The box–ball system and theN-soliton solution of the ultradiscrete KdV equation

Abstract

Any state of the box-ball system (BBS) together with its time evolution is described by the N-soliton solution (with appropriate choice of N) of the ultradiscrete KdV equation. It is shown that simultaneous elimination of all '10'-walls in a state of the BBS corresponds exactly to reducing the parameters that determine 'the size of a soliton' by one. This observation leads to an expression for the solution to the initial-value problem (IVP) for the BBS. Expressions for the solution to the IVP for the ultradiscrete Toda molecule equation and the periodic BBS are also presented.

Conserved quantities and generalized solutions of the ultradiscrete KdV equation

“…For the original BBS where every box capacity is one and only a finite number of balls exist, all the solutions are proved to be constructed by ultradiscretization from the soliton solutions of the discrete KdV equation ( 6) [16]. Although the ultradiscrete KdV equation ( 1) over arbitrary integers is shown to be equivalent to the BBS with larger box capacity, the solutions to them can not be the ultradiscrete limit of soliton solutions to (6), because the boundary condition of the BBS changes such that the number of balls becomes M for |n| → ∞.…”

Bethe Ansatz, Inverse Scattering Transform and Tropical Riemann Theta Function in a Periodic Soliton Cellular Automaton for An(1)

“…1. The solution of initial value problem of the n = 1 periodic box-ball system [24,20,21] has been reproduced partially by the procedure called 10-elimination [29]. By now the precise relation between the two approaches has been shown [17].…”

Correlation functions for a periodic box–ball system

“…where W i denotes the amplitude of the "soliton" corresponding to a i obtained by the procedure explained in [9].…”