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In this article we present the results obtained applying the multiple scale expansion up to the order ε6 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 seem to be new, depending at most on 2 parameters. [ABSTRACT FROM AUTHOR]

We use the conditions of linearizability by point and by Hopf-Cole transformations, introduced recently by the authors, to classify all real multilinear equations on a square lattice with at most quadratic nonlinearity. We find that, up to a linear transformation of the dependent variable and an exchange of the independent variables, only two equations in this class are linearizable by point transformations and none by Hopf-Cole transformations. [ABSTRACT FROM AUTHOR]

In this article, we consider the multiscale reduction around the harmonic solution of a general discrete nonlinear Schrodinger equation (dNLSE) depending on constant coefficients. According to the values of the coefficients we can have both integrable and non-integrable dNLSEs. For all values of the coefficients entering the dNLSE, non-secularity conditions provide an integrable NLSE at the lowest order in the perturbation parameter. However at higher order in the perturbation expansion the request that the expansion is compatible with the NLSE hierarchy gives integrability conditions which are not satisfied for the non-integrable dNLSEs. [ABSTRACT FROM AUTHOR]

In this paper we discuss the integrability properties of a nonlinear partial difference equation on the square obtained by the multiple scale integrability test from a class of multilinear dispersive equations defined on a four-point lattice. [ABSTRACT FROM AUTHOR]