Discrete Reductive Perturbation Technique

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We expand a partial difference equation (PΔE) on multiple lattices and obtain the PΔE which governs its far field behaviour. The perturbative--reductive approach is here performed on well known nonlinear PΔEs, both integrable and non integrable. We study the cases of the lattice modified Korteweg--de Vries (mKdV) equation, the Hietarinta equation, the lattice Volterra--Kac--Van Moerbeke (VKVM) equation and a non integrable lattice KdV equation. Such reductions allow us to obtain many new PΔEs of the nonlinear Schrödinger (NLS) type.