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<p>The ℤ<sub><i>N</i></sub>-graded Toda lattices are introduced and investigated under both infinite and periodic boundary conditions. Initially, a hierarchy of integrable ℤ<sub><i>N</i></sub>-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<sub><i>N</i></sub>-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<sub><i>N</i></sub>-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<sub><i>N</i></sub>-graded Toda lattice as a Hamiltonian system in the Liouville sense is established.</p>

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ℤN-Graded Toda Lattices

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Ruguang Zhou,
  • Huiyue Zhou,
  • Na Li,
  • Min Zhao

摘要

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.