A delay analogue of the box and ball system arising from the ultra-discretization of the delay discrete Lotka-Volterra equation

TL;DR

This paper introduces a delay-based cellular automaton derived from ultra-discretizing a delay Lotka-Volterra equation, revealing new soliton patterns including both normal and novel abnormal types.

Contribution

It presents a new delay analogue of the box and ball system, classifies solitons, and analytically constructs { au}-functions for 1-soliton cases.

Findings

Normal solitons relate to known BBS solitons with multiple ball types.

Abnormal solitons exhibit novel patterns not seen in existing BBS models.

Analytical { au}-functions are derived for 1-soliton abnormal cases.

Abstract

A delay analogue of the box and ball system (BBS) is presented. This new soliton cellular automaton is constructed by the ultra-discretization of the delay discrete Lotka-Volterra equation, which is an integrable delay analogue of the discrete Lotka-Volterra equation. Soliton patterns generated by this delay BBS are classified into normal solitons and abnormal solitons. Normal solitons have a clear relationship to the solitons of the BBS with K kinds of balls. On the other hand, abnormal solitons show various types of novel soliton patterns, which have not been observed in almost all known BBSs. We obtain them by numerical experiments, and then construct {\tau}-functions of them analytically in 1-soliton cases.