3,085 research outputs found
Computer algebra application for classification of integrable non-linear evolution equations
The application of computer algebra for classification of integrable non-linear evolution equations is discussed. Algorithms for testing conditions of formal integrability, to calculate the Lie-Bäcklund symmetries and conservation law densities are developed and implemented on the basis of the computer algebra system PL/FORMAC
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
Supersymmetric Representations and Integrable Fermionic Extensions of the Burgers and Boussinesq Equations
We construct new integrable coupled systems of N=1 supersymmetric equations
and present integrable fermionic extensions of the Burgers and Boussinesq
equations. Existence of infinitely many higher symmetries is demonstrated by
the presence of recursion operators. Various algebraic methods are applied to
the analysis of symmetries, conservation laws, recursion operators, and
Hamiltonian structures. A fermionic extension of the Burgers equation is
related with the Burgers flows on associative algebras. A Gardner's deformation
is found for the bosonic super-field dispersionless Boussinesq equation, and
unusual properties of a recursion operator for its Hamiltonian symmetries are
described. Also, we construct a three-parametric supersymmetric system that
incorporates the Boussinesq equation with dispersion and dissipation but never
retracts to it for any values of the parameters.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
A remark on nonlocal symmetries for the Calogero-Degasperis-Ibragimov-Shabat equation
We consider the Calogero-Degasperis-Ibragimov-Shabat (CDIS) equation and find
the complete set of its nonlocal symmetries depending on the local variables
and on the integral of the only local conserved density of the equation in
question. The Lie algebra of these symmetries turns out to be a central
extension of that of local generalized symmetries.Comment: arxiv version is already officia
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
An ultradiscrete matrix version of the fourth Painleve equation
We establish a matrix generalization of the ultradiscrete fourth Painlev\'e
equation (ud-PIV). Well-defined multicomponent systems that permit
ultradiscretization are obtained using an approach that relies on a group
defined by constraints imposed by the requirement of a consistent evolution of
the systems. The ultradiscrete limit of these systems yields coupled
multicomponent ultradiscrete systems that generalize ud-PIV. The dynamics,
irreducibility, and integrability of the matrix valued ultradiscrete systems
are studied.Comment: 12 pages, 12 figures, Latex2e, Submitted to J. Phys. A, corrections
mad
Classification of integrable super-systems using the SsTools environment
A classification problem is proposed for supersymmetric evolutionary PDE that
satisfy the assumptions of nonlinearity and nondegeneracy. Four classes of
nonlinear coupled boson-fermion systems are discovered under the homogeneity
assumption |f|=|b|=|D_t|=1/2. The syntax of the Reduce package SsTools, which
was used for intermediate computations, and the applicability of its procedures
to the calculus of super-PDE are described.Comment: MSC 35Q53,37K05,37K10,81T40; PACS 02.30.Ik,02.70.Wz,12.60.Jv; Comput.
Phys. Commun. (2007), 26 pages (accepted
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