Proof of solitonical nature of box and ball systems by means of inverse ultra-discretization

Abstract

A soliton cellular automaton, which represents movement of a finite number of balls in an array of boxes, is investigated. Its dynamics is described by an ultra-discrete equation obtained from an extended Toda molecule equation. The rules for soliton interactions and factorization property of the scattering matrices (Yang-Baxter relation) are proved by means of inverse ultradiscretization. The conserved quantities are also presented and used for another proof of the solitonical nature. § This name was given by B Grammaticos.

Tropical Spectral Curves, Fay's Trisecant Identity, and Generalized Ultradiscrete Toda Lattice

“…In closing the introduction, we make a brief remark on the interesting close relation between the ultradiscrete periodic Toda lattice and the periodic box and ball system (pBBS) [9], which is generalized to that between T(M, N ) and the pBBS of M kinds of balls [14,7]. When M = 1, the relation is explained at the level of tropical Jacobian [2].…”

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].…”

Untitled

“…The box-ball systems [24][25][26][27] are important examples of soliton cellular automata. They are discrete dynamical systems whose time evolution is expressed as a certain motion of balls along the one dimensional array of boxes.…”