An application of the max-plus spectral theory to an ultradiscrete analogue of the Lax pair

TL;DR

This paper applies max-plus spectral theory to analyze an ultradiscrete Lax pair, revealing how solutions can be expressed via eigenvectors and exploring the system's solvability.

Contribution

It introduces a max-plus spectral approach to the ultradiscrete Lax pair, providing a new framework for understanding solutions and soliton interactions.

Findings

Solutions are expressible as max-linear combinations of eigenvectors.

The spectral approach clarifies the undressing operation in soliton systems.

Conditions for the solvability of the ultradiscrete Lax system are identified.

Abstract

We study the ultradiscrete analogue of Lax pair proposed by Willox et al. This "pair" is a max-plus linear system comprising four equations. Our starting point is to treat this system as a combination of two max-plus eigenproblems, with two additional constraints. Though infinite-dimensional, these two eigenproblems can be treated by means of the "standard" max-plus spectral theory. In particular, any solution to the system can be described as a max-linear combination of fundamental eigenvectors associated with each soliton. We then describe the operation of undressing using pairs of fundamental eigenvectors. We also study the solvability of the complete system of four equations as proposed by Willox et al.