-dimensional and the(2 + 1)-dimensional discrete integrable systems.We also use the potential function to obtain a new difference-differential equation.In the third part,we find the Hamilton structure,genetic operator and symmetry of these integrable systems.In addition,we have established the B¨aclund transform of the isospectral equation hierarchy.A kind of isospectral equation is reduced to a new longwater wave equation.The Lie Group method is used to find similarity solutions,nonsimilarity solutions and non-linear self-adjoint solutions of these equations.Finally,we use the Variational method to analyze the infinite conservation law of the long-water wave equations.In Chapter 3,the R-matrix method is applied to derive a Toda lattice system that is widely used in statistical physics and quantum physics.First,a new discrete integrable system generation formula is constructed using the R-matrix,and the extended Toda lattice system and its Lax pair are obtained.Then we use this formula again to get the corresponding(2 + 1)-dimensional Toda lattice system and its extended discrete system,and find their Lax pair.Finally,we get the infinite conservation laws of a(1+1)-dimensional generalized Toda lattice system and a new(2 + 1)-dimensional lattice system.In Chapter 4,we reduce a class of generalized long water wave system to a standard water wave system,and further obtain Lax pairs,symmetric,invariant solutions,and sequence solutions of generalized shallow water waves.In addition,we also study the corresponding self-adjoining systems and conservation laws of the long-water wave system.In Chapter 5,the similarity solutions of the time fractional Burgers system are discussed.Using Lie-point symmetry and variational method,the fractional partial differential equations are transformed into Riemann-Liouville type fractional ordinary differential equations,and the similarity solutions and numerical solutions equations are obtained.In addition,the scale transformation is used to transform the fractional partial differential equation into a fractional ordinary differential equation in the Caputo sense.We find that its solution can be expressed by the β function.Finally,we also get this approximation Numerical solution of the equation.In Chapter 6,the primary of this dissertation and further work is prospected.There are 3 figures,3 tables and 169 references in this dissertation.更多还原