Lie-Theoretical Generalization and Discretization of the Pinney Equation

A Lie group approach is adopted to construct generalized Pinney equations of two distinct types which admit nonlinear superposition principles. The procedure also provides a route to discretizations of these Pinney equations which preserves the property of admittance of a nonlinear superposition principle. To conclude, underlying linearizations are placed in the context of results for C-integrable nonlinear Schr¨odinger equations