Symmetry and Complete Integrability of Nonlinear Difference Equations

本研究的目的是阐明非线性差分方程奇性约束条件的数学结构，将该条件推广到超离散系统，并研究离散可积系统的代数和几何结构。我们的部分结果如下所示。研究了连续Painleve方程和离散Painleve方程之间的关系，给出了它们的行列式解。我们还提出了一种统一描述Q-Painleve方程的方法。此外，我们还详细研究了它们的变换结构。我们提出了几个新的可积的非线性方程，并研究了耦合的KP方程和高阶非线性薛定谔方程的解。3.我们提出了一个超离散的Sine-Gordon方程，它被认为是一个具有孤子解的元胞自动机。我们还给出了一个超离散的修正KdV方程，并证明了该方程与箱球系统直接相关。我们研究了周期超离散系统，精确地确定了其解的基本周期，并给出了一个显式构造守恒量的算法。阐明了超离散系统守恒量与连续系统守恒量之间的关系。

The purpose of this research is to clarify the mathematical structure of singularity confinement condition for nonlinear difference equations, to extend the condition to ultradiscrete systems, and to study algebraic and geometrical structure of discrete integrable systems. Some of our results are as follows.1. We have studied the relation between continuous and discrete Painleve equations and given a determinant-type solution for them. We also presented a method to describe q-Painleve equations in a unified way. Furthermore, we studied their transformation structure in detail.2. We have presented a few of new integrable nonlinear equations and studied solutions of a coupled KP equation and the higher order nonlinear Schrodinger equations.3.We have proposed an ultradiscrete Sine-Gordon equation, which is considered to be a cellular automaton with soliton-type solutions. We have also given an ultradiscrete modified KdV equation and have shown that the equation directly relates with a box and ball system. Moreover, we proposed a method to ultradiscretize variables without positivity.4.We have studied periodic ultradiscrete systems and determined fundamental period of their solutions exactly and have given an algolithm to constitute the conserved quantities explicitly. We also clarify the relation between conserved quantities of ultradiscrete and continuous systems.