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

Christian Scimiterna ; Decio Levi

Symmetry, integrability and geometry: methods and applications, Tome 6 (2010)

Résumé

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 Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable.

Détail

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

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