Solitons in cellular automata

Jeffrey Giansiracusa - email: jeffrey.giansiracusa@durham.ac.uk

Description

A soliton is a travelling wave in some system that maintains its localised shape and doesn't disperse as it moves. In 1990 Takahashi and Satsuma invented an interesting discrete cellular automaton with soliton-like behaviour. This system is called the box-ball system (BBS). It consists of an infinite sequence of boxes that can each hold at most one ball. The update rule is that we move along and pick up each ball and then place it in the next available box.  

This deceptively simple rule turns out to lead to a richness of connections to various areas in mathematics, including tropical geometry and the theory of integrable systems, quantum groups, combinatorics, and probability theory. One such connection is with the KdV equation, which is a famous partial differential equation that was proposed to describe waves travelling in shallow water.  By discretising this equation and then applying a procedure called tropicalisation, the BBS appears!  Another pathway to BBS is via the famous Lotka-Volterra predator-prey model.

In this project we will explore the BBS and its many variations.  Work could include deriving it via tropicalisation, understanding algorithms for identifying solitons in it, writing computer code to simulate it, and exploring some of the links with group theory.

Prerequisites and corequisites

This project will build heavily on concepts from Algebra II (MATH2581).  Almost nothing else will be necessary, but whatever background you bring could be useful.

Resources

As far as I know, nobody has written a book on this topic yet, but there are many papers that should be accessible to undergraduates.

A nice little illustration: Reiho Sakamoto "Periodic Box-Ball System" (Wolfram Demonstrations Project) - http://demonstrations.wolfram.com/PeriodicBoxBallSystem/

Ferrari, P., Nguyen, C., Rolla, L., & Wang, M. (2021). Soliton Decomposition of the Box-Ball System. Forum of Mathematics, Sigma, 9, E60. doi:10.1017/fms.2021.49
https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/soliton-decomposition-of-the-boxball-system/69C2B6AE2CF71F3CC842DB5ABBA7DBAC/share/b38ddf9a8c94c774df45a413e7bc07b6093d33dd

A recorded lecture by David Croyden: https://www.mathtube.org/lecture/video/probabilistic-view-box-ball-system-and-other-discrete-integrable-systems-0

The original paper is: Daisuke Takahashi, Junkichi Satsuma, A Soliton Cellular Automaton, Journal of the Physical Society of Japan, 59, 3514-3519 (1990)
(available through Durham Library at https://journals-jps-jp.ezphost.dur.ac.uk/doi/pdf/10.1143/JPSJ.59.3514 )