Application of Soliton Theory to Engineering

本研究的目的是将孤子理论的思想应用到这一领域，为工程技术的发展做出贡献。(1)数值分析：研究了离散孤子方程与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：指出孤子理论中的tau函数在线性系统理论、Karmark方法和信息几何中也具有重要作用。此外，还发现离散Painleve方程的解与Toda方程的解密切相关。(3)信息工程：利用“超离散极限”的概念阐明了孤子元胞自动机的结构。结构分析：利用量子自旋系统中的可积性质，证明了光激子弦解存在的可能性。

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.