203 research outputs found
A generalization of \lambda-symmetry reduction for systems of ODEs: \sigma-symmetries
We consider a deformation of the prolongation operation, defined on sets of
vector fields and involving a mutual interaction in the definition of prolonged
ones. This maintains the "invariants by differentiation" property, and can
hence be used to reduce ODEs satisfying suitable invariance conditions in a
fully algorithmic way, similarly to what happens for standard prolongations and
symmetries.Comment: 32 page
Simple and collective twisted symmetries
After the introduction of -symmetries by Muriel and Romero, several
other types of so called "twisted symmetries" have been considered in the
literature (their name refers to the fact they are defined through a
deformation of the familiar prolongation operation); they are as useful as
standard symmetries for what concerns symmetry reduction of ODEs or
determination of special (invariant) solutions for PDEs and have thus attracted
attention. The geometrical relation of twisted symmetries to standard ones has
already been noted: for some type of twisted symmetries (in particular,
and -symmetries), this amounts to a certain kind of gauge
transformation.
In a previous review paper [G. Gaeta, "Twisted symmetries of differential
equations", {\it J. Nonlin. Math. Phys.}, {\bf 16-S} (2009), 107-136] we have
surveyed the first part of the developments of this theory; in the present
paper we review recent developments. In particular, we provide a unifying
geometrical description of the different types of twisted symmetries; this is
based on the classical Frobenius reduction applied to distribution generated by
Lie-point (local) symmetries.Comment: 40 pages; to appear in J. Nonlin. Math. Phys. 21 (2014), 593-62
Twisted symmetries of differential equations
We review the basic ideas lying at the foundation of the recently developed
theory of twisted symmetries of differential equations, and some of its
developments
On the geometry of lambda-symmetries, and PDEs reduction
We give a geometrical characterization of -prolongations of vector
fields, and hence of -symmetries of ODEs. This allows an extension to
the case of PDEs and systems of PDEs; in this context the central object is a
horizontal one-form , and we speak of -prolongations of vector fields
and -symmetries of PDEs. We show that these are as good as standard
symmetries in providing symmetry reduction of PDEs and systems, and explicit
invariant solutions
On the relation between standard and -symmetries for PDEs
We give a geometrical interpretation of the notion of -prolongations of
vector fields and of the related concept of -symmetry for partial
differential equations (extending to PDEs the notion of -symmetry for
ODEs). We give in particular a result concerning the relationship between
-symmetries and standard exact symmetries. The notion is also extended to
the case of conditional and partial symmetries, and we analyze the relation
between local -symmetries and nonlocal standard symmetries.Comment: 25 pages, no figures, latex. to be published in J. Phys.
Twisted symmetries and integrable systems
Symmetry properties are at the basis of integrability. In recent years, it
appeared that so called "twisted symmetries" are as effective as standard
symmetries in many respects (integrating ODEs, finding special solutions to
PDEs). Here we discuss how twisted symmetries can be used to detect
integrability of Lagrangian systems which are not integrable via standard
symmetries
Lambda and mu-symmetries
Lambda-symmetries of ODEs were introduced by Muriel and Romero, and discussed
by C. Muriel in her talk at SPT2001. Here we provide a geometrical
characterization of lambda-prolongations, and a generalization of these -- and
of lambda-symmetries -- to PDEs and systems thereof
Symmetries of nonlinear ordinary differential equations: the modified Emden equation as a case study
Lie symmetry analysis is one of the powerful tools to analyze nonlinear
ordinary differential equations. We review the effectiveness of this method in
terms of various symmetries. We present the method of deriving Lie point
symmetries, contact symmetries, hidden symmetries, nonlocal symmetries,
-symmetries, adjoint symmetries and telescopic vector fields of a
second-order ordinary differential equation. We also illustrate the algorithm
involved in each method by considering a nonlinear oscillator equation as an
example. The connections between (i) symmetries and integrating factors and
(ii) symmetries and integrals are also discussed and illustrated through the
same example. The interconnections between some of the above symmetries, that
is (i) Lie point symmetries and -symmetries and (ii) exponential
nonlocal symmetries and -symmetries are also discussed. The order
reduction procedure is invoked to derive the general solution of the
second-order equation.Comment: 31 pages, To appear in the proceedings of NMI workshop on nonlinear
integrable systems and their applications which was held at Centre for
Nonlinear Dynamics, Tiruchirappalli, Indi
Jacobi multipliers, non-local symmetries and nonlinear oscillators
Constants of motion, Lagrangians and Hamiltonians admitted by a family of
relevant nonlinear oscillators are derived using a geometric formalism. The
theory of the Jacobi last multiplier allows us to find Lagrangian descriptions
and constants of the motion. An application of the jet bundle formulation of
symmetries of differential equations is presented in the second part of the
paper. After a short review of the general formalism, the particular case of
non-local symmetries is studied in detail by making use of an extended
formalism. The theory is related to some results previously obtained by
Krasil'shchi, Vinogradov and coworkers. Finally the existence of non-local
symmetries for such two nonlinear oscillators is proved.Comment: 20 page
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