203 research outputs found
Monodromy transform and the integral equation method for solving the string gravity and supergravity equations in four and higher dimensions
The monodromy transform and corresponding integral equation method described
here give rise to a general systematic approach for solving integrable
reductions of field equations for gravity coupled bosonic dynamics in string
gravity and supergravity in four and higher dimensions. For different types of
fields in space-times of dimensions with commuting isometries
-- stationary fields with spatial symmetries, interacting waves or partially
inhomogeneous cosmological models, the string gravity equations govern the
dynamics of interacting gravitational, dilaton, antisymmetric tensor and any
number of Abelian vector gauge fields (all depending only on two
coordinates). The equivalent spectral problem constructed earlier allows to
parameterize the infinite-dimensional space of local solutions of these
equations by two pairs of \cal{arbitrary} coordinate-independent holomorphic
- and - matrix functions of a spectral parameter which constitute a complete set
of monodromy data for normalized fundamental solution of this spectral problem.
The "direct" and "inverse" problems of such monodromy transform --- calculating
the monodromy data for any local solution and constructing the field
configurations for any chosen monodromy data always admit unique solutions. We
construct the linear singular integral equations which solve the inverse
problem. For any \emph{rational} and \emph{analytically matched} (i.e.
and
) monodromy data the solution for string
gravity equations can be found explicitly. Simple reductions of the space of
monodromy data leads to the similar constructions for solving of other
integrable symmetry reduced gravity models, e.g. 5D minimal supergravity or
vacuum gravity in dimensions.Comment: RevTex 7 pages, 1 figur
Infinite hierarchies of exact solutions of the Einstein and Einstein-Maxwell equations for interacting waves and inhomogeneous cosmologies
For space-times with two spacelike isometries, we present infinite
hierarchies of exact solutions of the Einstein and Einstein--Maxwell equations
as represented by their Ernst potentials. This hierarchy contains three
arbitrary rational functions of an auxiliary complex parameter. They are
constructed using the so called `monodromy transform' approach and our new
method for the solution of the linear singular integral equation form of the
reduced Einstein equations. The solutions presented, which describe
inhomogeneous cosmological models or gravitational and electromagnetic waves
and their interactions, include a number of important known solutions as
particular cases.Comment: 7 pages, minor correction and reduction to conform with published
versio
Discrete singular integrals in a half-space
We consider Calderon -- Zygmund singular integral in the discrete half-space
, where is entire lattice () in ,
and prove that the discrete singular integral operator is invertible in
) iff such is its continual analogue. The key point for
this consideration takes solvability theory of so-called periodic Riemann
boundary problem, which is constructed by authors.Comment: 9 pages, 1 figur
Mode signature and stability for a Hamiltonian model of electron temperature gradient turbulence
Stability properties and mode signature for equilibria of a model of electron
temperature gradient (ETG) driven turbulence are investigated by Hamiltonian
techniques. After deriving the infinite families of Casimir invariants,
associated with the noncanonical Poisson bracket of the model, a sufficient
condition for stability is obtained by means of the Energy-Casimir method. Mode
signature is then investigated for linear motions about homogeneous equilibria.
Depending on the sign of the equilibrium "translated" pressure gradient, stable
equilibria can either be energy stable, i.e.\ possess definite linearized
perturbation energy (Hamiltonian), or spectrally stable with the existence of
negative energy modes (NEMs). The ETG instability is then shown to arise
through a Kre\u{\i}n-type bifurcation, due to the merging of a positive and a
negative energy mode, corresponding to two modified drift waves admitted by the
system. The Hamiltonian of the linearized system is then explicitly transformed
into normal form, which unambiguously defines mode signature. In particular,
the fast mode turns out to always be a positive energy mode (PEM), whereas the
energy of the slow mode can have either positive or negative sign
Monodromy-data parameterization of spaces of local solutions of integrable reductions of Einstein's field equations
For the fields depending on two of the four space-time coordinates only, the
spaces of local solutions of various integrable reductions of Einstein's field
equations are shown to be the subspaces of the spaces of local solutions of the
``null-curvature'' equations constricted by a requirement of a universal (i.e.
solution independent) structures of the canonical Jordan forms of the unknown
matrix variables. These spaces of solutions of the ``null-curvature'' equations
can be parametrized by a finite sets of free functional parameters -- arbitrary
holomorphic (in some local domains) functions of the spectral parameter which
can be interpreted as the monodromy data on the spectral plane of the
fundamental solutions of associated linear systems. Direct and inverse problems
of such mapping (``monodromy transform''), i.e. the problem of finding of the
monodromy data for any local solution of the ``null-curvature'' equations with
given canonical forms, as well as the existence and uniqueness of such solution
for arbitrarily chosen monodromy data are shown to be solvable unambiguously.
