204 research outputs found
Classification of conservation laws of compressible isentropic fluid flow in n>1 spatial dimensions
For the Euler equations governing compressible isentropic fluid flow with a
barotropic equation of state (where pressure is a function only of the
density), local conservation laws in spatial dimensions are fully
classified in two primary cases of physical and analytical interest: (1)
kinematic conserved densities that depend only on the fluid density and
velocity, in addition to the time and space coordinates; (2) vorticity
conserved densities that have an essential dependence on the curl of the fluid
velocity. A main result of the classification in the kinematic case is that the
only equation of state found to be distinguished by admitting extra
-dimensional conserved integrals, apart from mass, momentum, energy, angular
momentum and Galilean momentum (which are admitted for all equations of state),
is the well-known polytropic equation of state with dimension-dependent
exponent . In the vorticity case, no distinguished equations of
state are found to arise, and here the main result of the classification is
that, in all even dimensions , a generalized version of Kelvin's
two-dimensional circulation theorem is obtained for a general equation of
state.Comment: 24 pages; published version with misprints correcte
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 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
The converse problem for the multipotentialisation of evolution equations and systems
We propose a method to identify and classify evolution equations and systems
that can be multipotentialised in given target equations or target systems. We
refer to this as the {\it converse problem}. Although we mainly study a method
for -dimensional equations/system, we do also propose an extension of
the methodology to higher-dimensional evolution equations. An important point
is that the proposed converse method allows one to identify certain types of
auto-B\"acklund transformations for the equations/systems. In this respect we
define the {\it triangular-auto-B\"acklund transformation} and derive its
connections to the converse problem. Several explicit examples are given. In
particular we investigate a class of linearisable third-order evolution
equations, a fifth-order symmetry-integrable evolution equation as well as
linearisable systems.Comment: 31 Pages, 7 diagrams, submitted for consideratio
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
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
Isotropy, shear, symmetry and exact solutions for relativistic fluid spheres
The symmetry method is used to derive solutions of Einstein's equations for
fluid spheres using an isotropic metric and a velocity four vector that is
non-comoving. Initially the Lie, classical approach is used to review and
provide a connecting framework for many comoving and so shear free solutions.
This provides the basis for the derivation of the classical point symmetries
for the more general and mathematicaly less tractable description of Einstein's
equations in the non-comoving frame. Although the range of symmetries is
restrictive, existing and new symmetry solutions with non-zero shear are
derived. The range is then extended using the non-classical direct symmetry
approach of Clarkson and Kruskal and so additional new solutions with non-zero
shear are also presented. The kinematics and pressure, energy density, mass
function of these solutions are determined.Comment: To appear in Classical and Quantum Gravit
Symmetries of the near horizon of a Black Hole by Group Theoretic methods
We use group theoretic methods to obtain the extended Lie point symmetries of
the quantum dynamics of a scalar particle probing the near horizon structure of
a black hole. Symmetries of the classical equations of motion for a charged
particle in the field of an inverse square potential and a monopole, in the
presence of certain model magnetic fields and potentials are also studied. Our
analysis gives the generators and Lie algebras generating the inherent
symmetries.Comment: To appear in Int. J. Mod. Phys.
Symmetries, weak symmetries and related solutions of the Grad-Shafranov equation
We discuss a new family of solutions of the Grad--Shafranov (GS) equation
that describe D-shaped toroidal plasma equilibria with sharp gradients at the
plasma edge. These solutions have been derived by exploiting the continuous Lie
symmetry properties of the GS equation and in particular a special type of
"weak" symmetries. In addition, we review the continuous Lie symmetry
properties of the GS equation and present a short but exhaustive survey of the
possible choices for the arbitrary flux functions that yield GS equations
admitting some continuous Lie symmetry. Particular solutions related to these
symmetries are also discussed.Comment: 8 pages, 4 figure
Equivalence of conservation laws and equivalence of potential systems
We study conservation laws and potential symmetries of (systems of)
differential equations applying equivalence relations generated by point
transformations between the equations. A Fokker-Planck equation and the Burgers
equation are considered as examples. Using reducibility of them to the
one-dimensional linear heat equation, we construct complete hierarchies of
local and potential conservation laws for them and describe, in some sense, all
their potential symmetries. Known results on the subject are interpreted in the
proposed framework. This paper is an extended comment on the paper of J.-q. Mei
and H.-q. Zhang [Internat. J. Theoret. Phys., 2006, in press].Comment: 10 page
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