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

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.

25 pages, 3 figures

References in corpus (4)

• Enumerative tropical algebraic geometry in R2
• The Bethe ansatz in a periodic box-ball system and the ultradiscrete Riemann theta function
• Ultradiscretization of the solution of periodic Toda equation
• Tropical Jacobian and the generic fiber of the ultra-discrete periodic Toda lattice are isomorphic

Cited by in corpus (11)

• Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry
• Bethe Ansatz, Inverse Scattering Transform and Tropical Riemann Theta Function in a Periodic Soliton Cellular Automaton for A^{(1)}_n
• Relationships Between Two Approaches: Rigged Configurations and 10-Eliminations
• Toric networks, geometric $R$-matrices and generalized discrete Toda lattices
• Combinatorial aspects of the conserved quantities of the tropical periodic Toda lattice
• An ultradiscrete integrable map arising from a pair of tropical elliptic pencils
• Tropical Jacobian and the generic fiber of the ultra-discrete periodic Toda lattice are isomorphic
• Commuting Time Evolutions in the Tropical Periodic Toda Lattice
• A geometric realization of the periodic discrete Toda lattice and its tropicalization
• Tropical spectral curves, Fay's trisecant identity, and generalized ultradiscrete Toda lattice
• A geometric realization of the ultradiscrete periodic Toda lattice via tropical plane curves