A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice

TL;DR

This paper develops a tropical analogue of Fay's trisecant identity for hyperelliptic tropical curves and uses it to solve the ultra-discrete periodic Toda lattice via tropical Riemann's theta functions.

Contribution

It introduces a novel tropical analogue of Fay's identity and applies it to explicitly solve the ultra-discrete Toda lattice with periodic boundary conditions.

Findings

Established a tropical Fay's trisecant identity for hyperelliptic curves.

Derived the general solution of the ultra-discrete Toda lattice.

Connected tropical geometry with integrable systems through theta functions.

Abstract

We introduce a tropical analogue of Fay's trisecant identity for a special family of hyperelliptic tropical curves. We apply it to obtain the general solution of the ultra-discrete Toda lattice with periodic boundary conditions in terms of the tropical Riemann's theta function.