On the initial value problem of a periodic box-ball system

Abstract

We show that the initial value problem of a periodic box-ball system can be solved in an elementary way using simple combinatorial methods.A periodic box-ball system (PBBS) is a dynamical system of balls in an array of boxes with a periodic boundary condition [1,2]. The PBBS is obtained from the discrete KdV equation and the discrete Toda equation, both of which are known as typical integrable nonlinear discrete equations, through a limiting procedure called ultradiscretization [3,4]. Since the ultradiscretization preserves the main properties of the original discrete equations, and the solvability of the initial value problem being an important property of integrable equations, we expect that the initial value problem of the PBBS can also be solved. In fact, the initial value problem for the PBBS was first solved by inverse ultradiscretization combined with the method of inverse scattering transform of the discrete Toda equation [5] and recently by the Bethe ansatz for an integrable lattice model with quantum group symmetry at the deformation parameter q = 0 and q = 1 [6]. These two methods, however, require fairly specialized mathematical knowledge on algebraic curves or representation theory of quantum algebras.An important property which characterizes a state of the PBBS is the fundamental cycle of the state, i.e., the length of the trajectory to which it belongs. Its explicit formula as well as statistical distribution was obtained and its relation to the celebrated Riemann hypothesis was clarified [7,8,9]. To prove the formula for fundamental cycle, one of the key steps is to compare a state with its 'reduced states' constructed by the '10-elimination'. In this article, we show that the initial value problem of the PBBS is solved by simple combinatorial arguments -essentially given in Ref.[7] -with some remarkable features of the reduced states.First we quickly review the definition of the PBBS and its conserved quantities. Consider a one-dimensional array of boxes each with a capacity of one ball. A periodic boundary condition is imposed by assuming that the last box is adjacent to the first one. Let the number of boxes be N and that of balls be M . We assume M < N/2. An arrangement of M balls in N boxes is called a state of the PBBS. Denoting a vacant box by 0 and a filled box by 1, a state 1

Tropical Spectral Curves and Integrable Cellular Automata

“…However, due to technical difficulties, this method has been completed only in the case of genus 1. Thereafter the initial value problem of the pBBS is solved by a combinatoric way [7] and by Bethe ansatz using Kerov-Kirillov-Reshetikhin bijection [6].…”

Relationships Between Two Approaches: Rigged Configurations and 10-Eliminations

“…Since 90381 ≡ 13 (mod 32), we have T 90381 1 (p ′ ) = T 13 1 (p ′ ) = 11000111011000000011101110010000, which reconstructs the computation in the final part of [14].…”

“…In this paper, we describe precisely interrelations between these two approaches.Mathematics Subject Classification (2000) 17B37, 37K15, 05E15. Key words and phrases: crystal basis, periodic box-ball system, combinatorics.(1) 1 due to Kuniba-Takagi-Takenouchi [13] (KTT for short) and Mada-Idzumi-Tokihiro [14] (MIT for short). Our main result states that KTT ≈ MIT.The approach developed in [13] is based on the theory of the rigged configurations (RC for short -for details concerning the RC-bijection, see e.g.…”

Integrable structure of box–ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry

“…We note that the solution of the initial value problem based on the procedure called 10-elimination [58] is equivalent [44] to the preceding solution [57] explained here.…”