A discrete linearizability test based on multiscale analysis

TL;DR

This paper introduces a multiscale analysis method to classify dispersive linearizable partial difference equations on quad-graphs, identifying conditions that restrict parameters and allow linearization via Möbius transformations.

Contribution

It presents a novel classification approach for linearizable equations using multiscale reduction and explores linearization conditions on parameter-restricted subclasses.

Findings

A_1, A_2, and A_3 conditions restrict parameters

Equations passing A_3 C-integrability can be linearized by Möbius transformations

The method provides a discrete linearizability test based on multiscale analysis

Abstract

In this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad-graph by the multiple scale reduction around their harmonic solution. We show that the A_1, A_2 and A_3 linearizability conditions restrain the number of the parameters which enter into the equation. A subclass of the equations which pass the A_3 C-integrability conditions can be linearized by a Mobius transformation.