Toric networks, geometric \(R\)-matrices and generalized discrete Toda lattices. (English) Zbl 1352.37163
The combinatorics of toric networks and the double affine geometric \(R\)-matrix was used to define a three-parameter family of generalizations of the discrete Toda lattice. The integrals of motion and a spectral map for this system is constructed. 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.
MSC:
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37K60 | Lattice dynamics; integrable lattice equations |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 14H70 | Relationships between algebraic curves and integrable systems |
| 14H40 | Jacobians, Prym varieties |
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