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Mathematics on discrete and ultradiscrete integrable systems and its application

离散和超离散可积系统数学及其应用

基本信息

批准号：
16204009
负责人：
TOKIHIRO Tetsuji
金额：
$ 26.62万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16204009/
关键词：
discrete integrable system ultradiscrete integrable system Cellular Automaton Box-Ball System quantum algebra integrable lattice model tropical geometry Toda equation 可積分系離散系超離散系クリスタル逆超離散化セルオートマン

项目摘要

Among the systems of discrete equations, quantum mechanical lattice models and Cellular Automata (discrete dynamical systems which take finite number of states), there exist a class of systems named integrable systems in which exact solutions and/or statistical quantities can be expressed by analytic functions. In this research, we investigated mathematical aspects of these discrete integrable systems, in particular we clarified the mathematical structure of the Box-Ball system (BBS) which is a typical ultradiscrete system. The BBS is a dynamical system of balls moving in an array of boxes and shows solitonic behavior; it has soliton solutions, sufficient number of conserved quantities, and its initial value problem is solvable. Furthermore it has combinatorial features related to quantum integrable models; soliton scattering satisfies the Yang-Bater relation, its phase space and dynamics are related to some quantum algebra of deformation parameter q→0. For this BBS, we studied the ini … More tial value problem from various points of view such as elementary combinatorial methods, KKR bijection, soliton solutions of ultradiscrete KdV equation, and linealisation on the Jacobian variety. We found interesting relations between BBS and other mathematical objects such as Weyl groups, representation theory of quantum algebra, distribution of primes and so on. For example, conserved quantities of a generalized BBS are expressed by paths on a graph, the operation to determine the paths is equivalent to action of Weyl groups. We have also established the relation between fundamental cycles of the periodic BBS and eigenvalues of the transfer matrices of integrable lattice modes with quantum algebraic symmetry and showed that the string hypothesis of the Bethe ansatz equation is characterized by the conserved quantities of the periodic BBS. Furthermore we have made significant progress in the correlation functions of quantum integrable models, fundamental problems in ultradiscrete systems and application to traffic flows. Less

在离散方程系统、量子力学格点模型和元胞自动机（具有有限个状态的离散动力系统）中，存在一类称为可积系统的系统，其中精确解和/或统计量可以用解析函数表示。在这项研究中，我们研究了这些离散可积系统的数学方面，特别是我们澄清了Box-Ball系统（BBS），这是一个典型的超离散系统的数学结构。BBS是一个在盒子阵列中运动的球的动力学系统，表现出孤子行为;它有孤子解，足够数量的守恒量，其初值问题是可解的。此外，它还具有与量子可积模型相关的组合特征;孤子散射满足杨-巴特尔关系，其相空间和动力学与形变参数q→0的某些量子代数有关。为了这个论坛，我们研究了ini ...更多信息从不同的观点，如初等组合方法，KKR双射，超离散KdV方程的孤子解，和雅可比簇的线性化的初值问题。我们发现BBS与Weyl群、量子代数表示理论、素数分布等数学对象之间存在着有趣的联系，例如，广义BBS的守恒量可以用图上的路表示，确定路的运算等价于Weyl群的作用。我们还建立了周期BBS的基本周期与具有量子代数对称性的可积晶格模的转移矩阵的本征值之间的关系，并证明了Bethe方程的弦假设是由周期BBS的守恒量所表征的。此外，我们在量子可积模型的关联函数、超离散系统的基本问题以及在交通流中的应用等方面也取得了重大进展。少