MULTI-SCALE REDUCTION FOR DIFFERENTIAL DIFFERENCE EQUATIONS AND INTEGRABILITY

Perturbation methods has proved very successful in analyzing classes of partial differential equations, pinning out integrable cases and providing informations on their long time behavior. Here we present the state of art of this approach in the case of differential equations defined on a lattice. In particular we present the results of Leon and Manna on the multiscale analysis of discrete nonlinear evolution equations and show that their resulting equation has not the same integrability property as Ablowitz and Ladik Discrete Nonlinear Schrödinger equation.