Multiscale techniques for nonlinear difference equations

The aim of this thesis is the development of a multiscale reductive perturbation technique for discrete systems, that is systems described by partial difference equations. A guiding principle in such a programme should certainly be the requirement, if one starts from an integrable model, to maintain this integrability property for the reduced models. So, if for an integrable system the reduced equations should always be at all perturbative orders integrable (a member of an integrable hierarchy), for a nonintegrable one the result could be, up to any finite order, either integrable or not. Anyway for a nonintegrable system there should always exist an order at which we obtain a nonintegrable equation. Thus a properly developed multiscale technique should provide us as a by-product, besides approximate solutions to our equations of motion, an integrability test capable in principle to recognize a nonintegrable system.