On the Lie–Poisson structure of the nonlinearized discrete eigenvalue problem

Abstract

A 3×3 discrete eigenvalue problem and corresponding discrete soliton equations are proposed. Under a constraint between the potentials and eigenfunctions, the 3×3 discrete eigenvalue problem is nonlinearized into an integrable Poisson map with a Lie–Poisson structure. Further, a reduction of the Lie–Poisson structure on the co-adjoint orbit yields the standard symplectic structure. The Poisson map is reduced to the usual symplectic map when it is restricted on the leaves of the symplectic foliation.