Two dimensional periodic box-ball system and its fundamental cycle

TL;DR

This paper investigates a 2D periodic box-ball system, an ultradiscrete integrable model, and develops an algorithm based on tropical curve theory to compute its fundamental cycle, a key conserved quantity.

Contribution

It introduces a novel algorithm leveraging tropical geometry to calculate the fundamental cycle of the 2D Box-Ball system with periodic boundaries.

Findings

The algorithm effectively computes the fundamental cycle.

The 2D Box-Ball system is linked to ultradiscrete KP equations.

Tropical curve theory is applied to integrable systems.

Abstract

We study a 2-dimensional Box-Ball system which is a ultradiscrete analog of the discrete KP equation. We construct an algorithm to calculate the fundamental cycle, which is an important conserved quantity of the 2-dim. Box-Ball system with periodic boundary condition, by using the tropical curve theory.