The Liouville Integrable Lattice Equations Associated With a Discrete Three-by-Three Matrix Spectral Problem

Abstract

A hierarchy of integrable lattice equations with three potentials is constructed from a new discrete 3 × 3 matrix spectral problem. It is shown that the hierarchy possesses a Hamiltonian structure and a hereditary recursion operator, which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals.

Binary Bargmann symmetry constraint associated with 3×3 discrete matrix spectral problem

“…Then the compatibility conditions of (2.1) and (2.9) are 10) which implies the lattice solition equations…”

“…On the other hand, there appears much difficulty in handling the Liouville integrability of the so-called constrained flows generated from spectral problems, in the case of the third and fourth-order matrix spectral problems [5,10,15,16,29,34]. It is a challenging task to extend the theory of nonlinearization to the case of higher-order matrix spectral problems.…”