Generalization of the $\epsilon$-BBS and the Schensted insertion algorithm

TL;DR

This paper extends the $psilon$-BBS cellular automaton to multiple ball types, deriving conserved quantities via Schensted insertion and connecting continuous and discrete Toda systems.

Contribution

It introduces a generalized $psilon$-BBS with multiple ball types and establishes conserved quantities using combinatorial Schensted insertion, extending Toda orbit transformations.

Findings

Generalized $psilon$-BBS with many ball types.

Derived conserved quantities using Schensted insertion.

Extended birational transformations from continuous to discrete Toda orbits.

Abstract

The $\epsilon$-BBS is the family of solitonic cellular automata obtained via the ultradiscretization of the elementary Toda orbits, which is a parametrized family of integrable systems unifying the Toda equation and the relativistic Toda equation. In this paper, we derive the $\epsilon$-BBS with many kinds of balls and give its conserved quantities by the Schensted insertion algorithm which is introduced in combinatorics. To prove this, we extend birational transformations of the continuous elementary Toda orbits to the discrete hungry elementary Toda orbits.