Quasi-periodic solutions for some 2+1-dimensional discrete models

Some new 2+1-dimensional discrete models are proposed with the help of the 1+1-dimensional nonlinear network equations describing a Volterra system. The nonlinearization of the Lax pairs associated with the 1+1-dimensional nonlinear network equations leads to a new integrable symplectic map and a class of finite-dimensional integrable Hamiltonian systems. These 2+1-dimensional discrete models are decomposed into two Hamiltonian systems of ordinary differential equations plus the discrete flow generated by the symplectic map. The evolution of various flows is explicitly given through the Abel–Jacobi coordinates. Quasi-periodic solutions for these 2+1-dimensional discrete models are obtained resorting to the Riemann theta functions.