A Study of the A dependence of deepinelastic scattering of leptons and its implications fo.pdfVIP

A STUDY OF THE CONTINUOUS AND DISCRETE GAMBIER SYSTEMS 1 A. Ramani 0 CPT, Ecole Polytechnique 0 2 CNRS, UPR 14 r 91128 Palaiseau, France p A 6 B. Grammaticos ] LPN, Universit´e Paris VII I S Tour 24-14, 5eme` ´etage . n 75251 Paris, France i l n [ S. Lafortune† 1 LPTM et GMPIB, Universit´e Paris VII v e 7 Tour 24-14, 5 ´etage 1 75251 Paris, France 0 4 0 1 0 Abstract / n i We present a systematic study of the Gambier system, which in the continuous case is l n given by two Riccati equations in cascade. We derive the condition for its integrability : v and show that the generic Gambier system contains one free function. We also derive i X the Schlesinger transformations for this system which allows in principle the systematic r a construction of the integrable cases. The above procedure is carried over to a discrete setting. We show thus how the discrete Gambier system can be expressed as a system of two homographic mappings in cascade. The integrable cases are obtained through the singularity confinement discrete integrability criterion. Finally the discrete Schlesinger transformations are also derived giving a handle to the construction of the integrable Gambier mapping. 1. Introduction Linearisable equations play a very particular role in the domain of nonlinear sys- tems. Their nonlinearity is