494 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
Reduction of Algebraic Parametric Systems by Rectification of their Affine Expanded Lie Symmetries
Lie group theory states that knowledge of a -parameters solvable group of
symmetries of a system of ordinary differential equations allows to reduce by
the number of equations. We apply this principle by finding some
\emph{affine derivations} that induces \emph{expanded} Lie point symmetries of
considered system. By rewriting original problem in an invariant coordinates
set for these symmetries, we \emph{reduce} the number of involved parameters.
We present an algorithm based on this standpoint whose arithmetic complexity is
\emph{quasi-polynomial} in input's size.Comment: Before analysing an algebraic system (differential or not), one can
generally reduce the number of parameters defining the system behavior by
studying the system's Lie symmetrie
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
Nonlocal symmetries of Riccati and Abel chains and their similarity reductions
We study nonlocal symmetries and their similarity reductions of Riccati and
Abel chains. Our results show that all the equations in Riccati chain share the
same form of nonlocal symmetry. The similarity reduced order ordinary
differential equation (ODE), , in this chain yields
order ODE in the same chain. All the equations in the Abel chain also share the
same form of nonlocal symmetry (which is different from the one that exist in
Riccati chain) but the similarity reduced order ODE, , in
the Abel chain always ends at the order ODE in the Riccati chain.
We describe the method of finding general solution of all the equations that
appear in these chains from the nonlocal symmetry.Comment: Accepted for publication in J. Math. Phy
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
Use of Complex Lie Symmetries for Linearization of Systems of Differential Equations - II: Partial Differential Equations
The linearization of complex ordinary differential equations is studied by
extending Lie's criteria for linearizability to complex functions of complex
variables. It is shown that the linearization of complex ordinary differential
equations implies the linearizability of systems of partial differential
equations corresponding to those complex ordinary differential equations. The
invertible complex transformations can be used to obtain invertible real
transformations that map a system of nonlinear partial differential equations
into a system of linear partial differential equation. Explicit invariant
criteria are given that provide procedures for writing down the solutions of
the linearized equations. A few non-trivial examples are mentioned.Comment: This paper along with its first part ODE-I were combined in a single
research paper "Linearizability criteria for systems of two second-order
differential equations by complex methods" which has been published in
Nonlinear Dynamics. Due to citations of both parts I and II these are not
replaced with the above published articl
Lie group classifications and exact solutions for time-fractional Burgers equation
Lie group method provides an efficient tool to solve nonlinear partial
differential equations. This paper suggests a fractional Lie group method for
fractional partial differential equations. A time-fractional Burgers equation
is used as an example to illustrate the effectiveness of the Lie group method
and some classes of exact solutions are obtained.Comment: 9 pp, accepte
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