20 research outputs found
SLE_k: correlation functions in the coefficient problem
We apply the method of correlation functions to the coefficient problem in
stochastic geometry. In particular, we give a proof for some universal patterns
conjectured by M. Zinsmeister for the second moments of the Taylor coefficients
for special values of kappa in the whole-plane Schramm-Loewner evolution
(SLE_kappa). We propose to use multi-point correlation functions for the study
of higher moments in coefficient problem. Generalizations related to the
Levy-type processes are also considered. The exact multifractal spectrum of
considered version of the whole-plane SLE_kappa is discussed
Solitons and Normal Random Matrices
We discuss a general relation between the solitons and statistical mechanics
and show that the partition function of the normal random matrix model can be
obtained from the multi-soliton solutions of the two-dimensional Toda lattice
hierarchy in a special limit
Constrained Reductions of 2D dispersionless Toda Hierarchy, Hamiltonian Structure and Interface Dynamics
Finite-dimensional reductions of the 2D dispersionless Toda hierarchy,
constrained by the ``string equation'' are studied. These include solutions
determined by polynomial, rational or logarithmic functions, which are of
interest in relation to the ``Laplacian growth'' problem governing interface
dynamics. The consistency of such reductions is proved, and the Hamiltonian
structure of the reduced dynamics is derived. The Poisson structure of the
rationally reduced dispersionless Toda hierarchies is also derivedComment: 18 pages LaTex, accepted to J.Math.Phys, Significantly updated
version of the previous submissio
On Critical Velocities in Exciton Superfluidity
The presence of exciton phonon interactions is shown to play a key role in
the exciton superfluidity. We apply the Landau criterion for an exciton-phonon
condensate moving uniformly at zero temperature. It turns out that there are
essentially two critical velocities in the theory. Within the range of these
velocities the condensate can exist only as a bright soliton. The excitation
spectrum and differential equations for the wave function of this condensate
are derived.Comment: 7 pages, Latex; to be published in Phys.Rev.Lett (1997
Superfluidity of bosons on a deformable lattice
We study the superfluid properties of a system of interacting bosons on a
lattice which, moreover, are coupled to the vibrational modes of this lattice,
treated here in terms of Einstein phonon model. The ground state corresponds to
two correlated condensates: that of the bosons and that of the phonons. Two
competing effects determine the common collective soundwave-like mode with
sound velocity , arising from gauge symmetry breaking: i) The sound velocity
(corresponding to a weakly interacting Bose system on a rigid lattice) in
the lowest order approximation is reduced due to reduction of the repulsive
boson-boson interaction, arising from the attractive part of phonon mediated
interaction in the static limit. ii) the second order correction to the sound
velocity is enhanced as compared to the one of bosons on a rigid lattice when
the the boson-phonon interaction is switched on due to the retarded nature of
phonon mediated interaction. The overall effect is that the sound velocity is
practically unaffected by the coupling with phonons, indicating the robustness
of the superfluid state. The induction of a coherent state in the phonon
system, driven by the condensation of the bosons could be of experimental
significance, permitting spectroscopic detections of superfluid properties of
the bosons. Our results are based on an extension of the Beliaev - Popov
formalism for a weakly interacting Bose gas on a rigid lattice to that on a
deformable lattice with which it interacts.Comment: 12 pages, 14 figures, to appear in Phys. Rev.
Multiple sums and integrals as neutral BKP tau functions
We consider multiple sums and multi-integrals as tau functions of the BKP
hierarchy using neutral fermions as the simplest tool for deriving these. The
sums are over projective Schur functions for strict partitions
. We consider two types of such sums: weighted sums of over
strict partitions and sums over products . In this
way we obtain discrete analogues of the beta-ensembles ().
Continuous versions are represented as multiple integrals. Such sums and
integrals are of interest in a number of problems in mathematics and physics.Comment: 16 page
Deformed shape invariance and exactly solvable Hamiltonians with position-dependent effective mass
Known shape-invariant potentials for the constant-mass Schrodinger equation
are taken as effective potentials in a position-dependent effective mass (PDEM)
one. The corresponding shape-invariance condition turns out to be deformed. Its
solvability imposes the form of both the deformed superpotential and the PDEM.
A lot of new exactly solvable potentials associated with a PDEM background are
generated in this way. A novel and important condition restricting the
existence of bound states whenever the PDEM vanishes at an end point of the
interval is identified. In some cases, the bound-state spectrum results from a
smooth deformation of that of the conventional shape-invariant potential used
in the construction. In others, one observes a generation or suppression of
bound states, depending on the mass-parameter values. The corresponding
wavefunctions are given in terms of some deformed classical orthogonal
polynomials.Comment: 26 pages, no figure, reduced secs. 4 and 5, final version to appear
in JP
Bosons in a Lattice: Exciton-Phonon Condensate in Cu2O
We explore a nonlinear field model to describe the interplay between the
ability of excitons to be Bose-condensed and their interaction with other modes
of a crystal. We apply our consideration to the long-living para-excitons in
Cu2O. Taking into account the exciton-phonon interaction and introducing a
coherent phonon part of the moving condensate, we derive the dynamic equations
for the exciton-phonon condensate. These equations can support localized
solutions, and we discuss the conditions for the moving inhomogeneous
condensate to appear in the crystal. We calculate the condensate wave function
and energy, and a collective excitation spectrum in the semiclassical
approximation; the inside-excitations were found to follow the asymptotic
behavior of the macroscopic wave function exactly. The stability conditions of
the moving condensate are analyzed by use of Landau arguments, and Landau
critical parameters appear in the theory. Finally, we apply our model to
describe the recently observed interference and strong nonlinear interaction
between two coherent exciton-phonon packets in Cu2O.Comment: 34 pages, LaTeX, four figures (.ps) are incorporated by epsf.
Submitted to Phys. Rev.
Point vortices and classical orthogonal polynomials
Stationary equilibria of point vortices with arbitrary choice of circulations
in a background flow are studied. Differential equations satisfied by
generating polynomials of vortex configurations are derived. It is shown that
these equations can be reduced to a single one. It is found that polynomials
that are Wronskians of classical orthogonal polynomials solve the latter
equation. As a consequence vortex equilibria at a certain choice of background
flows can be described with the help of Wronskians of classical orthogonal
polynomials.Comment: 20 pages, 12 figure
Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations
Rational solutions and special polynomials associated with the generalized
K_2 hierarchy are studied. This hierarchy is related to the Sawada-Kotera and
Kaup-Kupershmidt equations and some other integrable partial differential
equations including the Fordy-Gibbons equation. Differential-difference
relations and differential equations satisfied by the polynomials are derived.
The relationship between these special polynomials and stationary
configurations of point vortices with circulations Gamma and -2Gamma is
established. Properties of the polynomials are studied. Differential-difference
relations enabling one to construct these polynomials explicitly are derived.
Algebraic relations satisfied by the roots of the polynomials are found.Comment: 23 pages, 8 figure