Mathematics on discrete and ultradiscrete integrable systems and its application

在离散方程系统、量子力学格子模型和元胞自动机系统(即有限状态的离散动力系统)中，存在一类可以用解析函数表示精确解和/或统计量的系统，称为可积系统。在本研究中，我们研究了这些离散可积系统的数学问题，特别是阐明了Box-Ball系统(BBS)这一典型的超离散系统的数学结构。BBS是一个球在盒子阵列中运动的动力学系统，表现出孤子行为；它有孤子解，有足够多的守恒量，并且它的初值问题是可解的。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-贝特尔关系，其相空间和动力学与形变参数Q→0的某些量子代数有关。对于这个论坛，我们研究了ini…从初等组合方法，KKR双射，超离散KdV方程的孤子解，以及在雅可比变换上的线性化等不同的角度讨论了更多的数值问题。我们发现BBS与其他数学对象如Weyl群、量子代数表示理论、素数分布等之间存在有趣的关系。例如，广义BBS的守恒量用图上的路来表示，确定路的运算等价于Weyl群的作用。我们还建立了周期BBS的基本圈与具有量子代数对称性的可积格子模的转移矩阵的本征值之间的关系，并证明了Bethe ansatz方程的弦假设是由周期BBS的守恒量来刻画的。此外，我们还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取得了重大进展。较少

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