Complex dynamical analysis of Soliton Systems

1. 经过三年的研究，本课题的主要目的已经达到。以下是对结果的总结。(1)首先，关于孤子系统，我们发现通过离散化Toda格得到的系统，在保持可积性的情况下，包含W_<1+*>代数。此外，还证明了这种对称性可以推广到Moyal代数。为了研究这种类型的系统，我们发现微分几何的离散模拟可以一致地表述，并在分析中起着核心作用。(2)证明了二维Toda点阵系统可以看作是由4个点阵点组成的Toda原子的集合。这一事实使我们能够独立考虑晶格的一小部分，并分析其解析性质，从而了解整个系统本身。(3) Toda原子的时间演化与Mobius图相同，因此是可积的。如果原子稍微变形，则在因变量的复平面上出现茱莉亚集。详细研究了Julia集的性质，特别是在Julia集消失而可积性恢复的临界点附近。结果表明，Julia集合均匀地聚集在极限的莫比乌斯图的点上。这一结果也用计算机进行了研究，并在数值上观察到了同样的现象。除已发表的论文外，还将结果报告在以下应提交发表的论文中。Saito，“双共振模型求解Yang-Baxter方程”。齐藤，《离散曲面与差分几何的对应关系》

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'