Algebro-geometric solutions for some (2+1)(2+1)-dimensional discrete systems

Starting from a discrete spectral problem, a discrete soliton hierarchy is derived. Some (2+1)(2+1)-dimensional discrete systems related to the hierarchy are proposed. The elliptic coordinates are introduced and the equations in the discrete soliton hierarchy are decomposed into solvable ordinary differential equations. The straightening out of the continuous flow and the discrete flow are exactly given through the Abel–Jacobi coordinates. As an application, explicit algebro-geometric solutions for the (2+1)(2+1)-dimensional discrete systems are obtained.