A new integrable symplectic map associated with lattice soliton equations

A method is developed that extends the nonlinearization technique to the hierarchy of lattice soliton equations associated with a discrete 3×3 matrix spectral problem. A new integrable symplectic map and its involutive system of conserved integrals are obtained by the nonlinearization of spatial parts and the time parts of Lax pairs and their adjoint Lax pairs of the hierarchy. Moreover, the solutions of the typical system of lattice equations in the hierarchy are reduced to the solutions of a system of ordinary differential equations and a simple iterative process of the symplectic map.