Exact discretization of the Ermakov-Pinney equation

Making use of the link with Schrodinger operators and the Darboux transform ation, a Backlund transformation (BT) for the (continuous) Ermakov-Pinney e quation is constructed. By considering two applications of the BT we obtain a second order discrete equation, which is naturally interpreted as the ex act discretization of the Ermakov-Pinney equation. Another second order equ ation with the same continuum limit is obtained by applying the BT to a dif ferent dependent variable. The two discretizations considered previously by Musette and Common are seen to be approximations to these two exact equati ons. We consider the connection with the discrete Schwarzian, the lineariza tion to a third order difference equation and the nonlinear superposition p rinciple relating the general solution to a discrete Schrodinger equation. Applications to finite-dimensional Hamiltonian systems are discussed. (C) 1 999 Elsevier Science B.V. All rights reserved.