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

超离散孤子方程的代数结构研究及其工程应用

• 批准号: 13640212
• 负责人: NAGAI Atsushi
• 金额: $ 2.43万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640212/
• 关键词: box and ball systems; integrable system; discrete, ultradiscrete; fractional derivative; MIttag-Leffler function; Green function; inverse ultradiscretization; RI chain; q差分; 非整数階微分、差分; 可解格子模型; 離散; 超離散; ソリトン; クリスタル; ソリトンセルオートマトン; 保存量; 4階微分

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.

本研究的目的是寻找微分方程、差分方程和超离散方程的代数和解析结构，并将数学结果应用到工程领域。主要研究结果如下： 1．我们利用超离散KdV和Lotka-Volterra方程及其修改版本，发现了具有任意盒子容量的盒子和球系统的守恒量。 2．采用逆超离散化方法，得到了与D型仿射李代数有关的晶体组合R的分段线性方程。 3．研究了一种称为 RI 链的离散可积系统，它被认为是 Toda 方程的推广。特别是，我们得到了它的双线性形式及其行列式解。还建立了与相对论Toda方程的关系。 4．我们已经获得了分数阶导数的离散和 q 离散版本以及称为 Mittag-Leffler 函数的特征函数。我们还构造了一个新的带分数差的可积映射。5．我们已经找到了磁盘上双调和算子的格林函数和泊松函数。还找到了它们的积分表示。