322 research outputs found
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
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
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
Conditional symmetry and spectrum of the one-dimensional Schr\"odinger equation
We develop an algebraic approach to studying the spectral properties of the
stationary Schr\"odinger equation in one dimension based on its high order
conditional symmetries. This approach makes it possible to obtain in explicit
form representations of the Schr\"odinger operator by matrices for
any and, thus, to reduce a spectral problem to a purely
algebraic one of finding eigenvalues of constant matrices. The
connection to so called quasi exactly solvable models is discussed. It is
established, in particular, that the case, when conditional symmetries reduce
to high order Lie symmetries, corresponds to exactly solvable Schr\"odinger
equations. A symmetry classification of Sch\"odinger equation admitting
non-trivial high order Lie symmetries is carried out, which yields a hierarchy
of exactly solvable Schr\"odinger equations. Exact solutions of these are
constructed in explicit form. Possible applications of the technique developed
to multi-dimensional linear and one-dimensional nonlinear Schr\"odinger
equations is briefly discussed.Comment: LaTeX-file, 31 pages, to appear in J.Math.Phys., v.37, N7, 199
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
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
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
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
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
Gravitating fluids with Lie symmetries
We analyse the underlying nonlinear partial differential equation which
arises in the study of gravitating flat fluid plates of embedding class one.
Our interest in this equation lies in discussing new solutions that can be
found by means of Lie point symmetries. The method utilised reduces the partial
differential equation to an ordinary differential equation according to the Lie
symmetry admitted. We show that a class of solutions found previously can be
characterised by a particular Lie generator. Several new families of solutions
are found explicitly. In particular we find the relevant ordinary differential
equation for all one-dimensional optimal subgroups; in several cases the
ordinary differential equation can be solved in general. We are in a position
to characterise particular solutions with a linear barotropic equation of
state.Comment: 13 pages, To appear in J. Phys. A: Math. Theo
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