ℤ_N-Graded Toda Lattices

Abstract

The ℤ_N-graded Toda lattices are introduced and investigated under both infinite and periodic boundary conditions. Initially, a hierarchy of integrable ℤ_N-graded Toda lattices is constructed using the technique of discrete zero curvature equations under infinite boundary conditions. The integrability of these lattices is demonstrated through their bi-Hamiltonian structures. Subsequently, particular emphasis is placed on the study of the ℤ_N-graded Toda lattice, the first nontrivial lattice in the hierarchy. It is discovered that this lattice can be represented in a Newtonian form with an exponential potential in the Flaschka-Manakov variables. Furthermore, the periodic ℤ_N-graded Toda lattice is identified as either a periodic Toda lattice or a set of independent periodic Toda lattices sharing the same periodicity. Finally, the complete integrability of the periodic ℤ_N-graded Toda lattice as a Hamiltonian system in the Liouville sense is established.

Keywords

ℤ_N-Graded Toda lattice / Zero curvature representation / Bi-Hamiltonian structure / Integrable Hamiltonian system / 37K10 / 37K60 / 37J35

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