Linearization of the box-ball system with box capacity L

TL;DR

This paper introduces a bijection that linearizes the box-ball system with arbitrary box capacity L, extending previous methods and enabling analysis of more general states.

Contribution

It constructs a new bijection linking the system's state to two sequences, allowing linearization even with negative or large values.

Findings

Bijection between system state and sequence pairs established

Linearization achieved for states with negative or large values

Method extends previous linearization techniques to broader cases

Abstract

We construct a bijection between the state of the box-ball system with box capacity L and a pair of two sequences. In time evolution, one of the sequences moves at speed 1, and the other follows the rules of the box-ball system with box capacity one, which can be linearized by the Kerov-Kirillov-Reshetikhin(KKR) bijection. Our method can be applied to a state including a negative value or a value greater than the box capacity.