274 research outputs found
MHD equilibria with incompressible flows: symmetry approach
We identify and discuss a family of azimuthally symmetric, incompressible,
magnetohydrodynamic plasma equilibria with poloidal and toroidal flows in terms
of solutions of the Generalized Grad Shafranov (GGS) equation. These solutions
are derived by exploiting the incompressibility assumption, in order to rewrite
the GGS equation in terms of a different dependent variable, and the continuous
Lie symmetry properties of the resulting equation and in particular a special
type of "weak" symmetries.Comment: Accepted for publication in Phys. Plasma
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.
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 notion of conditional symmetry of differential equations
Symmetry properties of PDE's are considered within a systematic and unifying
scheme: particular attention is devoted to the notion of conditional symmetry,
leading to the distinction and a precise characterization of the notions of
``true'' and ``weak'' conditional symmetry. Their relationship with exact and
partial symmetries is also discussed. An extensive use of ``symmetry-adapted''
variables is made; several clarifying examples, including the case of
Boussinesq equation, are also provided.Comment: 18 page
Local and nonlocal solvable structures in ODEs reduction
Solvable structures, likewise solvable algebras of local symmetries, can be
used to integrate scalar ODEs by quadratures. Solvable structures, however, are
particularly suitable for the integration of ODEs with a lack of local
symmetries. In fact, under regularity assumptions, any given ODE always admits
solvable structures even though finding them in general could be a very
difficult task. In practice a noteworthy simplification may come by computing
solvable structures which are adapted to some admitted symmetry algebra. In
this paper we consider solvable structures adapted to local and nonlocal
symmetry algebras of any order (i.e., classical and higher). In particular we
introduce the notion of nonlocal solvable structure
Weak Transversality and Partially Invariant Solutions
New exact solutions are obtained for several nonlinear physical equations,
namely the Navier-Stokes and Euler systems, an isentropic compressible fluid
system and a vector nonlinear Schroedinger equation. The solution methods make
use of the symmetry group of the system in situations when the standard Lie
method of symmetry reduction is not applicable.Comment: 23 pages, preprint CRM-284
Nonlocal aspects of -symmetries and ODEs reduction
A reduction method of ODEs not possessing Lie point symmetries makes use of
the so called -symmetries (C. Muriel and J. L. Romero, \emph{IMA J.
Appl. Math.} \textbf{66}, 111-125, 2001). The notion of covering for an ODE
is used here to recover -symmetries of as
nonlocal symmetries. In this framework, by embedding into a
suitable system determined by the function ,
any -symmetry of can be recovered by a local symmetry of
. As a consequence, the reduction method of Muriel and
Romero follows from the standard method of reduction by differential invariants
applied to .Comment: 13 page
Efeitos da inibição prolongada da enzima de conversĂŁo da angiotensina sobre as caracterĂsticas morfolĂłgicas e funcionais da hipertrofia ventricular esquerda em ratos com sobrecarga pressĂłrica persistente
Noether theorem for mu-symmetries
We give a version of Noether theorem adapted to the framework of
mu-symmetries; this extends to such case recent work by Muriel, Romero and
Olver in the framework of lambda-symmetries, and connects mu-symmetries of a
Lagrangian to a suitably modified conservation law. In some cases this
"mu-conservation law'' actually reduces to a standard one; we also note a
relation between mu-symmetries and conditional invariants. We also consider the
case where the variational principle is itself formulated as requiring
vanishing variation under mu-prolonged variation fields, leading to modified
Euler-Lagrange equations. In this setting mu-symmetries of the Lagrangian
correspond to standard conservation laws as in the standard Noether theorem. We
finally propose some applications and examples.Comment: 28 pages, to appear in J. Phys.
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