Discrete Reductive Perturbation Technique

TL;DR

This paper develops a discrete reductive perturbation technique to analyze the far field behavior of nonlinear partial difference equations, deriving new equations of the nonlinear Schrödinger type from well-known lattice models.

Contribution

It introduces a novel perturbative-reductive method applied to various lattice equations, including integrable and non-integrable cases, to derive new nonlinear Schrödinger type equations.

Findings

Derived new PΔEs of NLS type from known lattice equations.

Applied the method to both integrable and non-integrable equations.

Provided insights into the asymptotic behavior of lattice equations.

Abstract

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