On solutions of integrable discretizations of Ernst-type equations

Ernst-type equations are elegant reformulations of Einsteins vacuum equations of general relativity when the existence of two commuting Killing vector fields is assumed. Axisymmetric, stationary spacetimes such as rotating black holes and planar gravitational waves are examples of solutions of the Ernst-type equations. An essential mathematical property of the Ernst-type equations are that they are integrable nonlinear differential equations, in particular, there exists the nonlinear superposition principle for their solutions. A key focus is the discretizations of the Ernst-type equation (i.e. difference equations that in a continuum limit become Ernst-type equations) that exhibit all the features of integrability including the nonlinear superposition principle. In the literature, one can find two discretisations of this kind. I will present exact solutions to the discrete equations