1,262 research outputs found
Constructing Integrable Third Order Systems:The Gambier Approach
We present a systematic construction of integrable third order systems based
on the coupling of an integrable second order equation and a Riccati equation.
This approach is the extension of the Gambier method that led to the equation
that bears his name. Our study is carried through for both continuous and
discrete systems. In both cases the investigation is based on the study of the
singularities of the system (the Painlev\'e method for ODE's and the
singularity confinement method for mappings).Comment: 14 pages, TEX FIL
Integrable systems without the Painlev\'e property
We examine whether the Painlev\'e property is a necessary condition for the
integrability of nonlinear ordinary differential equations. We show that for a
large class of linearisable systems this is not the case. In the discrete
domain, we investigate whether the singularity confinement property is
satisfied for the discrete analogues of the non-Painlev\'e continuous
linearisable systems. We find that while these discrete systems are themselves
linearisable, they possess nonconfined singularities
Is my ODE a Painleve equation in disguise?
Painleve equations belong to the class y'' + a_1 {y'}^3 + 3 a_2 {y'}^2 + 3
a_3 y' + a_4 = 0, where a_i=a_i(x,y). This class of equations is invariant
under the general point transformation x=Phi(X,Y), y=Psi(X,Y) and it is
therefore very difficult to find out whether two equations in this class are
related. We describe R. Liouville's theory of invariants that can be used to
construct invariant characteristic expressions (syzygies), and in particular
present such a characterization for Painleve equations I-IV.Comment: 8 pages. Based on talks presented at NEEDS 2000, Gokova, Turkey, 29
June - 7 July, 2000, and at the AMS-HKMS joint meeting 13-16 December, 2000.
Submitted to J. Nonlin. Math. Phy
On a q-difference Painlev\'e III equation: I. Derivation, symmetry and Riccati type solutions
A q-difference analogue of the Painlev\'e III equation is considered. Its
derivations, affine Weyl group symmetry, and two kinds of special function type
solutions are discussed.Comment: arxiv version is already officia
The spectrum of large powers of the Laplacian in bounded domains
We present exact results for the spectrum of the Nth power of the Laplacian
in a bounded domain. We begin with the one dimensional case and show that the
whole spectrum can be obtained in the limit of large N. We also show that it is
a useful numerical approach valid for any N. Finally, we discuss implications
of this work and present its possible extensions for non integer N and for 3D
Laplacian problems.Comment: 13 pages, 2 figure
Quasi-point separation of variables for the Henon-Heiles system and a system with quartic potential
We examine the problem of integrability of two-dimensional Hamiltonian
systems by means of separation of variables. The systematic approach to
construction of the special non-pure coordinate separation of variables for
certain natural two-dimensional Hamiltonians is presented. The relations with
SUSY quantum mechanics are discussed.Comment: 11 pages, Late
Dark Matter Annihilation inside Large Volume Neutrino Detectors
New particles in theories beyond the standard model can manifest as stable
relics that interact strongly with visible matter and make up a small fraction
of the total dark matter abundance. Such particles represent an interesting
physics target since they can evade existing bounds from direct detection due
to their rapid thermalization in high-density environments. In this work we
point out that their annihilation to visible matter inside large-volume
neutrino telescopes can provide a new way to constrain or discover such
particles. The signal is the most pronounced for relic masses in the GeV range,
and can be efficiently constrained by existing Super-Kamiokande searches for
di-nucleon annihilation. We also provide an explicit realization of this
scenario in the form of secluded dark matter coupled to a dark photon, and we
show that the present method implies novel and stringent bounds on the model
that are complementary to direct constraints from beam dumps, colliders, and
direct detection experiments.Comment: v2: 7 pages, 2 figures. Conclusions Unchanged. Matches version
Published in Physical Review Letter
Linearisable Mappings and the Low-Growth Criterion
We examine a family of discrete second-order systems which are integrable
through reduction to a linear system. These systems were previously identified
using the singularity confinement criterion. Here we analyse them using the
more stringent criterion of nonexponential growth of the degrees of the
iterates. We show that the linearisable mappings are characterised by a very
special degree growth. The ones linearisable by reduction to projective systems
exhibit zero growth, i.e. they behave like linear systems, while the remaining
ones (derivatives of Riccati, Gambier mapping) lead to linear growth. This
feature may well serve as a detector of integrability through linearisation.Comment: 9 pages, no figur
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