Box and ball system as a realization of ultradiscrete nonautonomous KP equation

A geometric realization of the periodic discrete Toda lattice and its tropicalization

“…In recent years, many studies on ultradiscrete integrable systems, or integrable cellular automata have intensively been made by using various mathematical tools such as combinatorics [24,3,37,31], ultradiscretization procedure [35,27,34,14], crystal bases of quantum groups [9,7,8,11], and so forth. In particular, among many studies on geometric aspects of ultradiscrete integrable systems, several remarkable works have successively been made by Kimijima-Tokihiro [21], Inoue-Takenawa [12,13], and Iwao [17] in terms of tropical geometry and ultradiscretization procedure.…”

Inversible Max-Plus algebras and integrable systems

“…Recently, by the development of computer sciences, discrete and ultra-discrete versions of integral system have been attracting a great deal of attention. The ultra-discretization of soliton equations and other various important equations are intensively studied in [1,2,3,4,5,6,7,8]. Moreover, it is known that the various ultra-discrete soliton equations are obtained from solvable vertex model in statistical mechanics in a certain limiting procedure [9,10].…”

Euler–lagrange Correspondence of Generalized Burgers Cellular Automaton

“…In recent years, the authors have proposed and studied the so-called ultra-discrete method by which we obtain corresponding CA from difference equations [9,10,11,12,13,14]. Various soliton equations have been transformed into CA by this method and the solutions of the CA have also been obtained exactly without losing the mathematical properties of corresponding difference equations.…”