10,954 research outputs found
Integrability Test for Discrete Equations via Generalized Symmetries
In this article we present some integrability conditions for partial
difference equations obtained using the formal symmetries approach. We apply
them to find integrable partial difference equations contained in a class of
equations obtained by the multiple scale analysis of the general multilinear
dispersive difference equation defined on the square.Comment: Proceedings of the Symposium in Memoriam Marcos Moshinsk
Asymptotic symmetries of difference equations on a lattice
It is known that many equations of interest in Mathematical Physics display
solutions which are only asymptotically invariant under transformations (e.g.
scaling and/or translations) which are not symmetries of the considered
equation. In this note we extend the approach to asymptotic symmetries for the
analysis of PDEs, to the case of difference equations
Classification of five-point differential-difference equations
Using the generalized symmetry method, we carry out, up to autonomous point
transformations, the classification of integrable equations of a subclass of
the autonomous five-point differential-difference equations. This subclass
includes such well-known examples as the Itoh-Narita-Bogoyavlensky and the
discrete Sawada-Kotera equations. The resulting list contains 17 equations some
of which seem to be new. We have found non-point transformations relating most
of the resulting equations among themselves and their generalized symmetries.Comment: 29 page
Multiscale expansion and integrability properties of the lattice potential KdV equation
We apply the discrete multiscale expansion to the Lax pair and to the first
few symmetries of the lattice potential Korteweg-de Vries equation. From these
calculations we show that, like the lowest order secularity conditions give a
nonlinear Schroedinger equation, the Lax pair gives at the same order the
Zakharov and Shabat spectral problem and the symmetries the hierarchy of point
and generalized symmetries of the nonlinear Schroedinger equation.Comment: 10 pages, contribution to the proceedings of the NEEDS 2007
Conferenc
A discrete linearizability test based on multiscale analysis
In this paper we consider the classification of dispersive linearizable
partial difference equations defined on a quad-graph by the multiple scale
reduction around their harmonic solution. We show that the A_1, 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 Mobius transformation
Lie discrete symmetries of lattice equations
We extend two of the methods previously introduced to find discrete
symmetries of differential equations to the case of difference and
differential-difference equations. As an example of the application of the
methods, we construct the discrete symmetries of the discrete Painlev\'e I
equation and of the Toda lattice equation
A discrete integrability test based on multiscale analysis
In this article we present the results obtained applying the multiple scale
expansion up to the order \epsilon^6 to a dispersive multilinear class of
equations on a square lattice depending on 13 parameters. We show that the
integrability conditions given by the multiple scale expansion give rise to 4
nonlinear equations, 3 of which are new, depending at most on 2 parameters and
containing integrable sub cases. Moreover at least one sub case provides an
example of a new integrable system
The lattice Schwarzian KdV equation and its symmetries
In this paper we present a set of results on the symmetries of the lattice
Schwarzian Korteweg-de Vries (lSKdV) equation. We construct the Lie point
symmetries and, using its associated spectral problem, an infinite sequence of
generalized symmetries and master symmetries. We finally show that we can use
master symmetries of the lSKdV equation to construct non-autonomous
non-integrable generalized symmetries.Comment: 11 pages, no figures. Submitted to Jour. Phys. A, Special Issue SIDE
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