The linear singular integral equations solving the inverse problems and the
explicit forms of the monodromy data corresponding to the spaces of solutions
of the symmetry reduced Einstein's field equations are derived.Comment: LaTeX, 33 pages, 1 figure. Typos, language and reference correction
A comparison of pre-impact gas cushioning and Wagner theory for liquid-solid impacts
ACKNOWLEDGEMENTS Snizhana Ross was supported by the Development Trust of the University of Aberdeen.Peer reviewedPostprintPublisher PD
Large scale correlations in normal and general non-Hermitian matrix ensembles
We compute the large scale (macroscopic) correlations in ensembles of normal
random matrices with an arbitrary measure and in ensembles of general
non-Hermition matrices with a class of non-Gaussian measures. In both cases the
eigenvalues are complex and in the large limit they occupy a domain in the
complex plane. For the case when the support of eigenvalues is a connected
compact domain, we compute two-, three- and four-point connected correlation
functions in the first non-vanishing order in 1/N in a manner that the
algorithm of computing higher correlations becomes clear. The correlation
functions are expressed through the solution of the Dirichlet boundary problem
in the domain complementary to the support of eigenvalues. The two-point
correlation functions are shown to be universal in the sense that they depend
only on the support of eigenvalues and are expressed through the Dirichlet
Green function of its complement.Comment: 16 pages, 1 figure, LaTeX, submitted to J. Phys. A special issue on
random matrices, minor corrections, references adde
First passage and arrival time densities for L\'evy flights and the failure of the method of images
We discuss the first passage time problem in the semi-infinite interval, for
homogeneous stochastic Markov processes with L{\'e}vy stable jump length
distributions (),
namely, L{\'e}vy flights (LFs). In particular, we demonstrate that the method
of images leads to a result, which violates a theorem due to Sparre Andersen,
according to which an arbitrary continuous and symmetric jump length
distribution produces a first passage time density (FPTD) governed by the
universal long-time decay . Conversely, we show that for LFs the
direct definition known from Gaussian processes in fact defines the probability
density of first arrival, which for LFs differs from the FPTD. Our findings are
corroborated by numerical results.Comment: 8 pages, 3 figures, iopart.cls style, accepted to J. Phys. A (Lett
Cavity formation on the surface of a body entering water with deceleration
The two-dimensional water entry of a rigid symmetric body with account for cavity formation on the body surface is studied. Initially the liquid is at rest and occupies the lower half plane. The rigid symmetric body touches the liquid free surface at a single point and then starts suddenly to penetrate the liquid vertically with a time-varying speed. We study the effect of the body deceleration on the pressure distribution in the flow region. It is shown that, in addition to the high pressures expected from the theory of impact, the pressure on the body surface can later decrease to sub-atmospheric levels. The creation of a cavity due to such low pressures is considered. The cavity starts at the lowest point of the body and spreads along the body surface forming a thin space between a new free surface and the body. Within the linearised hydrodynamic problem, the positions of the two turnover points at the periphery of the wetted area are determined by Wagner’s condition. The ends of the cavity’s free surface are modelled by the Brillouin–Villat condition. The pressure in the cavity is assumed to be a prescribed constant, which is a parameter of the model. The hydrodynamic problem is reduced to a system of integral and differential equations with respect to several functions of time. Results are presented for constant deceleration of two body shapes: a parabola and a wedge. The general formulation made also embraces conditions where the body is free to decelerate under the total fluid force. Contrasts are drawn between results from the present model and a simpler model in which the cavity formation is suppressed. It is shown that the expansion of the cavity can be significantly slower than the expansion of the corresponding zone of sub-atmospheric pressure in the simpler model. For forced motion and cavity pressure close to atmospheric, the cavity grows until almost complete detachment of the fluid from the body. In the problem of free motion of the body, cavitation with vapour pressure in the cavity is achievable only for extremely large impact velocities
Analytical solution of second Stokes problem of behaviour of rarefied gas with Cercignani boundary accomodation conditions
Analytical solution of second Stokes problem of behaviour of rarefied gas
with Cercignani boundary accomodation conditions The second Stokes problem
about behaviour of rarefied gas filling half-space is analytically solved. A
plane, limiting half-space, makes harmonious fluctuations in the plane. The
kinetic BGK-equation (Bhatnagar, Gross, Krook) is used. The boundary
accomodation conditions of Cercignani of reflexion gaseous molecules from a
wall are considered. Distribution function of the gaseous molecules is
constructed. The velocity of gas in half-space is found, also its value direct
at a wall is found. The force resistance operating from gas on border is found.
Besides, the capacity of dissipation of the energy falling to unit of area of
the fluctuating plate limiting gas is obtained.Comment: 26 pages, 5 figure
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