Super-QRT and 4D-mappings reduced from the lattice super-KdV equation

Abstract

Starting from the complete integrable lattice super-KdV equation, two super-mappings are obtained by performing a travelling-wave reduction. The first one is linear and the second is a four dimensional super-QRT mapping containing both Grassmann commuting and anti-commuting dependent variables. Adapting the classical "staircase" method to the Lax super-matrices of the lattice super-KdV equation, we compute the Lax super-matrices of the mapping and the two invariants; the first one is a pure nilpotent commuting quantity and the second one is given by an elliptic curve containing nilpotent commuting Grassmann coefficients as well. In the case of finitely generated Grassmann algebra with two generators, the super-QRT mapping becomes a four-dimensional ordinary discrete dynamical system that has two invariants but does not satisfy singularity confinement criterion. It is also observed that the dynamical degree of this system grows quadratically.