Complete Integrability and Structure of Solutions for Nonlinear Discrete Equations

为了阐明“奇点约束”的结构，将其概念推广到超离散系统，揭示离散可积系统的代数和几何结构，我们得到了以下结果：(1)给出了非自治离散可积系统解方法的代数结构，并构造了该系统用特殊函数表示的解。我们还提出了解之间新的非线性关系。(2)对于一些离散Painleve方程，我们澄清了它们的自对偶性和解之间的非线性关系结构。(3)得到了Toda分子方程的超离散形式及其解的显式表达式。我们还研究了它的代数结构和对称性，并讨论了将该系统应用于数学工程领域的可能性。(4)将一类非线性积分-微分方程看作差分-微分方程，研究了它的可积结构。我们还研究了耦合非线性波动方程离散模拟解的结构。(5)我们提出了一种满足奇点约束准则但表现出混沌行为的方程的几何方法。给出了一种利用交数理论计算代数熵的方法。(6)研究了离散painlevel方程的几何结构，给出了Toda格方程的解空间与离散painlevel方程解的关系。

For the purpose of clarifying the structure of "singularity confinement", extending the concept to ultra-discrete system, and revealing algebraic and geometrical structure of discrete integrable systems, we have obtained the following results.(1) We have shown the algebraic structure of solution method for nonautonomous discrete integrable systems and constructed solutions expressed by special functions for the systems. We have also proposed new nonlinear relations among the solutions.(2) For some discrete Painleve equations, we have clarified their self-duality and the structure of nonlinear relations among the solutions.(3) We have obtained an ultra-discrete version of the Toda molecule equation and the explicit expression of its solutions. We have also studied its algebraic structure and symmetry, and discussed a possibility of applying the system to the field of mathematical engineering.(4) By considering a nonlinear integro-differential equation as a difference-differential equation, we have studied its integrable structure. We have also studied the structure of solutions for a discrete analogue of coupled nonlinear wave equations.(5) We have presented a geometrical approach to the equation which satisfies the singularity confinement criterion but exhibits chaotic behaviour. We have also given a method to caluculate the algebraic entropy by using the theory of intersection numbers.(6) We have studied geometrical structure of the discrete Painleve equations and shown the relation between the solution space of Toda lattice equation and the solutions of discrete Painleve equations.