Analysis and Application of Integrable Cellular Automaton

（1）证明了几乎所有的可积元胞自动机都是通过对非自治离散KP方程及其约化的超离散化而得到的。(2)利用多分量离散KP方程的τ函数构造了离散可积格（四边形格）。(3)证明了离散户田分子方程与加速收敛方法的ε算法是等价的，并对离散户田分子方程的收敛性进行了解析讨论. (4)盒-球系统是一个离散动力系统，其中CA的孤子时间演化模式表示为球在无限多个盒中的运动，其孤子模式的散射具有组合性质。对于广义box-ball系统，证明了它们是由离散KP方程（Hirota-Miwa方程）的1-约化超离散化得到的，并得到了孤子解的具体形式.利用超离散方法证明了广义户田分子方程的孤子性质和组合性质。构造了系统的守恒量，并给出了孤子性质的另一种证明。此外，我们还将箱球系统与A型量子可积格的对应关系应用于孤子性质的证明。然后构造了最一般的盒-球系统，其中盒的容量、载体和盒的速度都是完全任意的，并给出了系统的孤子性质的证明，构造了系统的元激发的显式解。

The main results obtained in the term are as follows :(1) We showed that almost all integrable Cellular Automata (CAs) are obtained by ultradiscretization of the nonautonomous discrete KP equation and its reductions.(2) We constructed discrete integrable lattices (quadrilateral lattices) using the τ functions of multi-component discrete KP equations.(3) We showed that discrete Toda molecule equation is equivalent to the ε algorithm for convergence acceleration methods, and discussed analytically about the convergence of the methods in terms of the discrete Toda molecule equation.(4) A box-ball system, a discrete dynamical system in which solitonic time evolution patterns of CAs are expressed as movement of balls in an infinite array of boxes, shows some combinatorial natures in the scattering of solitonic patterns. For generalized box-ball systems, we proved that they are obtained by ultra-discretization from 1-reduction of the discrete KP equation (Hirota-Miwa equation) and obtained concrete form of soliton solutions. We proved the solitonic natures and the combinatorial properties with ultradiscretization of the generalized Toda molecule equation. We also constructed the conserved quantities of the system and gave another proof for the solitonic nature. Furthermore we applied the correspondence between box-ball system and quantum integrable lattices of A type to the proof of solitonic natures. Then we constructed the most general box-ball system in which the capacity of boxes, carriers, and spedies of boxes are completely arbitrary, and gave the proof of solitonic natures of the system and constructed explicit solutions to the elementary excitations of the system.