Complex dynamical analysis of Soliton Systems
Project/Area Number  06835023 
Research Category 
GrantinAid for Scientific Research (C)

Section  時限 
Research Field 
非線形科学

Research Institution  TOKYO METROPOLITAN UNIVERSITY 
Principal Investigator 
SAITO Satoru Faculty of Science, TOKYO METROPOLITAN UNIVERSITY Assistant Professor, 理学部, 助教授 (90087099)

Project Fiscal Year 
1994 – 1996

Project Status 
Completed(Fiscal Year 1996)

Budget Amount *help 
¥1,900,000 (Direct Cost : ¥1,900,000)
Fiscal Year 1996 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1995 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1994 : ¥900,000 (Direct Cost : ¥900,000)

Keywords  Solitons / Julia set / Complex Dynamical Systems / Logistic map / ソリトン / ジュリア集合 / 複素力学系 / ロジスティック・写像 / ロジスティック写像 / カオス 
Research Abstract 
1. After the research of three years the main part of the purpose of this project has been achieved. The followings are the summary of the results. (1) First, about the soliton systems, we found that the systems obtained by discretization of the Toda lattice, but still preserving integrability, include the W_<1+*> algebra. Moreover this symmetry is shown to be extended to the Moyal algebra. In order to study this type of systems we found that a discrete analogue of differential geometry can be formulated consistently and plays the central role in the analysis. (2) Besides the generalization by discretizing the space, it was proved that the 2 dimensional Toda lattice system can be regarded as a collection of Toda atoms which are formed by 4 lattice points. This fact enables us to consider a small piece of the lattice independently from the rest and analyze its analytical properties to know the total system itself. (3) The time evolution of the Toda atom is the same as the Mobius map, hence is integrable. If one deforms the atom a little, a Julia set appears on the complex plane of the dependent variable. The behavior of the Julia set was investigated in detail, in particular near the critical point where the Julia set disappears and the integrability is recovered. As a result the Julia set was shown to accumlate uniformly into the points of the Mobius map in the limit. This result was also studied using computors and the same phenomenon is observed numerically. 2. In addition to the papers already published, the results are reported in the following papers which should be submitted for publication. S.Saito, 'Dual Resonance Model Solves the YangBaxter Equation' S.Saito, 'The Correspondence between Discrete Surface and Difference Geometry'

Report
(5results)
Research Output
(16results)