1999 Fiscal Year Final Research Report Summary
Analysis and Application of Integrable Cellular Automaton
| Project/Area Number |
09640245
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | The University of Tokyo |
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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)
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| Project Period (FY) |
1997 – 1999
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| Keywords | cellular automaton / integrable system / soliton / nonautonomous KP equation / box-ball system |
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| 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 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.
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