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

We continue study of equilibrium of two species of 2d coulomb charges (or point vortices in 2d ideal fluid) started in [20]. 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.

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.

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.

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.

We find a wide class of Levy-Loewner evolutions for which the value of integral means beta-spectrum beta(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.

We consider scattering of electromagnetic waves from a distant point source by the gravitational field of the sun, taking the field oblateness due to the quadrupole moment of the sun into account. The effects of field oblateness can play an important role in high-resolution solar gravitational lens imaging in the sub-micrometer wavelength range of the electromagnetic spectrum.

We consider a class of radial Levy–Loewner evolutions and compute exactly sets of points in their average means beta-spectrum. The methods we use originate from our previous work on harmonic measure of the whole-plane SLEκ, where β(q)-spectrum has been computed along infinite sets of curves in the parametric (q, κ)-plane.

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.

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.

We consider the unbounded version of the whole-plane Schramm–Loewner evolution. Using exact solutions of differential equations for moments of derivatives of conformal mappings, we determine its average integral means beta spectrum.

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.

We consider the whole-plane Shramm-Loewner evolution. Using exact solutions of fundamental equations for moments of derivative of conformal mappings we determine its average integral means beta-spectrum.

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.

We introduce a competitive model of Laplacian growth in both stochastic and deterministic versions. This defines two different aggregation laws with probabilities λ and 1−λ. The parameter λ varying from 0 to 1 is used to weight a ratio between the inner and outer harmonic measures that leads to a competition between the Eden-like process and the DLA solved with site-sticking conditions. We perform numerical and qualitative analysis of the competitive growth.

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.

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.

We study integrable generalizations of the Laplacian growth, describing flows in inhomogeneous porous media. The boundary is driven by a field satisfying an elliptic PDE, that is not generally reduced to a Beltrami–Laplace equation (“Non-Laplacian” growth). These turn out to be PDEs of the Calogero–Moser type, related to finite reflection groups as well as their integrable deformations.