Application of Soliton Theory to Engineering

孤子理论在工程中的应用

• 批准号: 06302034
• 负责人: SATSUMA Junkichi
• 金额: $ 3.46万
• 依托单位: University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Co-operative Research (A)
• 财政年份: 1994
• 资助国家: 日本
• 起止时间: 1994 至 1995
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-06302034/
• 关键词: soliton; integrability; QR method; cellular automaton; inverse scattering; Toda equation; Volterra system; system theory; 数列の加速

The purpose of this research is to contribute to the development of engineering by applying the idea of soliton theory to the field. The following results are obtained through the research.(1)Numerical Analysis : Relationship of discrete soliton equations with LR-and QR methods is investigated. It is shown that the symmetry properties of soliton equations are extracted from the LR method. Relationships with accelaration method and extrapolation formula are also studied and a new accelaration scheme is proposed.(2)Engineering Mathematics : It is pointed out that the tau function in the soliton theory plays important roles also in the linear system theory, Karmarker method and information geometry. Moreover, it is revealed that the solutions of discrete Painleve equations are closely related to those of the Toda equation.(3)Information Engineering : The structure of soliton cellular automata is clarified by employing a concept of "ultra-discrete limit". New insight on storage and compression of solitons in the discrete electrical circuit is also obtained.(4)Structural Analysis : The possibility of existence of string solution for optical exiton is shown by applying the integrable property in the quantum spin system.

本研究的目的是将孤子理论的思想应用于该领域，为工程的发展做出贡献。通过研究取得了以下成果。（1）数值分析：研究离散孤立子方程与LR和QR方法的关系。结果表明，从LR方法中提取了孤子方程的对称性。研究了加速方法与外推公式的关系，提出了一种新的加速方案。（2）工程数学：指出孤立子理论中的τ函数在线性系统理论、Karmarker方法和信息几何中也起着重要的作用。此外，还揭示了离散Painleve方程的解与户田方程的解密切相关。（3）信息工程：引入“超离散极限”概念，阐明了孤子元胞自动机的结构。对离散电路中孤子的存储和压缩也有了新的认识。（4）结构分析：利用量子自旋系统的可积性质，证明了光激子弦解存在的可能性。

项目成果

K.Kajiwara: "Casorati Defcrminant Solutions for the Discrete Painleve-II equation" J.Phys.A27. 915-922 (1994)

K.Kajiwara：“离散 Painleve-II 方程的 Casorati Defcrminant 解”J.Phys.A27。

Y.Nakamura: "Jacobi algorithm for symmetric eigenvalue problem and integrable gradient system of Lax form" Japan J.Indust・Appl.Math. (掲載予定).

Y.Nakamura：“对称特征值问题的雅可比算法和Lax形式的可积梯度系统”日本J.Indust・Appl.Math（待出版）。

R.Hirota: "Discrete 2N-wave interaction" Mathematics and Computers in Simulation. 37. 371 (1994)

R.Hirota：“离散 2N 波相互作用”模拟中的数学和计算机。

K.Okamoto: "An addition Formula for cot (x), Aitken-Steffensen Acceleration, and Cauchy Matrix" (preprint).

K.Okamoto：“cot (x)、Aitken-Steffensen 加速和柯西矩阵的加法公式”（预印本）。

Y.Nakamura: "Stochastic Lax representations and random collision models" J.Phys.Soc.Jpn.63. 827-829 (1994)

Y.Nakamura：“随机宽松表示和随机碰撞模型”J.Phys.Soc.Jpn.63。