Toric networks, geometric $R$-matrices and generalized discrete Toda lattices

Abstract

We use the combinatorics of toric networks and the double affine geometric $R$-matrix to define a three-parameter family of generalizations of the discrete Toda lattice. We construct the integrals of motion and a spectral map for this system. The family of commuting time evolutions arising from the action of the $R$-matrix is explicitly linearized on the Jacobian of the spectral curve. The solution to the initial value problem is constructed using Riemann theta functions.

54 pages

References in corpus (3)

• Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry
• Solution of the genaralized periodic discrete Toda equation
• On the cluster nature and quantization of geometric $R$-matrices

Cited by in corpus (3)

• Cluster Toda chains and Nekrasov functions
• The Schwarzian octahedron recurrence (dSKP equation) II: geometric systems
• Discrete dynamics in cluster integrable systems from geometric $R$-matrix transformations