High order multiscale analysis of discrete integrable equations

In this article we present the results obtained applying the multiple scale expansion up to the order $\varepsilon^6$ to a dispersive multilinear class of equations on a square lattice depending on 13 parameters. We show that the integrability conditions given by the multiple scale expansion give rise to 4 nonlinear equations, 3 of which are new, depending at most on 2 parameters and containing integrable sub cases. Moreover at least one sub case provides an example of a new integrable system.