3,127 research outputs found
A list of all integrable 2D homogeneous polynomial potentials with a polynomial integral of order at most 4 in the momenta
We searched integrable 2D homogeneous polynomial potential with a polynomial
first integral by using the so-called direct method of searching for first
integrals. We proved that there exist no polynomial first integrals which are
genuinely cubic or quartic in the momenta if the degree of homogeneous
polynomial potentials is greater than 4.Comment: 22 pages, no figures, to appear in J. Phys. A: Math. Ge
Instability of toroidal magnetic field in jets and plerions
Jets and pulsar-fed supernova remnants (plerions) tend to develop highly
organized toroidal magnetic field. Such a field structure could explain the
polarization properties of some jets, and contribute to their lateral
confinement. A toroidal field geometry is also central to models for the Crab
Nebula - the archetypal plerion - and leads to the deduction that the Crab
pulsar's wind must have a weak magnetic field. Yet this `Z-pinch' field
configuration is well known to be locally unstable, even when the magnetic
field is weak and/or boundary conditions slow or suppress global modes. Thus,
the magnetic field structures imputed to the interiors of jets and plerions are
unlikely to persist.
To demonstrate this, I present a local analysis of Z-pinch instabilities for
relativistic fluids in the ideal MHD limit. Kink instabilities dominate,
destroying the concentric field structure and probably driving the system
toward a more chaotic state in which the mean field strength is independent of
radius (and in which resistive dissipation of the field may be enhanced). I
estimate the timescales over which the field structure is likely to be
rearranged and relate these to distances along relativistic jets and radii from
the central pulsar in a plerion.
I conclude that a concentric toroidal field is unlikely to exist well outside
the Crab pulsar's wind termination shock. There is thus no dynamical reason to
conclude that the magnetic energy flux carried by the pulsar wind is much
weaker than the kinetic energy flux. Abandoning this inference would resolve a
long-standing puzzle in pulsar wind theory.Comment: 28 pages, plain TeX. Accepted for publication in Ap
Algorithmic Integrability Tests for Nonlinear Differential and Lattice Equations
Three symbolic algorithms for testing the integrability of polynomial systems
of partial differential and differential-difference equations are presented.
The first algorithm is the well-known Painlev\'e test, which is applicable to
polynomial systems of ordinary and partial differential equations. The second
and third algorithms allow one to explicitly compute polynomial conserved
densities and higher-order symmetries of nonlinear evolution and lattice
equations.
The first algorithm is implemented in the symbolic syntax of both Macsyma and
Mathematica. The second and third algorithms are available in Mathematica. The
codes can be used for computer-aided integrability testing of nonlinear
differential and lattice equations as they occur in various branches of the
sciences and engineering. Applied to systems with parameters, the codes can
determine the conditions on the parameters so that the systems pass the
Painlev\'e test, or admit a sequence of conserved densities or higher-order
symmetries.Comment: Submitted to: Computer Physics Communications, Latex, uses the style
files elsart.sty and elsart12.st
Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations
The truncation method is a collective name for techniques that arise from
truncating a Laurent series expansion (with leading term) of generic solutions
of nonlinear partial differential equations (PDEs). Despite its utility in
finding Backlund transformations and other remarkable properties of integrable
PDEs, it has not been generally extended to ordinary differential equations
(ODEs). Here we give a new general method that provides such an extension and
show how to apply it to the classical nonlinear ODEs called the Painleve
equations. Our main new idea is to consider mappings that preserve the
locations of a natural subset of the movable poles admitted by the equation. In
this way we are able to recover all known fundamental Backlund transformations
for the equations considered. We are also able to derive Backlund
transformations onto other ODEs in the Painleve classification.Comment: To appear in Nonlinearity (22 pages
Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property
The integrability (solvability via an associated single-valued linear
problem) of a differential equation is closely related to the singularity
structure of its solutions. In particular, there is strong evidence that all
integrable equations have the Painlev\'e property, that is, all solutions are
single-valued around all movable singularities. In this expository article, we
review methods for analysing such singularity structure. In particular, we
describe well known techniques of nonlinear regular-singular-type analysis,
i.e. the Painlev\'e tests for ordinary and partial differential equations. Then
we discuss methods of obtaining sufficiency conditions for the Painlev\'e
property. Recently, extensions of \textit{irregular} singularity analysis to
nonlinear equations have been achieved. Also, new asymptotic limits of
differential equations preserving the Painlev\'e property have been found. We
discuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer
School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B.
Grammaticos and K. Tamizhman
Rifts in Spreading Wax Layers
We report experimental results on the rift formation between two freezing wax
plates. The plates were pulled apart with constant velocity, while floating on
the melt, in a way akin to the tectonic plates of the earth's crust. At slow
spreading rates, a rift, initially perpendicular to the spreading direction,
was found to be stable, while above a critical spreading rate a "spiky" rift
with fracture zones almost parallel to the spreading direction developed. At
yet higher spreading rates a second transition from the spiky rift to a zig-zag
pattern occurred. In this regime the rift can be characterized by a single
angle which was found to be dependent on the spreading rate. We show that the
oblique spreading angles agree with a simple geometrical model. The coarsening
of the zig-zag pattern over time and the three-dimensional structure of the
solidified crust are also discussed.Comment: 4 pages, Postscript fil
Multiple-Time Higher-Order Perturbation Analysis of the Regularized Long-Wavelength Equation
By considering the long-wave limit of the regularized long wave (RLW)
equation, we study its multiple-time higher-order evolution equations. As a
first result, the equations of the Korteweg-de Vries hierarchy are shown to
play a crucial role in providing a secularity-free perturbation theory in the
specific case of a solitary-wave solution. Then, as a consequence, we show that
the related perturbative series can be summed and gives exactly the
solitary-wave solution of the RLW equation. Finally, some comments and
considerations are made on the N-soliton solution, as well as on the
limitations of applicability of the multiple scale method in obtaining uniform
perturbative series.Comment: 15 pages, RevTex, no figures (to appear in Phys. Rev. E
On the formation of black holes in non-symmetric gravity
It has been recently suggested that the Non-symmetric Gravitational Theory
(NGT) is free of black holes. Here, we study the linear version of NGT. We find
that even with spherical symmetry the skew part of the metric is generally
non-static. In addition, if the skew field is initially regular, it will remain
regular everywhere and, in particular, at the horizon. Therefore, in the
fully-nonlinear theory, if the initial skew-field is sufficiently small, the
formation of a black hole is to be anticipated.Comment: 9 pages, ordinary LaTex
Extending the scope of microscopic solvability: Combination of the Kruskal-Segur method with Zauderer decomposition
Successful applications of the Kruskal-Segur approach to interfacial pattern
formation have remained limited due to the necessity of an integral formulation
of the problem. This excludes nonlinear bulk equations, rendering convection
intractable. Combining the method with Zauderer's asymptotic decomposition
scheme, we are able to strongly extend its scope of applicability and solve
selection problems based on free boundary formulations in terms of partial
differential equations alone. To demonstrate the technique, we give the first
analytic solution of the problem of velocity selection for dendritic growth in
a forced potential flow.Comment: Submitted to Europhys. Letters, No figures, 5 page
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