非线性微分方程与超离散方程的若干求解和可积性问题研究
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摘要
本文以数学机械化思想和AC=BD理论为指导,以构造性变换及符号计算软件为辅助工具,从代数曲线和Riemann theta函数的角度来研究非线性偏微分方程,离散方程的精确解,超椭圆函数解,拟周期解；超离散方程的Lax可积性和热带Riemann theta函数解等相关问题.
第一章介绍数学机械化、孤立子理论、可积性与代数几何解、超离散方程的求解与可积,几种求解数学物理方程的构造性方法的历史发展和研究现状,并介绍本文的选题及主要工作.
第二章首先介绍了张鸿庆教授提出的数学机械化中的AC=BD模式和应用,其次在C-D对的理论框架下,利用微分伪带余除法,构造性给出了求方程间的变换的方法.并通过研究一种类型的算子D,给出Dv=0更多形式的解,从而推广了一类辅助方程展开法,给出了一类非线性发展方程更多形式的精确解.
第三章基于超椭圆函数和代数曲线的相关知识,通过亏格为2和3的超椭圆函数所满足的等式关系,利用构造性方法求解非线性方程的超椭圆函数解,得到了几类非线性发展方程的2亏格和3亏格的超椭圆函数解.
第四章首先基于具有有理特征的Riemann theta函数,推广了双线性和Riemann theta函数相结合的方法,给出了一类具有两个或两个以上因变量非线性方程(组)的拟周期解；并将这一方法应用到一类微分差分方程中.其次,利用四类特殊的具有有理特征Riemann theta函数所满足的恒等式关系,运用直接构造性方法,给出了两个离散方程的拟周期解.
第五章首先运用超离散化方法给出晶格修正的KdV方程的超离散方程,并通过协调条件证明了超离散方程的Lax可积.其次运用热带化方法,给出约化的非自治超离散Kadomtsev-Petviashvili方程(rndKP)的热带谱曲线,并利用热带形式下的Fay三度割线恒等式给出其热带Riemann theta函数形式的解.
In this dissertation, some topics are studied based on the mathematical mechanization and AC= BD model, including exact solutions, hyperelliptic functions solutions and quasi-period solutions of nonlinear evolution equations and discrete equations, Lax integrability and tropical Riemann theta functions solutions of ultra-discrete equations.
In Chapter 1, we introduce the history and development of mathematical mechaniza-tion, soliton theory, integrability of nonlinear evolution equations and algebraic geometry solutions, soliton celluar automaton, solutions of the mathematical and physical equations, as well as the main work of this dissertation.
In Chapter 2, the main content and idea are AC= BD model and C-D pair theory. According to these, the mechanized construction of the transformations of two differential equations is presented. Based on the research on a class operator of "D", we get more solutions of this "Dv=0", so we can get more exact solutions of some nonlinear evolution equations.
In Chapter 3, based on the theory of hyperelliptic functions and algebraic curve, we extend the direct method of solving the nonlinear evolution equations. By the identities of the hyperelliptic functions of two genus and three genus, we obtain the solutions of two genus and three genus hyperelliptic functions of some classes of the nonlinear evolution equations.
In Chapter 4, first, we extend the method of combined the bilinear with the Rie-mann theta functions to the Riemann theta functions with rational character, and give the quasi-periodic solutions of some nonlinear equation(s) which has (have) above two de- pendent variables and apply this method to the differential difference equations. Second, we give quasi-periodic solutions of two discrete equations by the direct method based on the identities of the Riemann theta functions with rational character.
In Chapter 5, we obtain the ultradiscrete equation of the discrete mKdV equation and prove its Lax integrability by the ultradiscretization. And By the tropicalization, we give the tropical spectral curve of the ultradiscrete rndKP equation and show that its solution can be got by the tropical Riemann theta function.
第一章介绍数学机械化、孤立子理论、可积性与代数几何解、超离散方程的求解与可积,几种求解数学物理方程的构造性方法的历史发展和研究现状,并介绍本文的选题及主要工作.
第二章首先介绍了张鸿庆教授提出的数学机械化中的AC=BD模式和应用,其次在C-D对的理论框架下,利用微分伪带余除法,构造性给出了求方程间的变换的方法.并通过研究一种类型的算子D,给出Dv=0更多形式的解,从而推广了一类辅助方程展开法,给出了一类非线性发展方程更多形式的精确解.
第三章基于超椭圆函数和代数曲线的相关知识,通过亏格为2和3的超椭圆函数所满足的等式关系,利用构造性方法求解非线性方程的超椭圆函数解,得到了几类非线性发展方程的2亏格和3亏格的超椭圆函数解.
第四章首先基于具有有理特征的Riemann theta函数,推广了双线性和Riemann theta函数相结合的方法,给出了一类具有两个或两个以上因变量非线性方程(组)的拟周期解；并将这一方法应用到一类微分差分方程中.其次,利用四类特殊的具有有理特征Riemann theta函数所满足的恒等式关系,运用直接构造性方法,给出了两个离散方程的拟周期解.
第五章首先运用超离散化方法给出晶格修正的KdV方程的超离散方程,并通过协调条件证明了超离散方程的Lax可积.其次运用热带化方法,给出约化的非自治超离散Kadomtsev-Petviashvili方程(rndKP)的热带谱曲线,并利用热带形式下的Fay三度割线恒等式给出其热带Riemann theta函数形式的解.
In this dissertation, some topics are studied based on the mathematical mechanization and AC= BD model, including exact solutions, hyperelliptic functions solutions and quasi-period solutions of nonlinear evolution equations and discrete equations, Lax integrability and tropical Riemann theta functions solutions of ultra-discrete equations.
In Chapter 1, we introduce the history and development of mathematical mechaniza-tion, soliton theory, integrability of nonlinear evolution equations and algebraic geometry solutions, soliton celluar automaton, solutions of the mathematical and physical equations, as well as the main work of this dissertation.
In Chapter 2, the main content and idea are AC= BD model and C-D pair theory. According to these, the mechanized construction of the transformations of two differential equations is presented. Based on the research on a class operator of "D", we get more solutions of this "Dv=0", so we can get more exact solutions of some nonlinear evolution equations.
In Chapter 3, based on the theory of hyperelliptic functions and algebraic curve, we extend the direct method of solving the nonlinear evolution equations. By the identities of the hyperelliptic functions of two genus and three genus, we obtain the solutions of two genus and three genus hyperelliptic functions of some classes of the nonlinear evolution equations.
In Chapter 4, first, we extend the method of combined the bilinear with the Rie-mann theta functions to the Riemann theta functions with rational character, and give the quasi-periodic solutions of some nonlinear equation(s) which has (have) above two de- pendent variables and apply this method to the differential difference equations. Second, we give quasi-periodic solutions of two discrete equations by the direct method based on the identities of the Riemann theta functions with rational character.
In Chapter 5, we obtain the ultradiscrete equation of the discrete mKdV equation and prove its Lax integrability by the ultradiscretization. And By the tropicalization, we give the tropical spectral curve of the ultradiscrete rndKP equation and show that its solution can be got by the tropical Riemann theta function.
引文
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