251 research outputs found
Meromorphic Solutions to a Differential--Difference Equation Describing Certain Self-Similar Potentials
In this paper we prove the existence of meromorphic solutions to a nonlinear
differential difference equation that describe certain self-similar potentials
for the Schroedinger operator.Comment: 10 pages, LaTeX, uses additional package
Chaotic hysteresis in an adiabatically oscillating double well
We consider the motion of a damped particle in a potential oscillating slowly
between a simple and a double well. The system displays hysteresis effects
which can be of periodic or chaotic type. We explain this behaviour by
computing an analytic expression of a Poincar'e map.Comment: 4 pages RevTeX, 3 PS figs, uses psfig.sty. Submitted to Phys. Rev.
Letters. PS file also available at
http://dpwww.epfl.ch/instituts/ipt/berglund.htm
Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism
A method is presented for solving elastodynamic problems in radially
inhomogeneous elastic materials with spherical anisotropy, i.e.\ materials such
that in a spherical coordinate system
. The time harmonic displacement field is expanded in a separation of variables form with dependence on
described by vector spherical harmonics with -dependent
amplitudes. It is proved that such separation of variables solution is
generally possible only if the spherical anisotropy is restricted to transverse
isotropy with the principal axis in the radial direction, in which case the
amplitudes are determined by a first-order ordinary differential system.
Restricted forms of the displacement field, such as ,
admit this type of separation of variables solutions for certain lower material
symmetries. These results extend the Stroh formalism of elastodynamics in
rectangular and cylindrical systems to spherical coordinates.Comment: 15 page
Quantum geometrodynamics for black holes and wormholes
The geometrodynamics of the spherical gravity with a selfgravitating thin
dust shell as a source is constructed. The shell Hamiltonian constraint is
derived and the corresponding Schroedinger equation is obtained. This equation
appeared to be a finite differences equation. Its solutions are required to be
analytic functions on the relevant Riemannian surface. The method of finding
discrete spectra is suggested based on the analytic properties of the
solutions. The large black hole approximation is considered and the discrete
spectra for bound states of quantum black holes and wormholes are found. They
depend on two quantum numbers and are, in fact, quasicontinuous.Comment: Latex, 32 pages, 5 fig
Riemann-Hilbert problem for Hurwitz Frobenius manifolds: regular singularities
In this paper we study the Fuchsian Riemann-Hilbert (inverse monodromy)
problem corresponding to Frobenius structures on Hurwitz spaces. We find a
solution to this Riemann-Hilbert problem in terms of integrals of certain
meromorphic differentials over a basis of an appropriate relative homology
space, study the corresponding monodromy group and compute the monodromy
matrices explicitly for various special cases.Comment: final versio
Memory Effects and Scaling Laws in Slowly Driven Systems
This article deals with dynamical systems depending on a slowly varying
parameter. We present several physical examples illustrating memory effects,
such as metastability and hysteresis, which frequently appear in these systems.
A mathematical theory is outlined, which allows to show existence of hysteresis
cycles, and determine related scaling laws.Comment: 28 pages (AMS-LaTeX), 18 PS figure
An algorithm to obtain global solutions of the double confluent Heun equation
A procedure is proposed to construct solutions of the double confluent Heun
equation with a determinate behaviour at the singular points. The connection
factors are expressed as quotients of Wronskians of the involved solutions.
Asymptotic expansions are used in the computation of those Wronskians. The
feasibility of the method is shown in an example, namely, the Schroedinger
equation with a quasi-exactly-solvable potential
Autoresonance in a Dissipative System
We study the autoresonant solution of Duffing's equation in the presence of
dissipation. This solution is proved to be an attracting set. We evaluate the
maximal amplitude of the autoresonant solution and the time of transition from
autoresonant growth of the amplitude to the mode of fast oscillations.
Analytical results are illustrated by numerical simulations.Comment: 22 pages, 3 figure
Stability conditions and Stokes factors
Let A be the category of modules over a complex, finite-dimensional algebra.
We show that the space of stability conditions on A parametrises an
isomonodromic family of irregular connections on P^1 with values in the Hall
algebra of A. The residues of these connections are given by the holomorphic
generating function for counting invariants in A constructed by D. Joyce.Comment: Very minor changes. Final version. To appear in Inventione
Existence and Uniqueness of Tri-tronqu\'ee Solutions of the second Painlev\'e hierarchy
The first five classical Painlev\'e equations are known to have solutions
described by divergent asymptotic power series near infinity. Here we prove
that such solutions also exist for the infinite hierarchy of equations
associated with the second Painlev\'e equation. Moreover we prove that these
are unique in certain sectors near infinity.Comment: 13 pages, Late
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