Complex dynamical analysis of Soliton Systems

点击翻译按钮获取中文摘要

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 Yang-Baxter Equation'S.Saito, 'The Correspondence between Discrete Surface and Difference Geometry'