Combinatorial Bethe ansatz and generalized periodic box-ball system

TL;DR

This paper reformulates the combinatorial Bethe ansatz for the periodic box-ball system using crystal base theory, providing a new interpretation and solving the initial value problem with an inverse scattering method.

Contribution

It introduces a novel crystal interpretation of the KKR bijection and applies it to solve a higher spin generalization of the box-ball system.

Findings

Complete crystal interpretation of the KKR bijection for U_q(sl}_2)

Solution of the initial value problem using ultradiscrete Riemann theta functions

Extension to higher spin periodic box-ball systems

Abstract

We reformulate the Kerov-Kirillov-Reshetikhin (KKR) map in the combinatorial Bethe ansatz from paths to rigged configurations by introducing local energy distribution in crystal base theory. Combined with an earlier result on the inverse map, it completes the crystal interpretation of the KKR bijection for U_q(\hat{sl}_2). As an application, we solve an integrable cellular automaton, a higher spin generalization of the periodic box-ball system, by an inverse scattering method and obtain the solution of the initial value problem in terms of the ultradiscrete Riemann theta function.