33 research outputs found
Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg-de Vries Equation
A new class of linear second order hyperbolic partial differential operators
satisfying Huygens' principle in Minkowski spaces is presented. The
construction reveals a direct connection between Huygens' principle and the
theory of solitary wave solutions of the Korteweg-de Vries equation.Comment: 23 pages, LaTeX, to be published in Comm.Math.Phys (1997
On Bose-Einstein condensate inside moving exciton-phonon droplets
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 paraexcitons in
Cu2O. Taking into account the exciton-phonon interaction and introducing a
coherent phonon part of the moving condensate, we solve quasi-stationary
equations for the exciton-phonon condensate. These equations support localized
solutions, and we discuss the conditions for the inhomogeneous condensate to
appear in the crystal. Allowable values of the characteristic width of
ballistic condensates are estimated. The stability conditions of the moving
condensate are analyzed by use of Landau arguments, and Landau critical
parameters appear in the theory. It follows that, under certain conditions, the
condensate can move through the crystal as a stable droplet. To separate the
coherent and non-coherent parts of the exciton-phonon packet, we suggest to
turn off the phonon wind by the changes in design of the 3D crystal and
boundary conditions for the moving droplet.Comment: 13 pages, LaTeX, three eps figures are incorporated by epsf.
submitted to Phys. Letters
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
A class of exactly solvable free-boundary inhomogeneous porous medium flows
We describe a class of inhomogeneous two-dimensional porous medium flows, driven by a finite number of multipole sources; the free boundary dynamics can be parametrized by polynomial conformal maps
Bose-Einstein Condensation of Excitons: Reply to Tikhodeev's Criticism
The extended version of our reply to Comment on ``Critical Velocities in
Exciton Superfluidity'' by S. G. Tikhodeev (Phys. Rev. Lett., 84 (2000), 3502
or from http://prl.aps.org/) is presented here. The principal question is
discussed: does the moving exciton-phonon packet contain the coherent
`nucleus', or the exciton-phonon condensate?Comment: 3 pages in LaTe
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
Vortices ans Polynomials: Nonuniqueness of the Adler-Moser polynomials for the Tkachenko equation
Stationary and translating relative equilibria of point vortices in the plane
are studied. It is shown that stationary equilibria of a system containing
point vortices with arbitrary choice of circulations can be described with the
help of the Tkachenko equation. It is obtained that the Adler - Moser
polynomial are not unique polynomial solutions of the Tkachenko equation. A
generalization of the Tkachenko equation to the case of translating relative
equilibria is derived. It is shown that the generalization of the Tkachenko
equation possesses polynomial solutions with degrees that are not triangular
numbers.Comment: 15 pages, 2 figure