三角曲线在离散可积系统中的应用

发布时间：2018-04-10 23:34

本文选题：三角曲线 + 离散可积系统　；参考：《郑州大学》2017年博士论文

【摘要】：本文基于三角曲线的理论来研究几个与3×3离散矩阵谱问题相联系的孤子方程族的有限亏格解,分别是Itoh Narita Bogoyavlensky晶格族,修正Belov Chaltikian晶格族,四分量Toda晶格族,Merola Ragnisco Tu晶格族.从3×3离散矩阵谱问题出发,利用离散的零曲率方程和Lenard递推序列导出与之相对应的孤子方程族.借助于驻定情况的Lax矩阵的特征多项式引入三角曲线,对其紧致化后产生一个亏格为2)的三叶Riemann面2).在Riemann面2)上定义Baker Akhiezer函数以及与Baker Akhiezer函数紧密相联系的亚纯函数,通过定义亚纯函数的零点和极点引入椭圆变量,从而将离散孤子方程族分解成可解的Dubrovin-型常微分方程组.进一步分析得到亚纯函数和Baker Akhiezer函数的渐近性质和因子.引入三类Abelian微分,利用Riemann Roch定理构造亚纯函数和Baker Akhiezer函数的Riemann theta函数表示,再结合其渐近性质得到离散孤子方程族的有限亏格解.这四个问题的共同点是与3×3离散矩阵谱问题相联系,需要在三叶Riemann面上去考虑.与3×3连续矩阵谱问题不同,在分析亚纯函数和Baker Akhiezer函数的渐近展式时,需要同时考虑无穷远点和零点.与此同时,每个问题又有各自的特点.在第二章中,Riemann面有两个无穷远点(一个是二重分支点,一个不是分支点)和一个三重零点;第三章的Riemann面有三个无穷远点(均不是分支点)和一个三重零点;第四章的Riemann面有三个无穷远点,亚纯函数和Baker Akhiezer函数在零点处无奇性;第五章的Riemann面有三个无穷远点和三个零点.因此在局部坐标的选取,亏格的计算以及亚纯函数和Baker Akhiezer函数的渐近性质分析上都不同.
[Abstract]:Based on the theory of trigonometric curves, the finite genus solutions of several soliton equations related to 3 脳 3 discrete matrix spectral problems are studied in this paper. They are Itoh Narita Bogoyavlensky lattice family, modified Belov Chaltikian lattice family and four-component Toda lattice family.Starting from the spectral problem of 3 脳 3 discrete matrix, the corresponding family of soliton equations is derived by using the discrete zero curvature equation and the Lenard recurrence sequence.The trigonometric curve is introduced by means of the characteristic polynomial of the stationary Lax matrix. After compactness, a trivalent Riemann plane with genus 2) is generated.The Baker Akhiezer function and the meromorphic function closely related to the Baker Akhiezer function are defined on the Riemann plane 2. By defining the zeros and poles of the meromorphic function, the elliptic variables are introduced to decompose the family of discrete soliton equations into a system of solvable Dubrovin-type ordinary differential equations.Further, the asymptotic properties and factors of meromorphic functions and Baker Akhiezer functions are obtained.In this paper, we introduce three kinds of Abelian differential, construct Riemann theta function representation of meromorphic function and Baker Akhiezer function by using Riemann Roch theorem, and obtain finite genus solution of discrete soliton equation family by combining its asymptotic property.The common point of these four problems is that they are related to the spectral problem of 3 脳 3 discrete matrix and need to be considered on the trefoil Riemann surface.Different from the spectral problem of 3 脳 3 continuous matrices, the asymptotic expansions of meromorphic functions and Baker Akhiezer functions need to be considered at the same time.At the same time, each problem has its own characteristics.In chapter 2, there are two infinite points (one is a double bifurcation point, one is a non-branching point) and a triple zero point, and the third chapter has three infinite points (all non-branching points) and a triple zero point on the Riemann surface.In chapter 4, the Riemann surface has three infinite points, and the meromorphic function and the Baker Akhiezer function have no singularity at the zero point, and the Riemann surface in chapter 5 has three infinite points and three zeros.Therefore, the selection of local coordinates, the calculation of genus and the analysis of asymptotic properties of meromorphic functions and Baker Akhiezer functions are different.
【学位授予单位】：郑州大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175.5

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