The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation

• In this paper, we construct a discrete matrix spectral and obtain a discrete integrable system which is Liouville integrable.
• Compared to the continue integrable system, we get the integrable symplectic map of the discrete integrable system.
• We get the solution of the discrete equation by the symmetry method which is different from the traditional approaches such as Backlund transformation, Hirota approach and Darboux transformation.

A discrete matrix spectral problem is presented and the hierarchy of discrete integrable systems is derived. Their Hamiltonian structures are established. As to the discrete integrable system, nonlinearization of the spatial parts of the Lax pairs and the adjoint Lax pairs generate a new integrable symplectic map. Based on the theory, a new integrable symplectic map and a family of finite-dimension completely integrable systems are given. Especially, two explicit equations are obtained under the Bargmann constraint. Finally, the symmetry of the discrete equation is provided according to the recursion operator and the seed symmetry. Although the solutions of the discrete equations have been gained by many methods, there are few articles that solving the discrete equation via the symmetry. So the solution of the discrete lattice equation is obtained through the symmetry theory.