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、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。主要研究结果如下：1.利用超离散KdV方程和Lotka-Volterra方程（包括其修正形式），我们得到了具有任意盒容量的盒球系统的守恒量.通过逆超离散化得到了D型仿射李代数对应的晶体组合R的分段线性方程。研究了一个称为RI链的离散可积系统，它被认为是户田方程的推广。特别地，我们得到了它的双线性形式和它的行列式解。并与相对论性户田方程建立了联系.我们已经获得了离散和q-离散版本的分数阶导数连同它的本征函数称为Mittag-Leffler函数。我们还构造了一个新的具有分数阶差分的可积映射.我们找到了圆盘上双调和算子的绿色函数和泊松函数。还找到了它们的积分表示。

项目成果

Y.Kametaka, K.Takemura, Y.Suzuki, A.Nagai: "Positivity and hierarchical structure of Green's functions of 2-point Boundary value problems for bending of a beam"Japan Journal of Industrial and Applied Mathematics. 18. 543-566 (2001)

Y.Kametaka、K.Takemura、Y.Suzuki、A.Nagai：“梁弯曲的 2 点边值问题的格林函数的正性和层次结构”日本工业应用数学杂志。

M.Okado, A.Schlling, M.Shimozono: "Crystal bases and q-identities"Comtemporary Mathematics. 291. 29-53 (2001)

M.Okado、A.Schlling、M.Shimozono：“晶体基和 q 恒等式”当代数学。

M.NIshizawa, Y.Ohta, S.Tsujimoto: "Some aspects of Toda molecule"Glasgow Math.J.. (to appear).

M.NIshizawa、Y.Ohta、S.Tsujimoto：“Toda 分子的某些方面”Glasgow Math.J.（待出现）。

A.Kuniba, M.Okado, T.Takagi, Y.Yamada: "Geometric crystal and tropical R for D _n^(1)"Internat.Math.Res.Notices. 48. 2565-2620 (2003)

A.Kuniba、M.Okado、T.Takagi、Y.Yamada：“D _n^(1) 的几何晶体和热带 R”Internat.Math.Res.Notices。

永井 敦: "ミッタークレフラー関数の差分化-非整数階差分とは?"数理科学. 483. 50-55 (2003)

Atsushi Nagai：“Mitterkrefler 函数的微分微分 - 什么是分数差分？”483. 50-55 (2003)