TY - JOUR

T1 - A Path-Counting Analysis of Phase Shifts in Box-Ball Systems

AU - Ercolani, Nicholas M.

AU - Ramalheira-Tsu, Jonathan

N1 - Funding Information:
This work was supported by NSF grant DMS-1615921. We thank the referees for their very careful reading of the manuscript.
Publisher Copyright:
© 2022, Institute of Mathematics. All rights reserved.

PY - 2022

Y1 - 2022

N2 - In this paper, we perform a detailed analysis of the phase shift phenomenon of the classical soliton cellular automaton known as the box-ball system, ultimately resulting in a statement and proof of a formula describing this phase shift. This phenomenon has been observed since the nineties, when the system was first introduced by Takahashi and Satsuma, but no explicit global description was made beyond its observation. By using the Gessel–Viennot–Lindström lemma and path-counting arguments, we present here a novel proof of the classical phase shift formula for the continuous-time Toda lattice, as discovered by Moser, and use this proof to derive a discrete-time Toda lattice analogue of the phase shift phenomenon. By carefully analysing the connection between the box-ball system and the discrete-time Toda lattice, through the mechanism of tropicalisation/dequantisation, we translate this discrete-time Toda lattice phase shift formula into our new formula for the box-ball system phase shift.

AB - In this paper, we perform a detailed analysis of the phase shift phenomenon of the classical soliton cellular automaton known as the box-ball system, ultimately resulting in a statement and proof of a formula describing this phase shift. This phenomenon has been observed since the nineties, when the system was first introduced by Takahashi and Satsuma, but no explicit global description was made beyond its observation. By using the Gessel–Viennot–Lindström lemma and path-counting arguments, we present here a novel proof of the classical phase shift formula for the continuous-time Toda lattice, as discovered by Moser, and use this proof to derive a discrete-time Toda lattice analogue of the phase shift phenomenon. By carefully analysing the connection between the box-ball system and the discrete-time Toda lattice, through the mechanism of tropicalisation/dequantisation, we translate this discrete-time Toda lattice phase shift formula into our new formula for the box-ball system phase shift.

KW - Gessel–Viennot–Lind-ström lemma

KW - box-ball system

KW - soliton phase shifts

KW - ultradiscretization

UR - http://www.scopus.com/inward/record.url?scp=85137316961&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85137316961&partnerID=8YFLogxK

U2 - 10.3842/SIGMA.2022.063

DO - 10.3842/SIGMA.2022.063

M3 - Article

AN - SCOPUS:85137316961

VL - 18

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SN - 1815-0659

M1 - 63

ER -