Multiscale Expansion and Integrability Properties of the Lattice Potential KdV Equation

Heredero, Rafael Hernández; Levi, Decio; Petrera, Matteo; Scimiterna, Christian

doi:10.2991/jnmp.2008.15.s3.31

Cited by 6 publications

(9 citation statements)

References 15 publications

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“…Then, by a paradox, they showed that a C-integrable equation must reduce to a linear equation or another C-integrable equation as the Eckhaus equation [4,22]. In the case of discrete equations it has been shown [26,1,16,18,17,19,10,11,12] that a similar situation is also true. One presents the equivalent of the Calogero-Eckhaus theorem stating that a necessary condition for a nonlinear dispersive partial difference equation to be S-integrable is that the lowest order multiple scale expansion on C (∞) functions give rise to integrable NLSE.…”

Section: Introductionmentioning

confidence: 99%

C-Integrability Test for Discrete Equations via Multiple Scale Expansions

Scimiterna

Levi

2010

SIGMA

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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 nonsecularity condition is given by a non-integrable nonlinear Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable.

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Section: Introductionmentioning

confidence: 99%

C-Integrability Test for Discrete Equations via Multiple Scale Expansions

Scimiterna

Levi

2010

SIGMA

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“…As was shown in [10,11,17], the introduction of multiple scales on a lattice reduces the given discrete equation either to a local nonlinear PDE, by imposing a slow-varying condition, or to a PDE of infinite order when dealing with analytic functions. Here we choose the second alternative because, as shown in [10,11], only in this case are the integrability conditions of the discrete equation preserved.…”

Section: Introductionmentioning

confidence: 99%

A discrete linearizability test based on multiscale analysis

Heredero¹,

Levi²,

Scimiterna³

2010

J. Phys. A: Math. Theor.

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In this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad-graph by the múltiple scale reduction around their harmonic solution. We show that the Ai, 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 Móbius transformation.

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“…An integrable partial differential equation, as the NLS equation ( 1), has an asymptotic integrability of infinite order. Some attempts to extend this approach to discrete equations have been proposed [6,[16][17][18][23][24][25]. In [23][24][25], a multiscale technique for dispersive Z 2 lattice equations has been developed which is based on the dilation transformations of discrete shift operators.…”

Section: Introductionmentioning

confidence: 99%