997 research outputs found
An annotated bibliography of aquatic sediment traps and trapping methods
This annotated bibliography is intended to give as reasonably complete a review of the existing literature as possible, and to offer some practical guidance in the selection and operation of sediment traps in future monitoring programmes
Combinatorial Bethe ansatz and ultradiscrete Riemann theta function with rational characteristics
The U_q(\hat{sl}_2) vertex model at q=0 with periodic boundary condition is
an integrable cellular automaton in one-dimension. By the combinatorial Bethe
ansatz, the initial value problem is solved for arbitrary states in terms of an
ultradiscrete analogue of the Riemann theta function with rational
characteristics.Comment: 9 page
Boundary value problems for the stationary axisymmetric Einstein equations: a disk rotating around a black hole
We solve a class of boundary value problems for the stationary axisymmetric
Einstein equations corresponding to a disk of dust rotating uniformly around a
central black hole. The solutions are given explicitly in terms of theta
functions on a family of hyperelliptic Riemann surfaces of genus 4. In the
absence of a disk, they reduce to the Kerr black hole. In the absence of a
black hole, they reduce to the Neugebauer-Meinel disk.Comment: 46 page
The variational Poisson cohomology
It is well known that the validity of the so called Lenard-Magri scheme of
integrability of a bi-Hamiltonian PDE can be established if one has some
precise information on the corresponding 1st variational Poisson cohomology for
one of the two Hamiltonian operators. In the first part of the paper we explain
how to introduce various cohomology complexes, including Lie superalgebra and
Poisson cohomology complexes, and basic and reduced Lie conformal algebra and
Poisson vertex algebra cohomology complexes, by making use of the corresponding
universal Lie superalebra or Lie conformal superalgebra. The most relevant are
certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology
complexes, which we identify (non-canonically) with the generalized de Rham
complex and the generalized variational complex. In the second part of the
paper we compute the cohomology of the generalized de Rham complex, and, via a
detailed study of the long exact sequence, we compute the cohomology of the
generalized variational complex for any quasiconstant coefficient Hamiltonian
operator with invertible leading coefficient. For the latter we use some
differential linear algebra developed in the Appendix.Comment: 130 pages, revised version with minor changes following the referee's
suggestion
Why the Hamilton operator alone is not enough
In the many worlds community seems to exist a belief that the physics of a
quantum theory is completely defined by it's Hamilton operator given in an
abstract Hilbert space, especially that the position basis may be derived from
it as preferred using decoherence techniques.
We show, by an explicit example of non-uniqueness, taken from the theory of
the KdV equation, that the Hamilton operator alone is not sufficient to fix the
physics. We need the canonical operators p, q as well. As a consequence, it is
not possible to derive a "preferred basis" from the Hamilton operator alone,
without postulating some additional structure like a "decomposition into
systems". We argue that this makes such a derivation useless for fundamental
physics
Quantum and Classical Integrable Systems
The key concept discussed in these lectures is the relation between the
Hamiltonians of a quantum integrable system and the Casimir elements in the
underlying hidden symmetry algebra. (In typical applications the latter is
either the universal enveloping algebra of an affine Lie algebra, or its
q-deformation.) A similar relation also holds in the classical case. We discuss
different guises of this very important relation and its implication for the
description of the spectrum and the eigenfunctions of the quantum system.
Parallels between the classical and the quantum cases are thoroughly discussed.Comment: 59 pages, LaTeX2.09 with AMS symbols. Lectures at the CIMPA Winter
School on Nonlinear Systems, Pondicherry, January 199
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
A new variational approach to the stability of gravitational systems
We consider the three dimensional gravitational Vlasov Poisson system which
describes the mechanical state of a stellar system subject to its own gravity.
A well-known conjecture in astrophysics is that the steady state solutions
which are nonincreasing functions of their microscopic energy are nonlinearly
stable by the flow. This was proved at the linear level by several authors
based on the pioneering work by Antonov in 1961. Since then, standard
variational techniques based on concentration compactness methods as introduced
by P.-L. Lions in 1983 have led to the nonlinear stability of subclasses of
stationary solutions of ground state type.
In this paper, inspired by pioneering works from the physics litterature
(Lynden-Bell 94, Wiechen-Ziegler-Schindler MNRAS 88, Aly MNRAS 89), we use the
monotonicity of the Hamiltonian under generalized symmetric rearrangement
transformations to prove that non increasing steady solutions are local
minimizer of the Hamiltonian under equimeasurable constraints, and extract
compactness from suitable minimizing sequences. This implies the nonlinear
stability of nonincreasing anisotropic steady states under radially symmetric
perturbations
Symmetries of a class of nonlinear fourth order partial differential equations
In this paper we study symmetry reductions of a class of nonlinear fourth
order partial differential equations \be u_{tt} = \left(\kappa u + \gamma
u^2\right)_{xx} + u u_{xxxx} +\mu u_{xxtt}+\alpha u_x u_{xxx} + \beta u_{xx}^2,
\ee where , , , and are constants. This
equation may be thought of as a fourth order analogue of a generalization of
the Camassa-Holm equation, about which there has been considerable recent
interest. Further equation (1) is a ``Boussinesq-type'' equation which arises
as a model of vibrations of an anharmonic mass-spring chain and admits both
``compacton'' and conventional solitons. A catalogue of symmetry reductions for
equation (1) is obtained using the classical Lie method and the nonclassical
method due to Bluman and Cole. In particular we obtain several reductions using
the nonclassical method which are no} obtainable through the classical method
Dynamical stability of infinite homogeneous self-gravitating systems: application of the Nyquist method
We complete classical investigations concerning the dynamical stability of an
infinite homogeneous gaseous medium described by the Euler-Poisson system or an
infinite homogeneous stellar system described by the Vlasov-Poisson system
(Jeans problem). To determine the stability of an infinite homogeneous stellar
system with respect to a perturbation of wavenumber k, we apply the Nyquist
method. We first consider the case of single-humped distributions and show
that, for infinite homogeneous systems, the onset of instability is the same in
a stellar system and in the corresponding barotropic gas, contrary to the case
of inhomogeneous systems. We show that this result is true for any symmetric
single-humped velocity distribution, not only for the Maxwellian. If we
specialize on isothermal and polytropic distributions, analytical expressions
for the growth rate, damping rate and pulsation period of the perturbation can
be given. Then, we consider the Vlasov stability of symmetric and asymmetric
double-humped distributions (two-stream stellar systems) and determine the
stability diagrams depending on the degree of asymmetry. We compare these
results with the Euler stability of two self-gravitating gaseous streams.
Finally, we determine the corresponding stability diagrams in the case of
plasmas and compare the results with self-gravitating systems
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