Nonlinear gravitons from the initial value constraints of GR in Ashtekar variables

In this paper we provide a possible realization of Penrose's idea of nonlinear gravitons by constructing a solution to the initial value constraints in Ashtekar variables. The solution inputs are a spatial SU(2) connection and two free functions of position, and can be constructed as a formal operatorial expansion in powers of the cosmological constant about spacetimes of Petrov Type O. We first present the linear case, and then provide a simple nonlinear example to first order using a spatially homogeneous connection.