Tropical Jacobian and the generic fiber of the ultra-discrete periodic Toda lattice are isomorphic

Abstract

We prove that the general isolevel set of the ultra-discrete periodic Toda lattice is isomorphic to the tropical Jacobian associated with the tropical spectral curve. This result implies that the theta function solution obtained in the authors' previous paper is the complete solution. We also propose a method to solve the initial value problem.

arXiv comment14 pages

References in corpus (6)

• Fundamental Cycle of a Periodic Box-Ball System
• On a Periodic Soliton Cellular Automaton
• Bethe ansatz and inverse scattering transform in a periodic box-ball system
• The Bethe ansatz in a periodic box-ball system and the ultradiscrete Riemann theta function
• Ultradiscretization of the solution of periodic Toda equation
• A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice

Cited by in corpus (3)

• Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry
• A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice
• Tropical spectral curves, Fay's trisecant identity, and generalized ultradiscrete Toda lattice