The multiscale expansions of difference equations in the small lattice spacing regime, and a vicinity and integrability test: I

We propose an algorithmic procedure (i) to study the 'distance' between an integrable PDE and any discretization of it, in the small lattice spacing epsilon regime, and, at the same time, (ii) to test the ( asymptotic) integrability properties of such discretization. This method should provide, in particular, useful and concrete information on how good is any numerical scheme used to integrate a given integrable PDE. The procedure, illustrated on a fairly general ten-parameter family of discretizations of the nonlinear Schrodinger equation, consists of the following three steps: ( i) the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in epsilon, following Degasperis et al (1997 Physica D 100 187-211); ( ii) the application, to such an expansion, of the Degasperis-Procesi (DP) integrability test ( Degasperis A and Procesi M 1999 Asymptotic integrability Symmetry and Perturbation Theory, SPT98, ed A Degasperis and G Gaeta ( Singapore: World Scientific) pp 23-37; Degasperis A 2001 Multiscale expansion and integrability of dispersive wave equations Lectures given at the Euro Summer School: 'What is integrability?' ( Isaac Newton Institute, Cambridge, UK, 13-24 August); Integrability ( Lecture Notes in Physics vol 767) ed A Mikhailov ( Berlin: Springer)), to test the asymptotic integrability properties of the discrete system and its 'distance' from its continuous limit; (iii) the use of the main output of the DP test to construct infinitely many approximate symmetries and constants of motion of the discrete system, through novel and simple formulas.