46 research outputs found
Four Points Linearizable Lattice Schemes
We provide conditions for a lattice scheme defined on a four points lattice
to be linearizable by a point transformation. We apply the obtained conditions
to a symmetry preserving difference scheme for the potential Burgers introduced
by Dorodnitsyn \cite{db} and show that it is not linearizable
Linearizability and fake Lax pair for a consistent around the cube nonlinear non-autonomous quad-graph equation
We discuss the linearization of a non-autonomous nonlinear partial difference
equation belonging to the Boll classification of quad-graph equations
consistent around the cube. We show that its Lax pair is fake. We present its
generalized symmetries which turn out to be non-autonomous and depending on an
arbitrary function of the dependent variables defined in two lattice points.
These generalized symmetries are differential difference equations which, in
some case, admit peculiar B\"acklund transformations.Comment: arXiv admin note: text overlap with arXiv:1311.2406 by other author
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
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
On the Integrability of the Discrete Nonlinear Schroedinger Equation
In this letter we present an analytic evidence of the non-integrability of
the discrete nonlinear Schroedinger equation, a well-known discrete evolution
equation which has been obtained in various contexts of physics and biology. We
use a reductive perturbation technique to show an obstruction to its
integrability.Comment: 4 pages, accepted in EP
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 A1, A2 and A3 linearizability conditions restrain the number of the parameters which enter into the equation. A subclass of the equations which pass the A3 C-integrability conditions can be linearized by a Möbius transformation
Contact transformations for difference schemes
We define a class of transformations of the dependent and independent
variables in an ordinary difference scheme. The transformations leave the
solution set of the system invariant and reduces to a group of contact
transformations in the continuous limit. We use a simple example to show that
the class is not empty and that such "contact transformations for discrete
systems" genuinely exist
Integrability of Differential-Difference Equations with Discrete Kinks
In this article we discuss a series of models introduced by Barashenkov,
Oxtoby and Pelinovsky to describe some discrete approximations to the \phi^4
theory which preserve travelling kink solutions. We show, by applying the
multiple scale test that they have some integrability properties as they pass
the A_1 and A_2 conditions. However they are not integrable as they fail the
A_3 conditions.Comment: submitted to the Proceedings of the workshop "Nonlinear Physics:
Theory and Experiment.VI" in a special issue di Theoretical and Mathematical
Physic
On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations
We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Bäcklund transformations for some particular cases of the discrete Krichever-Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate Bäcklund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability