Discrete Reductive Perturbation Technique

Abstract

We expand a partial difference equation (P$Δ$E) on multiple lattices and obtain the P$Δ$E which governs its far field behaviour. The perturbative--reductive approach is here performed on well known nonlinear P$Δ$Es, 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 P$Δ$Es of the nonlinear Schrödinger (NLS) type.

18 pages, 1 figure. submitted to Journal of Mathematical Physics