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Abstract
The ℤN-graded Toda lattices are introduced and investigated under both infinite and periodic boundary conditions. Initially, a hierarchy of integrable ℤN-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 ℤN-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 ℤN-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 ℤN-graded Toda lattice as a Hamiltonian system in the Liouville sense is established.
Keywords
ℤN-Graded Toda lattice
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Zero curvature representation
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Bi-Hamiltonian structure
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Integrable Hamiltonian system
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37K10
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37K60
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37J35
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Ruguang Zhou, Huiyue Zhou, Na Li, Min Zhao.
ℤN-Graded Toda Lattices.
Chinese Annals of Mathematics, Series B 1-12 DOI:10.1007/s11401-025-0037-9
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