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This article is cited in 4 scientific papers (total in 4 papers)
$C$-Integrability Test for Discrete Equations via Multiple Scale Expansions
Christian Scimiternaab, Decio Leviab a Sezione INFN
b Dipartimento di Ingegneria Elettronica, Università degli Studi Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
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 Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable.
Keywords:
linearizable discrete equations; linearizability theorem; multiple scale expansion; obstructions to linearizability; discrete Burgers.
Received: May 29, 2010; in final form August 20, 2010; Published online August 31, 2010
Citation:
Christian Scimiterna, Decio Levi, “$C$-Integrability Test for Discrete Equations via Multiple Scale Expansions”, SIGMA, 6 (2010), 070, 17 pp.
Linking options:
https://www.mathnet.ru/eng/sigma528 https://www.mathnet.ru/eng/sigma/v6/p70
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