Russian Journal of Mathematical Physics最新文献

The Cauchy problem for a one-dimensional (nonlinear) shallow water equations over a variable bottom (D(x)) is considered in an extended basin bounded from two sides by shores (where the bottom degenerates, (D(a)=0)), or by a shore and a wall. The short-wave asymptotics of the linearized system in the form of a propagating localized wave is constructed. After applying to the constructed functions a simple parametric or explicit change of variables proposed in recent papers (Dobrokhotov, Minenkov, Nazaikinsky, 2022 and Dobrokhotov, Kalinichenko, Minenkov, Nazaikinsky, 2023), we obtain the asymptotics of the original nonlinear problem. On the constructed families of functions, the ratio of the amplitude and the wavelength is studied for which hte wave does not collapse when running up to the shore.

DOI 10.1134/S1061920823040143

In the paper, three-dimensional Nijenhuis operators are studied that have differential singularities, i.e., such points at which the coefficients of the characteristic polynomials are dependent. The case is studied in which the differentials of all invariants of the Nijenhuis operator are proportional, as well as the case when two invariants are functionally independent and the third defines a fold-type singularity. In particular, new examples of three-dimensional Nijenhuis operators with singularities of the specified type are constructed.

DOI 10.1134/S1061920823040015

It is proved that if (G) is a connected solvable group and (pi) is a (not necessarily continuous) representation of (G) in a finite-dimensional vector space (E), then there is a basis in (E) in which the matrices of the representation operators of (pi) have upper triangular form. The assertion is extended to connected solvable locally compact groups (G) having a connected normal subgroup for which the quotient group is a Lie group.

DOI 10.1134/S1061920823040180

The aim of this paper is to study probabilistic versions of the degenerate Stirling numbers of the second kind and the degenerate Bell polynomials, namely the probabilisitc degenerate Stirling numbers of the second kind associated with (Y) and the probabilistic degenerate Bell polynomials associated with (Y), which are also degenerate versions of the probabilisitc Stirling numbers of the second and the probabilistic Bell polynomials considered earlier. Here (Y) is a random variable whose moment generating function exists in some neighborhood of the origin. We derive some properties, explicit expressions, certain identities and recurrence relations for those numbers and polynomials. In addition, we treat the special cases that (Y) is the Poisson random variable with parameter (alpha (>0)) and the Bernoulli random variable with probability of success (p).

DOI 10.1134/S106192082304009X

In this paper, equations describing a double-dimensional flow along a curved smooth plate with small periodic irregularities are derived. The parameters of the irregularities are chosen so that the flow has a double-deck structure. The equations describing the terms of the asymptotic solution are written in the original coordinate system, which required changes in the form of the usual ansatz.

DOI 10.1134/S1061920823040040

The Pauli–Jordan–Dirac anticommutator mean-square formula is presented.

DOI 10.1134/S1061920823040088

In (L_2(mathbb{R}^d;mathbb{C}^n)), we consider a matrix elliptic second order differential operator (B_varepsilon >0). Coefficients of the operator (B_varepsilon) are periodic with respect to some lattice in (mathbb{R}^d) and depend on (mathbf{x}/varepsilon). We study the quantitative homogenization for the solutions of the hyperbolic system (partial _t^2mathbf{u}_varepsilon =-B_varepsilonmathbf{u}_varepsilon). In operator terms, we are interested in approximations of the operators (cos (tB_varepsilon ^{1/2})) and (B_varepsilon ^{-1/2}sin (tB_varepsilon ^{1/2})) in suitable operator norms. Approximations for the resolvent (B_varepsilon ^{-1}) have been already obtained by T.A. Suslina. So, we rewrite hyperbolic equation as a system for the vector with components (mathbf{u}_varepsilon ) and (partial _tmathbf{u}_varepsilon), and consider the corresponding unitary group. For this group, we adapt the proof of the Trotter-Kato theorem by introduction of some correction term and derive hyperbolic results from elliptic ones.

DOI 10.1134/S106192082304012X

We study a system of two integro-differential equations that arises as the result of linearization of Boltzmann–Maxwell’s kinetic equations, where the collision integral is chosen in the Bhatnagar–Gross–Krook approximation, and the unperturbed state of the plasma is characterized by the Fermi–Dirac distribution. The unknown functions are the linear parts of the perturbations of the distribution function of the charged particles and the electric field strength in plasma. In the paper, an analytical representation for the general solution of this system is found. When deriving this representation, some new results were applied to Fourier transforms of distributions (generalized functions).

DOI 10.1134/S1061920823040039

We study an initial boundary value problem of axially symmetric one-dimensional unsteady shear in the viscoplastic space (a Bingham solid) initiated by a rectilinear vortex thread located along the symmetry axis. The force intensity of the thread is represented by a given monotone piecewise continuous function of time. The density and the dynamical viscosity of the medium are constant, and the yield point is a given piecewise continuous function of radius. We find similar and quasisimilar expressions for the tangent stress and for the rotating component of the velocity both in viscoplastic shear domains and in rigid zones. We show that the vortex thread with time-bounded force intensity may generate a viscoplastic shear only inside a cylinder of certain radius. If the thread intensity growth linearly with time, then the radius of the shear domain grows proportionally to (sqrt t).

We study the operator acting in (L_2(mathbb{R})) by the formula (( mathcal{A} psi)(x)=psi(x+omega)+psi(x-omega)+ lambda e^{-2pi i x} psi(x)), where (xinmathbb R) is a variable, and (lambda>0) and (omegain(0,1)) are parameters. It is related to the simplest quasi-periodic operator introduced by P. Sarnak in 1982. We investigate ( mathcal{A} ) using the monodromization method, the Buslaev–Fedotov renormalization approach, which arose when trying to extend the Bloch–Floquet theory to difference equations on ( mathbb{R} ). Within this approach, the analysis of ( mathcal{A} ) turns out to be very natural and transparent. We describe the geometry of the spectrum and calculate the Lyapunov exponent.