Scalarization of stationary semiclassical problems for systems of equations and its application in plasma physics

We propose a method for determining asymptotic solutions of stationary problems for pencils of differential (and pseudodifferential) operators whose symbol is a self-adjoint matrix. We show that in the case of constant multiplicity, the problem of constructing asymptotic solutions corresponding to a distinguished eigenvalue (called an effective Hamiltonian, term, or mode) reduces to studying objects related only to the determinant of the principal matrix symbol and the eigenvector corresponding to a given (numerical) value of this effective Hamiltonian. As an example, we show that stationary solutions can be effectively calculated in the problem of plasma motion in a tokamak.     

Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations

The result of this paper is that any fast-decaying function can be represented as an integral over the canonical Maslov operator, on a special Lagrangian manifold, acting on a specific function. This representation enables one to construct effective explicit formulas for asymptotic solutions of a vast class of linear hyperbolic systems with variable coefficients. In memoriam L. R. Volevich (1934–2007)     

Asymptotic solutions of the Cauchy problem with localized initial conditions for linearized two-dimensional Boussinesq-type equations with variable coefficients

In the paper, asymptotic solutions of the Cauchy problem with localized initial data for the two-dimensional wave equation (with variable speed) which is also perturbed by (spatially) variable weakly dispersive components are constructed. We consider both the case of normal dispersion occurring in the linearized Boussinesq equation for water waves over smoothly changing bottom and the case of anomalous dispersion arising when studying the wave equation with rapidly oscillating velocity. With regard to the fact that the front of the solution has focal points and self-intersection points, we present formulas based on the modified Maslov canonical operator in the case of initial perturbations of a rather general form which decrease at infinity. For perturbations of special form, we express the asymptotic behavior of a solution in the vicinity of the front, using derivatives of the sum of squares of the Airy functions Ai and Bi.     

Calculation of Integrals of the Hugoniot–Maslov Chain for Singular Vortical Solutions of the Shallow-Water Equation

We discuss the problems of the Hugoniot–Maslov chain integrability for singular vortical solutions of the shallow-water equations on the plane. We show that the complex variables used to derive the chain automatically give most of the integrals of the complete and the truncated chains. We also study how some of these integrals are related to the Lagrangian invariant (potential vorticity). We discuss how to choose solutions of the chain that can be used to describe the actual trajectories of tropical cyclones.     

The local central limit theorem for a Gibbs random field

We extend the validity of the implication of a local limit theorem from an integral one. Our extension eliminates the finite range assumption present in the previous works by using the cluster expansion to analyze the contribution from the tail of the potential.     

Maslov complex germ and high-frequency Gaussian beams for cold plasma in a toroidal domain

Asymptotic vector solutions describing, in the linear approximation, the passage of high-frequency Gaussian beams through an electroneutral plasma occupying a toroidal domain T (modeling a tokamak chamber) are constructed in a fairly effective form by using the Maslov complex germ theory. The particle density and the magnetic field in T are assumed to be given. Based on Radon transforms, the reconstruction of the particle density and the magnetic field from measurements of the characteristics of Gaussian beams after their passage through T is discussed.