1,364 research outputs found
Algorithmic Integrability Tests for Nonlinear Differential and Lattice Equations
Three symbolic algorithms for testing the integrability of polynomial systems
of partial differential and differential-difference equations are presented.
The first algorithm is the well-known Painlev\'e test, which is applicable to
polynomial systems of ordinary and partial differential equations. The second
and third algorithms allow one to explicitly compute polynomial conserved
densities and higher-order symmetries of nonlinear evolution and lattice
equations.
The first algorithm is implemented in the symbolic syntax of both Macsyma and
Mathematica. The second and third algorithms are available in Mathematica. The
codes can be used for computer-aided integrability testing of nonlinear
differential and lattice equations as they occur in various branches of the
sciences and engineering. Applied to systems with parameters, the codes can
determine the conditions on the parameters so that the systems pass the
Painlev\'e test, or admit a sequence of conserved densities or higher-order
symmetries.Comment: Submitted to: Computer Physics Communications, Latex, uses the style
files elsart.sty and elsart12.st
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
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 Schr\"odinger spectral problem associated with
Volterra-type equations. We show that the ABS equations correspond to
B\"acklund 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 B\"acklund transformations for
the YdKN equation. The higher order generalized symmetries we construct in the
present paper confirm their integrability.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Lie symmetries of multidimensional difference equations
A method is presented for calculating the Lie point symmetries of a scalar
difference equation on a two-dimensional lattice. The symmetry transformations
act on the equations and on the lattice. They take solutions into solutions and
can be used to perform symmetry reduction. The method generalizes one presented
in a recent publication for the case of ordinary difference equations. In turn,
it can easily be generalized to difference systems involving an arbitrary
number of dependent and independent variables
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