Exact discretization of the Ermakov-Pinney equation

Making use of the link with Schrodinger operators and the Darboux transformation, a Backlund transformation (BT) for the (continuous) Ermakov-Pinney equation is constructed. By considering two applications of the BT we obtain a second order discrete equation, which is naturally interpreted as the exact discretization of the Ermakov-Pinney equation. Another second order equation with the same continuum limit is obtained by applying the BT to a different dependent variable. The two discretizations considered previously by Musette and Common are seen to be approximations to these two exact equations. We consider the connection with the discrete Schwarzian, the linearization to a third order difference equation and the nonlinear superposition principle relating the general solution to a discrete Schrodinger equation. Applications to finite-dimensional Hamiltonian systems are discussed. (C) 1999 Elsevier Science B.V. All rights reserved.