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

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.