### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 063, 42 pages      arXiv:2106.07129      https://doi.org/10.3842/SIGMA.2022.063

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

Nicholas M. Ercolani and Jonathan Ramalheira-Tsu
Department of Mathematics, University of Arizona, USA

Received April 08, 2022, in final form August 20, 2022; Published online August 25, 2022

Abstract
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.

Key words: soliton phase shifts; box-ball system; ultradiscretization; Gessel-Viennot-Lindström lemma.

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