12,774 research outputs found
Recursion operators for a class of integrable third-order evolution equations
We consider with and , for those functional
forms of for which the equation is integrable in the sense
of an infinite number of Lie-B\"acklund symmetries. Local - and
-independent recursion operators that generate these infinite sets of
symmetries are obtained for the equations. A combination of potential forms,
hodograph transformations and -generalised hodograph transformations are
applied to the obtained equations
Linearisable third order ordinary differential equations and generalised Sundman transformations
We calculate in detail the conditions which allow the most general third
order ordinary differential equation to be linearised in X'''(T)=0 under the
transformation X(T)=F(x,t), dT=G(x,t)dt. Further generalisations are
considered.Comment: 33 page
On the complete integrability and linearization of nonlinear ordinary differential equations - Part II: Third order equations
We introduce a method for finding general solutions of third-order nonlinear
differential equations by extending the modified Prelle-Singer method. We
describe a procedure to deduce all the integrals of motion associated with the
given equation so that the general solution follows straightforwardly from
these integrals. The method is illustrated with several examples. Further, we
propose a powerful method of identifying linearizing transformations. The
proposed method not only unifies all the known linearizing transformations
systematically but also introduces a new and generalized linearizing
transformation (GLT). In addition to the above, we provide an algorithm to
invert the nonlocal linearizing transformation. Through this procedure the
general solution for the original nonlinear equation can be obtained from the
solution of the linear ordinary differential equation.Comment: Submitted to Proceedings of the Royal Society London Series A, 21
page
A tree of linearisable second-order evolution equations by generalised hodograph transformations
We present a list of (1+1)-dimensional second-order evolution equations all
connected via a proposed generalised hodograph transformation, resulting in a
tree of equations transformable to the linear second-order autonomous evolution
equation. The list includes autonomous and nonautonomous equations.Comment: arXiv version is already officia
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
Electromagnetic Deflection of Spinning Particles
We show that it is possible to obtain self-consistent and physically
acceptable relativistic classical equations of motion for a point-like
spin-half particle possessing an electric charge and a magnetic dipole moment,
directly from a manifestly covariant Lagrangian, if the classical degrees of
freedom are appropriately chosen. It is shown that the equations obtained
encompass the well-tested Lorentz force and Thomas--Bargmann--Michel--Telegdi
spin equations, as well as providing a definite specification of the classical
_magnetic_dipole_ force_, whose exact form has been the subject of recent
debate. Radiation reaction---the force and torque on an accelerated particle
due to its self-interaction---is neglected at this stage.Comment: 18 pp. (latex, uses revtex 3), UM-P-92/9
Generalised action-angle coordinates defined on island chains
Straight-field-line coordinates are very useful for representing magnetic
fields in toroidally confined plasmas, but fundamental problems arise regarding
their definition in 3-D geometries because of the formation of islands and
chaotic field regions, ie non-integrability. In Hamiltonian dynamical systems
terms these coordinates are a form of action-angle variables, which are
normally defined only for integrable systems. In order to describe 3-D magnetic
field systems, a generalisation of this concept was proposed recently by the
present authors that unified the concepts of ghost surfaces and
quadratic-flux-minimising (QFMin) surfaces. This was based on a simple
canonical transformation generated by a change of variable , where and are poloidal and toroidal
angles, respectively, with a new poloidal angle chosen to give
pseudo-orbits that are a) straight when plotted in the plane and
b) QFMin pseudo-orbits in the transformed coordinate. These two requirements
ensure that the pseudo-orbits are also c) ghost pseudo-orbits. In the present
paper, it is demonstrated that these requirements do not \emph{uniquely}
specify the transformation owing to a relabelling symmetry. A variational
method of solution that removes this lack of uniqueness is proposed.Comment: 10 pages. Accepted by Plasma Physics and Controlled Fusion as part of
a cluster of refereed papers in a special issue containing papers arising
from the Joint International Stellarator & Heliotron Workshop and
Asia-Pacific Plasma Theory Conference, held in Canberra and Murramarang
Resort, Australia, 30 January - 3 February, 201
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