238 research outputs found
The Bianchi Ix (MIXMASTER) Cosmological Model is Not Integrable
The perturbation of an exact solution exhibits a movable transcendental
essential singularity, thus proving the nonintegrability. Then, all possible
exact particular solutions which may be written in closed form are isolated
with the perturbative Painlev\'e test; this proves the inexistence of any
vacuum solution other than the three known ones.Comment: 14 pages, no figure
G3-homogeneous gravitational instantons
We provide an exhaustive classification of self-dual four-dimensional
gravitational instantons foliated with three-dimensional homogeneous spaces,
i.e. homogeneous self-dual metrics on four-dimensional Euclidean spaces
admitting a Bianchi simply transitive isometry group. The classification
pattern is based on the algebra homomorphisms relating the Bianchi group and
the duality group SO(3). New and general solutions are found for Bianchi III.Comment: 24 pages, few correction
Ricci flows and expansion in axion-dilaton cosmology
We study renormalization-group flows by deforming a class of conformal
sigma-models. We consider overall scale factor perturbation of Einstein spaces
as well as more general anisotropic deformations of three-spheres. At leading
order in alpha, renormalization-group equations turn out to be Ricci flows. In
the three-sphere background, the latter is the Halphen system, which is exactly
solvable in terms of modular forms. We also analyze time-dependent deformations
of these systems supplemented with an extra time coordinate and time-dependent
dilaton. In some regimes time evolution is identified with
renormalization-group flow and time coordinate can appear as Liouville field.
The resulting space-time interpretation is that of a homogeneous isotropic
Friedmann-Robertson-Walker universe in axion-dilaton cosmology. We find as
general behaviour the superposition of a big-bang (polynomial) expansion with a
finite number of oscillations at early times. Any initial anisotropy disappears
during the evolution.Comment: 22 page
Solutions of the sDiff(2)Toda equation with SU(2) Symmetry
We present the general solution to the Plebanski equation for an H-space that
admits Killing vectors for an entire SU(2) of symmetries, which is therefore
also the general solution of the sDiff(2)Toda equation that allows these
symmetries. Desiring these solutions as a bridge toward the future for yet more
general solutions of the sDiff(2)Toda equation, we generalize the earlier work
of Olivier, on the Atiyah-Hitchin metric, and re-formulate work of Babich and
Korotkin, and Tod, on the Bianchi IX approach to a metric with an SU(2) of
symmetries. We also give careful delineations of the conformal transformations
required to ensure that a metric of Bianchi IX type has zero Ricci tensor, so
that it is a self-dual, vacuum solution of the complex-valued version of
Einstein's equations, as appropriate for the original Plebanski equation.Comment: 27 page
Detection and construction of an elliptic solution to the complex cubic-quintic Ginzburg-Landau equation
In evolution equations for a complex amplitude, the phase obeys a much more
intricate equation than the amplitude. Nevertheless, general methods should be
applicable to both variables. On the example of the traveling wave reduction of
the complex cubic-quintic Ginzburg-Landau equation (CGL5), we explain how to
overcome the difficulties arising in two such methods: (i) the criterium that
the sum of residues of an elliptic solution should be zero, (ii) the
construction of a first order differential equation admitting the given
equation as a differential consequence (subequation method).Comment: 12 pages, no figure, to appear, Theoretical and Mathematical Physic
Projective dynamics and classical gravitation
Given a real vector space V of finite dimension, together with a particular
homogeneous field of bivectors that we call a "field of projective forces", we
define a law of dynamics such that the position of the particle is a "ray" i.e.
a half-line drawn from the origin of V. The impulsion is a bivector whose
support is a 2-plane containing the ray. Throwing the particle with a given
initial impulsion defines a projective trajectory. It is a curve in the space
of rays S(V), together with an impulsion attached to each ray. In the simplest
example where the force is identically zero, the curve is a straight line and
the impulsion a constant bivector. A striking feature of projective dynamics
appears: the trajectories are not parameterized.
Among the projective force fields corresponding to a central force, the one
defining the Kepler problem is simpler than those corresponding to other
homogeneities. Here the thrown ray describes a quadratic cone whose section by
a hyperplane corresponds to a Keplerian conic. An original point of view on the
hidden symmetries of the Kepler problem emerges, and clarifies some remarks due
to Halphen and Appell. We also get the unexpected conclusion that there exists
a notion of divergence-free field of projective forces if and only if dim V=4.
No metric is involved in the axioms of projective dynamics.Comment: 20 pages, 4 figure
Elliptic Solitons and Groebner Bases
We consider the solution of spectral problems with elliptic coefficients in
the framework of the Hermite ansatz. We show that the search for exactly
solvable potentials and their spectral characteristics is reduced to a system
of polynomial equations solvable by the Gr\"obner bases method and others. New
integrable potentials and corresponding solutions of the Sawada-Kotera,
Kaup-Kupershmidt, Boussinesq equations and others are found.Comment: 18 pages, no figures, LaTeX'2
Meromorphic traveling wave solutions of the complex cubic-quintic Ginzburg-Landau equation
We look for singlevalued solutions of the squared modulus M of the traveling
wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using
Clunie's lemma, we first prove that any meromorphic solution M is necessarily
elliptic or degenerate elliptic. We then give the two canonical decompositions
of the new elliptic solution recently obtained by the subequation method.Comment: 14 pages, no figure, to appear, Acta Applicandae Mathematica
Blurred constitutive laws and bipotential convex covers
In many practical situations, incertitudes affect the mechanical behaviour
that is given by a family of graphs instead of a single one. In this paper, we
show how the bipotential method is able to capture such blurred constitutive
laws, using bipotential convex covers
Explicit solutions of the four-wave mixing model
The dynamical degenerate four-wave mixing is studied analytically in detail.
By removing the unessential freedom, we first characterize this system by a
lower-dimensional closed subsystem of a deformed Maxwell-Bloch type, involving
only three physical variables: the intensity pattern, the dynamical grating
amplitude, the relative net gain. We then classify by the Painleve' test all
the cases when singlevalued solutions may exist, according to the two essential
parameters of the system: the real relaxation time tau, the complex response
constant gamma. In addition to the stationary case, the only two integrable
cases occur for a purely nonlocal response (Real(gamma)=0), these are the
complex unpumped Maxwell-Bloch system and another one, which is explicitly
integrated with elliptic functions. For a generic response (Re(gamma) not=0),
we display strong similarities with the cubic complex Ginzburg-Landau equation.Comment: 16 pages, J Phys A Fast track communication, to appear 200
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