Research on algebraic structures of ultradiscrete soliton equations and applications to engineering

本研究的目的是寻找微分、差分和超离散方程的代数和解析结构，并将数学结果应用于工程领域。得到的主要结果如下：1。我们利用超离散KdV和Lotka-Volterra方程及其修正版本，找到了具有任意箱容的箱球系统的守恒量。利用逆超离散方法，得到了与D型仿射李代数相关的晶体组合R的分段线性方程。研究了离散可积系统RI链，它被认为是Toda方程的推广。特别地，我们得到了它的双线性形式和行列式解。并找到了与相对论Toda方程的关系。我们得到了离散和q离散版本的分数阶导数，以及称为Mittag-Leffler函数的特征函数。我们还构造了一个新的分数阶差分可积映射。我们找到了圆盘上双调和算子的格林函数和泊松函数。并给出了它们的积分表示。

The purpose of this research is to find algebraic and analytical structures of differential, difference and ultradiscrete equations, and to apply, the mathematical results to the field of engineering. The main results obtained are as follows.1. We have found conserved quantities of box and ball systems with arbitrary box capacities by making use of ultradiscrete KdV and Lotka-Volterra equation including their modified version.2. We have obtained piecewise linear equation of combinatorial R of crystals associated with D type affine Lie algebra by employing inverse ultra-discretization.3. A discrete integrable system called the RI chain, which is considered as a generalization of the Toda equation, is studied. In particular, we have obtained its bilinear form and its determinant solution. The relation with relativistic Toda equation is also found.4. We have obtained discrete and q-discrete versions of fractional derivative together with ite eigen function called the Mittag-Leffler function. We have also constructed a new integrable mapping with fractional difference.5. We have found Green and Poisson functions for a biharmonic operator on a disk. Their integral representations are also found.