C-Integrability Test for Discrete Equations via Multiple Scale Expansions

TL;DR

This paper extends integrability tests for difference equations using multiple scale expansions, demonstrating that a discretized Burgers equation derived from a linearizable differential-difference equation is non-integrable.

Contribution

It develops a new C-integrability test for discrete equations via multiple scale expansions and applies it to show non-integrability of a discretized Burgers equation.

Findings

Discretized Burgers equation is not linearizable.

Multiple scale expansion reveals non-integrability.

Discretization affects integrability properties.

Abstract

In this paper we are extending the well known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Hopf-Cole transformation reduces to a linear differential difference equation. In this case the equation satisfies the $A_1$, $A_2$ and $A_3$ linearizability conditions. We then consider its discretization. To get a dispersive equation we substitute the time derivative by its symmetric discretization. When we apply to this nonlinear partial difference equation the multiple scale expansion we find out that the lowest order non-secularity condition is given by a non-integrable nonlinear Schr\"odinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable.