Complex dynamical analysis of Soliton Systems

孤子系统的复杂动力学分析

• 批准号: 06835023
• 负责人: SAITO Satoru
• 金额: $ 1.22万
• 依托单位: TOKYO METROPOLITAN UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1994
• 资助国家: 日本
• 起止时间: 1994 至 1996
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-06835023/
• 关键词: Solitons; Julia set; Complex Dynamical Systems; Logistic map; ロジスティック写像; カオス

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'

1.经过三年的研究，本项目的主要目标已经实现。主要结果如下：(1)首先，关于孤子系统，我们发现通过离散化的Toda格得到的系统仍保持可积性，其中包含W_&lt；1+*&gt；代数。此外，这种对称性还被推广到MoYal代数。为了研究这类系统，我们发现微分几何的离散类比可以一致地表示出来，并在分析中起到中心作用。(2)除了空间离散化的推广外，还证明了二维Toda晶格系统可以看作是由4个晶格点组成的Toda原子的集合。这一事实使我们能够独立地考虑晶格的一小部分，并分析其分析性质，以了解整个系统本身。(3)Toda原子的时间演化与Mobius映射相同，因此是可积的。如果人们使原子稍有形变，则在因变量的复平面上会出现Julia集。详细研究了Julia集的行为，特别是在Julia集消失和恢复可积性的临界点附近。结果表明，Julia集均匀地累加到Mobius映射点的极限中。这一结果也用计算机进行了研究，并从数值上观察到了同样的现象。除了已发表的论文外，结果还将在下列论文中报告，这些论文应提交发表。S.Saito，《对偶共振模型求解杨-巴克斯特方程》S.Saito，《离散曲面与差分几何之间的对应》

项目成果

R.Kemmoku and S.Saito: "'Difference Operator Approach to the Moyal Quantization'" J.Phys. Soc. Jpn.Vol. 65. 1881-1884 (1996)

R.Kemmoku 和 S.Saito：“‘Moyal 量化的差分算子方法’”J.Phys。

R.Kemmoku: "W_<1+∞> as a discretization of Virasoro algebra" J.Physics A:Math.Gen.29. 4141-4148 (1996)

R.Kemmoku：“W_<1+∞> 作为 Virasoro 代数的离散化”J.Physics A:Math.Gen.29 (1996)。

R.Kemmoku: "Difference Operator Approach to the Moyal Quantization" J.Physical Society of Japan. 65. 1881-1884 (1996)

R.Kemmoku：“Moyal 量化的差分算子方法”J.日本物理学会。

N.Saitoh: "An Analysis of a family of Rational maps containing integrable and non integrable difference aralogue of the logistic equation" J.Physics A:Math.Gen.29. 1831-1840 (1996)

N.Saitoh：“包含逻辑方程可积和不可积差值模拟的有理图系列的分析”J.Physics A:Math.Gen.29。

R.Kemmoku: "Difference-Opetator Approach to the Moyal Quantization" J.Physical Society of Japan. 65. 1881-1884 (1996)

R.Kemmoku：“Moyal 量子化的差分算子方法”J.日本物理学会。