Searcharxiv⌕ Search

arXiv subjects

Oksana Yermolayeva

Publications and source records attributed to Oksana Yermolayeva.

10 recordsLinked to original sources
arXiv2025

Equilibrium of Charges and Differential Equations Solved by Polynomials II

We continue study of equilibrium of two species of 2d coulomb charges (or point vortices in 2d ideal fluid) started in (Igor Loutsenko, J. Phys. A: Math. Gen. 37, 1309, 2004). Although for two species of vortices with circulation ratio -1 the relationship between the equilibria and the factorization/Darboux transformation of the Schrodinger operator was established a long ago, the question about similar relationship for the ratio -2 remained unanswered. Here we present the answer: One has to consider Darboux-type transformations of third order differential operators rather than second order Schrodinger operators. Furthermore, we show that such transformations can also generate equilibrium configurations where an additional charge of a third specie is present. Relationship with integrable hierarchies is briefly discussed.

math-ph↗
arXiv2025

Vortices and Factorization

We review applications of factorization methods to the problem of finding stationary point vortex patterns in two-dimensional fluid mechanics. Then we present a new class of patterns related to periodic analogs of Schrodinger operators from the ``even" bi-spectral family. We also show that patterns related to soliton solutions of the KdV hierarchy constitute complete solution of the problem for certain classes of vortex systems. Keywords: Point vortices in ideal fluid, Factorization of second- and third-order differential operators, KdV and Sawada-Kotera hierarchies, Bispectral problem, Locus configurations

math-ph↗
arXiv2021

Quasi Band-Limited Coronagraph for Extended Sources

We propose a class of graded coronagraphic "amplitude" image masks for a high throughput Lyot-type coronagraph that transmits light from an annular region around an extended source and suppresses light, with extremely high ratio, from elsewhere. The interior radius of the region is comparable with its exterior radius. The masks are designed using an idea inspired by approach due M.J. Kuchner and W.A. Traub ("band-limited" masks) and approach to optimal apodization by D.Slepian. One potential application of our masks is direct high-resolution imaging of exo-planets with the help of the Solar Gravitational Lens, where apparent radius of the "Einstein ring" image of a planet is of the order of an arc-second and is comparable with the apparent radius of the sun and solar corona. Keywords: Coronagraphy, Optimal Band-Limiting, Exo-Planets, Solar Gravitational Lens

astro-ph.IM↗
arXiv2019

On Integrability and Exact Solvability in Deterministic and Stochastic Laplacian Growth

We review applications of theory of classical and quantum integrable systems to the free-boundary problems of fluid mechanics as well as to corresponding problems of statistical mechanics. We also review important exact results obtained in the theory of multi-fractal spectra of the stochastic models related to the Laplacian growth: Schramm-Loewner and Levy-Loewner evolutions.

math-ph↗
arXiv2019

Stochastic Loewner Evolutions, Fuchsian Systems and Orthogonal Polynomials

We find a wide class of Levy-Loewner evolutions for which the value of integral means beta-spectrum $β(q)$ at $q=2$ is the maximal real eigenvalue of a three-diagonal matrix. The second moments of derivatives of corresponding conformal mappings are expressed through solutions of matrix Fuchsian systems with three singular points.

math-ph↗
arXiv2014

On Harmonic Measure of the Whole Plane Levy-Loewner Evolution

Levy-Loewner evolution (LLE) is a generalization of the Schramm-Loewner evolution (SLE) where the branching is possible in a course of growth process. We consider a class of radial Levy-Loewner evolutions for which sets of points of the average means beta-spectrum can be found exactly. In this paper we show how to overcome difficulties arised in previous works on multi-fractal analysis of SLE/LLE.

math-ph↗
arXiv2013

New Applications of Quantum Algebraically Integrable Systems in Fluid Dynamics

The rational quantum algebraically integrable systems are non-trivial generalizations of Laplacian operators to the case of elliptic operators with variable coefficients. We study corresponding extensions of Laplacian growth connected with algebraically integrable systems, describing viscous free-boundary flows in non-homogenous media. We introduce a class of planar flows related with application of Adler-Moser polynomials and construct solutions for higher-dimensional cases, where the conformal mapping technique is unavailable.

nlin.SI↗
arXiv2008

Discrete Laplacian Growth: Linear Stability vs Fractal Formation

We introduce stochastic Discrete Laplacian Growth and consider its deterministic continuous version. These are reminiscent respectively to well-known Diffusion Limited Aggregation and Hele-Shaw free boundary problem for the interface propagation. We study correlation between stability of deterministic free-boundary problem and macroscopic fractal growth in the corresponding discrete problem. It turns out that fractal growth in the discrete problem is not influenced by stability of its deterministic version. Using this fact one can easily provide a qualitative analytic description of the Discrete Laplacian Growth.

cond-mat.stat-mech↗
arXiv2006

Non-Laplacian growth, algebraic domains and finite reflection groups

Dynamics of planar domains with moving boundaries driven by the gradient of a scalar field that satisfies an elliptic PDE is studied. We consider the question: For which kind of PDEs the domains are algebraic, provided the field has singularities at a fixed point inside the domain? The construction reveals a direct connection with the theory of the Calogero-Moser systems related to finite reflection groups and their integrable deformations.

math-ph↗