11,996 research outputs found
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
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
Lie Symmetries and Exact Solutions of First Order Difference Schemes
We show that any first order ordinary differential equation with a known Lie
point symmetry group can be discretized into a difference scheme with the same
symmetry group. In general, the lattices are not regular ones, but must be
adapted to the symmetries considered. The invariant difference schemes can be
so chosen that their solutions coincide exactly with those of the original
differential equation.Comment: Minor changes and journal-re
Difference schemes with point symmetries and their numerical tests
Symmetry preserving difference schemes approximating second and third order
ordinary differential equations are presented. They have the same three or
four-dimensional symmetry groups as the original differential equations. The
new difference schemes are tested as numerical methods. The obtained numerical
solutions are shown to be much more accurate than those obtained by standard
methods without an increase in cost. For an example involving a solution with a
singularity in the integration region the symmetry preserving scheme, contrary
to standard ones, provides solutions valid beyond the singular point.Comment: 26 pages 7 figure
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
VI
A new two-dimensional lattice model that is "consistent around a cube"
For two-dimensional lattice equations one definition of integrability is that
the model can be naturally and consistently extended to three dimensions, i.e.,
that it is "consistent around a cube" (CAC). As a consequence of CAC one can
construct a Lax pair for the model. Recently Adler, Bobenko and Suris conducted
a search based on this principle and certain additional assumptions. One of
those assumptions was the "tetrahedron property", which is satisfied by most
known equations. We present here one lattice equation that satisfies the
consistency condition but does not have the tetrahedron property. Its Lax pair
is also presented and some basic properties discussed.Comment: 8 pages in LaTe
Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV
We consider multiple lattices and functions defined on them. We introduce
slow varying conditions for functions defined on the lattice and express the
variation of a function in terms of an asymptotic expansion with respect to the
slow varying lattices.
We use these results to perform the multiple--scale reduction of the lattice
potential Korteweg--de Vries equation.Comment: 17 pages. 1 figur
Continuous Symmetries of Difference Equations
Lie group theory was originally created more than 100 years ago as a tool for
solving ordinary and partial differential equations. In this article we review
the results of a much more recent program: the use of Lie groups to study
difference equations. We show that the mismatch between continuous symmetries
and discrete equations can be resolved in at least two manners. One is to use
generalized symmetries acting on solutions of difference equations, but leaving
the lattice invariant. The other is to restrict to point symmetries, but to
allow them to also transform the lattice.Comment: Review articl
Discrete derivatives and symmetries of difference equations
We show on the example of the discrete heat equation that for any given
discrete derivative we can construct a nontrivial Leibniz rule suitable to find
the symmetries of discrete equations. In this way we obtain a symmetry Lie
algebra, defined in terms of shift operators, isomorphic to that of the
continuous heat equation.Comment: submitted to J.Phys. A 10 Latex page
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
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