Finite dimensional Hamiltonian system related to Lax pair with symplectic and cyclic symmetries

TL;DR

This paper constructs finite dimensional Hamiltonian systems from 1+1D Lax pairs with symplectic and cyclic symmetries, proving their integrability and linking solutions to original PDEs, exemplified by the hyperbolic $C_n^{(1)}$ Toda equation.

Contribution

It introduces a unified approach to derive finite dimensional Hamiltonian systems from Lax pairs with symplectic and cyclic symmetries, establishing their Liouville integrability.

Findings

Finite dimensional Hamiltonian systems are constructed from the Lax pair.

Liouville integrability of these systems is proved.

Solutions of these systems correspond to solutions of the original PDE.

Abstract

For the 1+1 dimensional Lax pair with a symplectic symmetry and cyclic symmetries, it is shown that there is a natural finite dimensional Hamiltonian system related to it by presenting a unified Lax matrix. The Liouville integrability of the derived finite dimensional Hamiltonian systems is proved in a unified way. Any solution of these Hamiltonian systems gives a solution of the original PDE. As an application, the two dimensional hyperbolic $C_n^{(1)}$ Toda equation is considered and the finite dimensional integrable Hamiltonian system related to it is obtained from the general results.