Symmetry Constraint of the Differential-difference KP Hierarchy and a Second Discretization of the ZS-AKNS System

In this paper we construct a squared-eigenfunction symmetry of the scalar differential-difference KP hierarchy. Through a constraint of the symmetry, Lax triad of the differential-difference KP hierarchy is reduced to a known discrete spectral problem and a semidiscrete AKNS hierarchy. The discrete spectral problem corresponds to a bidirectional discretization of the derivatives phi(1,x) and phi(2,x) in the ZS-AKNS spectral problem and therefore it is a discretization of the later. The discrete spectral problem is also known as a Darboux transformation of the ZS-AKNS spectral problem. Isospectral and nonisospectral flows derived from the spectral problem compose a Lie algebra. Infinitely many symmetries of the nonisospectral hierarchy are obtained. By considering infinite dimensional subalgebras of the algebra and continuum limit of recursion operator, three semi-discrete AKNS hierarchies are constructed.