866 research outputs found
Topological Phenomena in the Real Periodic Sine-Gordon Theory
The set of real finite-gap Sine-Gordon solutions corresponding to a fixed
spectral curve consists of several connected components. A simple explicit
description of these components obtained by the authors recently is used to
study the consequences of this property. In particular this description allows
to calculate the topological charge of solutions (the averaging of the
-derivative of the potential) and to show that the averaging of other
standard conservation laws is the same for all components.Comment: LaTeX, 18 pages, 3 figure
On Integrable Systems and Supersymmetric Gauge Theories
The properties of the N=2 SUSY gauge theories underlying the Seiberg-Witten
hypothesis are discussed. The main ingredients of the formulation of the
finite-gap solutions to integrable equations in terms of complex curves and
generating 1-differential are presented, the invariant sense of these
definitions is illustrated. Recently found exact nonperturbative solutions to
N=2 SUSY gauge theories are formulated using the methods of the theory of
integrable systems and where possible the parallels between standard quantum
field theory results and solutions to integrable systems are discussed.Comment: LaTeX, 38 pages, no figures; based on the lecture given at INTAS
School on Advances in Quantum Field Theory and Statistical Mechanics, Como,
Italy, 1996; minor changes, few references adde
Invariant description of solutions of hydrodynamic type systems in hodograph space: hydrodynamic surfaces
Hydrodynamic surfaces are solutions of hydrodynamic type systems viewed as
non-parametrized submanifolds of the hodograph space. We propose an invariant
differential-geometric characterization of hydrodynamic surfaces by expressing
the curvature form of the characteristic web in terms of the reciprocal
invariants.Comment: 12 page
Weakly-nonlocal Symplectic Structures, Whitham method, and weakly-nonlocal Symplectic Structures of Hydrodynamic Type
We consider the special type of the field-theoretical Symplectic structures
called weakly nonlocal. The structures of this type are in particular very
common for the integrable systems like KdV or NLS. We introduce here the
special class of the weakly nonlocal Symplectic structures which we call the
weakly nonlocal Symplectic structures of Hydrodynamic Type. We investigate then
the connection of such structures with the Whitham averaging method and propose
the procedure of "averaging" of the weakly nonlocal Symplectic structures. The
averaging procedure gives the weakly nonlocal Symplectic Structure of
Hydrodynamic Type for the corresponding Whitham system. The procedure gives
also the "action variables" corresponding to the wave numbers of -phase
solutions of initial system which give the additional conservation laws for the
Whitham system.Comment: 64 pages, Late
Slow flows of an relativistic perfect fluid in a static gravitational field
Relativistic hydrodynamics of an isentropic fluid in a gravitational field is
considered as the particular example from the family of Lagrangian
hydrodynamic-type systems which possess an infinite set of integrals of motion
due to the symmetry of Lagrangian with respect to relabeling of fluid particle
labels. Flows with fixed topology of the vorticity are investigated in
quasi-static regime, when deviations of the space-time metric and the density
of fluid from the corresponding equilibrium configuration are negligibly small.
