Analysis and Application of Integrable Cellular Automaton
Project/Area Number  09640245 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  The University of Tokyo 
Principal Investigator 
TOKIHIRO Tetsuji Graduate School of Mathematical Sciences, The University of Tokyo, Professor, 大学院・数理科学研究科, 教授 (10163966)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥2,800,000 (Direct Cost : ¥2,800,000)
Fiscal Year 1999 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1998 : ¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1997 : ¥1,000,000 (Direct Cost : ¥1,000,000)

Keywords  cellular automaton / integrable system / soliton / nonautonomous KP equation / boxball system / セルオートマトン / 可積分系 / ソリトン / 非自立離散KP方程式 / 箱玉系 / 可解格子模型 / BoxBall system / 超離散化 / YangBaxter 関係式 / 特殊函数 / Darboux変換 
Research Abstract 
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 multicomponent 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 boxball 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 boxball systems, we proved that they are obtained by ultradiscretization from 1reduction of the discrete KP equation (HirotaMiwa 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 boxball system and quantum integrable lattices of A type to the proof of solitonic natures. Then we constructed the most general boxball 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.

Report
(5results)
Research Output
(19results)