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.
Bibliographical noteFunding Information:
P. Pylyavskyy was partially supported by NSF Grants DMS-1148634, DMS-1351590, and Sloan Fellowship.
T. Lam was partially supported by NSF Grants DMS-1160726, DMS-1464693, and a Simons Fellowship.
R. Inoue was partially supported by JSPS KAKENHI Grant Number 26400037.
© 2016, Springer-Verlag Berlin Heidelberg.