On the base of the variational principle for frozen-in vortex lines dynamics,
the equation of motion for a thin relativistic vortex filament is derived in
the local induction approximation.Comment: 4 pages, revtex, no figur
Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2+1 dimensions
Hamiltonian systems of hydrodynamic type occur in a wide range of
applications including fluid dynamics, the Whitham averaging procedure and the
theory of Frobenius manifolds. In 1+1 dimensions, the requirement of the
integrability of such systems by the generalised hodograph transform implies
that integrable Hamiltonians depend on a certain number of arbitrary functions
of two variables. On the contrary, in 2+1 dimensions the requirement of the
integrability by the method of hydrodynamic reductions, which is a natural
analogue of the generalised hodograph transform in higher dimensions, leads to
finite-dimensional moduli spaces of integrable Hamiltonians. In this paper we
classify integrable two-component Hamiltonian systems of hydrodynamic type for
all existing classes of differential-geometric Poisson brackets in 2D,
establishing a parametrisation of integrable Hamiltonians via
elliptic/hypergeometric functions. Our approach is based on the Godunov-type
representation of Hamiltonian systems, and utilises a novel construction of
Godunov's systems in terms of generalised hypergeometric functions.Comment: Latex, 34 page
On Bohr-Sommerfeld bases
This paper combines algebraic and Lagrangian geometry to construct a special
basis in every space of conformal blocks, the Bohr-Sommerfeld (BS) basis. We
use the method of [D. Borthwick, T. Paul and A. Uribe, Legendrian distributions
with applications to the non-vanishing of Poincar\'e series of large weight,
Invent. math, 122 (1995), 359-402, preprint hep-th/9406036], whereby every
vector of a BS basis is defined by some half-weighted Legendrian distribution
coming from a Bohr-Sommerfeld fibre of a real polarization of the underlying
symplectic manifold. The advantage of BS bases (compared to bases of theta
functions in [A. Tyurin, Quantization and ``theta functions'', Jussieu preprint
216 (Apr 1999), e-print math.AG/9904046, 32pp.]) is that we can use information
from the skillful analysis of the asymptotics of quantum states. This gives
that Bohr-Sommerfeld bases are unitary quasi-classically. Thus we can apply
these bases to compare the Hitchin connection with the KZ connection defined by
the monodromy of the Knizhnik-Zamolodchikov equation in combinatorial theory
(see, for example, [T. Kohno, Topological invariants for 3-manifolds using
representations of mapping class group I, Topology 31 (1992), 203-230; II,
Contemp. math 175} (1994), 193-217]).Comment: 43 pages, uses: latex2e with amsmath,amsfonts,theore
Systems of Hess-Appel'rot Type and Zhukovskii Property
We start with a review of a class of systems with invariant relations, so
called {\it systems of Hess--Appel'rot type} that generalizes the classical
Hess--Appel'rot rigid body case. The systems of Hess-Appel'rot type carry an
interesting combination of both integrable and non-integrable properties.
Further, following integrable line, we study partial reductions and systems
having what we call the {\it Zhukovskii property}: these are Hamiltonian
systems with invariant relations, such that partially reduced systems are
completely integrable. We prove that the Zhukovskii property is a quite general
characteristic of systems of Hess-Appel'rote type. The partial reduction
neglects the most interesting and challenging part of the dynamics of the
systems of Hess-Appel'rot type - the non-integrable part, some analysis of
which may be seen as a reconstruction problem. We show that an integrable
system, the magnetic pendulum on the oriented Grassmannian has
natural interpretation within Zhukovskii property and it is equivalent to a
partial reduction of certain system of Hess-Appel'rot type. We perform a
classical and an algebro-geometric integration of the system, as an example of
an isoholomorphic system. The paper presents a lot of examples of systems of
Hess-Appel'rot type, giving an additional argument in favor of further study of
this class of systems.Comment: 42 page
Rational Solutions of the Painleve' VI Equation
In this paper, we classify all values of the parameters , ,
and of the Painlev\'e VI equation such that there are
rational solutions. We give a formula for them up to the birational canonical
transformations and the symmetries of the Painlev\'e VI equation.Comment: 13 pages, 1 Postscript figure Typos fixe
Temperature, composition and age of the Kara Sea Shelf sediments in the area of the Marre-Sale Geocryological Station
The paper presents results of the study of the uppermost 20 m-thick layer of the near-Yamal shelf bottom
sediments, penetrated in May 2014 by two VSEGINGEO boreholes equipped with LРС loggers, with an aim
of the temperature regime dynamics monitoring in the nearshore bottom sediments, both for the research purposes
and in as much as the data add value to the forthcoming hydrocarbon resource development on the Russian
continental shelf. On the basis of the temperature variation observations during three summer months of 2014,
it has been established that marine silty clays and aleurites composing the bottom sediment section, represent
relict frozen deposits subjected to cryogenic metamorphism in the subaerial exposure environment. Diatom
assemblages occurring in aleurite and clayey deposits consist exclusively of the marine extinct species typical of
the Early Eocene Pyxilla gracilis diatom zone. A modern marine sublittoral diatom assemblage is found inhabiting
the sands of the upper part of the onshore borehole section
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