A New Fully Discrete Integrable System Related To The AKNS Hierarchy

A new discrete integrable partial difference equation with two discrete variablesand its finite genus solution is constructed by using two Darboux transformations of thewell-known AKNS hierarchy of equations.Darboux transformation is a powerful tool to construct explicit solutions of thesoliton equations. It has shown that the AKNS hierarchy of equations have two kindsof elementary Darboux transformations which are linear in the parameter of transfor-mations. Such Darboux transformations can be regarded as discrete spectral problems,and the compatible condition of the spectral problem gives rise to a new fully discreteintegrable system with two discrete variables. To solve the discrete integrable system,we investigate the nonlinearizations of spectral problems or the Darboux transforma-tions, and obtain two integrable symplectic maps which share a Lax matrix and thusthe invariants, the commutative flows as well as. In particular, the common solutionof such discrete flows may generate a solution of the fully discrete integrable system.Explicitly, we introduce a set of Abel-Jacobi coordinates, linearize the discrete flowsin the Jacobi variety and make the Riemann-Jacobi reversion. Finally, a finite genussolution of the new system is presented in terms of the Riemann-Theta function.