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Analysis and Application of Integrable Cellular Automaton

可积元胞自动机的分析与应用

基本信息

批准号：
09640245
负责人：
TOKIHIRO Tetsuji
金额：
$ 1.79万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640245/
关键词：
cellular automaton integrable system soliton nonautonomous KP equation box-ball system 可解格子模型 Box-Ball system 超離散化 Yang-Baxter 関係式特殊函数 Darboux変換

项目摘要

The main results obtained in the term are as follows :(1) We showed that almost all integrable Cellular Automata (CAs) are obtained by ultradiscretization of the nonautonomous discrete KP equation and its reductions.(2) We constructed discrete integrable lattices (quadrilateral lattices) using the τ functions of multi-component discrete KP equations.(3) We showed that discrete Toda molecule equation is equivalent to the ε algorithm for convergence acceleration methods, and discussed analytically about the convergence of the methods in terms of the discrete Toda molecule equation.(4) A box-ball system, a discrete dynamical system in which solitonic time evolution patterns of CAs are expressed as movement of balls in an infinite array of boxes, shows some combinatorial natures in the scattering of solitonic patterns. For generalized box-ball systems, we proved that they are obtained by ultra-discretization from 1-reduction of the discrete KP equation (Hirota-Miwa equation) and obtained concrete form of soliton solutions. We proved the solitonic natures and the combinatorial properties with ultradiscretization of the generalized Toda molecule equation. We also constructed the conserved quantities of the system and gave another proof for the solitonic nature. Furthermore we applied the correspondence between box-ball system and quantum integrable lattices of A type to the proof of solitonic natures. Then we constructed the most general box-ball system in which the capacity of boxes, carriers, and spedies of boxes are completely arbitrary, and gave the proof of solitonic natures of the system and constructed explicit solutions to the elementary excitations of the system.

本期取得的主要成果如下：（1）我们证明了几乎所有可积元胞自动机（CA）都是通过非自治离散KP方程及其约简的超离散化得到的。（2）我们利用多分量离散KP方程的τ函数构造了离散可积格子（四边形格子）。（3）我们证明了离散Toda分子方程等价于收敛加速方法的ε算法，并根据离散Toda分子方程分析讨论了该方法的收敛性。(4)盒球系统是一种离散动力系统，其中CA的孤子时间演化模式表示为无限阵列盒子中球的运动，在孤子散射中表现出一些组合性质模式。对于广义盒球系统，我们证明了它们是由离散KP方程（Hirota-Miwa方程）的1-约简得到的超离散化，并得到了孤子解的具体形式。我们用广义Toda分子方程的超离散化证明了孤子性质和组合性质。我们还构造了系统的守恒量，并给出了孤子性质的另一个证明。此外，我们将盒子球系统与A型量子可积晶格之间的对应关系应用于孤子性质的证明。然后我们构造了最一般的盒子-球系统，其中盒子的容量、载体和盒子的速度是完全任意的，并给出了系统的孤子性质的证明，并构造了系统的基本激励的显式解。