Multiscale expansion on the lattice and integrability of partial difference equations

Abstract

We conjecture an integrability and linearizability test for dispersive -lattice equations by using a discrete multiscale analysis. The lowest order secularity conditions from the multiscale expansion give a partial differential equation of the form of a nonlinear Schrödinger (NLS) equation. If the starting lattice equation is integrable then the resulting NLS-type equation turns out to be integrable, while if the starting equation is linearizable we get a linear Schrödinger equation. On the other hand, if we start with a non-integrable lattice equation the resulting equation can be both integrable and non-integrable. This conjecture is confirmed by many examples.