### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 070, 17 pages      arXiv:1005.5288     https://doi.org/10.3842/SIGMA.2010.070

### C-Integrability Test for Discrete Equations via Multiple Scale Expansions

Christian Scimiterna and Decio Levi
Dipartimento di Ingegneria Elettronica, Università degli Studi Roma Tre and Sezione INFN, Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy

Received May 29, 2010, in final form August 20, 2010; Published online August 31, 2010

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 A1, A2 and A3 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 Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable.

Key words: linearizable discrete equations; linearizability theorem; multiple scale expansion; obstructions to linearizability; discrete Burgers.

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