514 research outputs found
[SADE] A Maple package for the Symmetry Analysis of Differential Equations
We present the package SADE (Symmetry Analysis of Differential Equations) for
the determination of symmetries and related properties of systems of
differential equations. The main methods implemented are: Lie, nonclassical,
Lie-B\"acklund and potential symmetries, invariant solutions, first-integrals,
N\"other theorem for both discrete and continuous systems, solution of ordinary
differential equations, reduction of order or dimension using Lie symmetries,
classification of differential equations, Casimir invariants, and the
quasi-polynomial formalism for ODE's (previously implemented in the package
QPSI by the authors) for the determination of quasi-polynomial first-integrals,
Lie symmetries and invariant surfaces. Examples of use of the package are
given
Classification of 3-dimensional integrable scalar discrete equations
We classify all integrable 3-dimensional scalar discrete quasilinear
equations Q=0 on an elementary cubic cell of the 3-dimensional lattice. An
equation Q=0 is called integrable if it may be consistently imposed on all
3-dimensional elementary faces of the 4-dimensional lattice.
Under the natural requirement of invariance of the equation under the action
of the complete group of symmetries of the cube we prove that the only
nontrivial (non-linearizable) integrable equation from this class is the
well-known dBKP-system. (Version 2: A small correction in Table 1 (p.7) for n=2
has been made.) (Version 3: A few small corrections: one more reference added,
the main statement stated more explicitly.)Comment: 20 p. LaTeX + 1 EPS figur
Some symmetry classifications of hyperbolic vector evolution equations
Motivated by recent work on integrable flows of curves and 1+1 dimensional
sigma models, several O(N)-invariant classes of hyperbolic equations for an -component vector are considered. In each
class we find all scaling-homogeneous equations admitting a higher symmetry of
least possible scaling weight. Sigma model interpretations of these equations
are presented.Comment: Revision of published version, incorporating errata on geometric
aspects of the sigma model interpretations in the case of homogeneous space
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
WDVV equations: symbolic computations of Hamiltonian operators
We describe software for symbolic computations that we developed in or-
der to find Hamiltonian operators for Witten–Dijkgraaf–Verlinde–Verlinde (WDVV)
equations, and verify their compatibility. The computation involves nonlocal (integro-
differential) operators, for which specific canonical forms and algorithms have been
used
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
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