<i>N</i>-Dimensional Lattice Integrable Systems and Their bi-Hamiltonian Structure on the Time Scale Using the <i>R</i>-Matrix Approach

Abstract

A time scale is a special measure chain that can unify continuous and discrete spaces, enabling the construction of integrable equations. In this paper, with the Lax operator generated by the displacement operator, N-dimensional lattice integrable systems on the time scale are given by the R-matrix approach. The recursion operators of the lattice systems are derived on the time scale. Finally, two integrable hierarchies of the discrete chain with a bi-Hamiltonian structure are obtained. In particular, we give the structure of two-field and four-field systems.

Keywords

• <i>N</i>-dimensional lattice integrable systems
• time scale
• <i>R</i>-matrix approach
• bi-Hamiltonian structure