Том 191, № 1 (2017)

We study initial-boundary value problems for a model differential equation in a bounded region with a quadratic nonlinearity of a special type typical for the theory of conductors. Using the test function method, we show that such a nonlinearity can lead to global unsolvability with respect to time, which from the physical standpoint means an electrical breakdown of the conductor in a finite time. For the simplest test functions, we obtain sufficient conditions for the unsolvability of the model problems and estimates of the blowup rate and time. With concrete examples, we demonstrate the possibility of using the method for one-, two- and three-dimensional problems with classical and nonclassical boundary conditions. We separately consider the Neumann and Navier problems in bounded R^N regions (N ≥ 2).

We find an analytic solution of the Schrödinger equation for an electron in a two-mode quantized electromagnetic field. The obtained solution allows calculating the spectra of photoelectrons and atom ionization rates in strong electromagnetic fields.

We consider the scattering problem for a system of three nonrelativistic particles in the case of energies below the threshold of the system breakup into three free particles. We assume that the interaction potentials can be represented as a sum of two terms, one of which is a small perturbation. We develop a perturbation theory scheme for solving the scattering problem based on the three-particle Faddeev equations.

We consider an ensemble of particles not interacting with each other and randomly walking in the d-dimensional Euclidean space ℝ^d. The individual moves of each particle are governed by the same distribution, but after the completion of each such move of a particle, its position in the medium is “marked” as a region in the form of a ball of diameter r₀, which is not available for subsequent visits by this particle. As a result, we obtain the corresponding ensemble in ℝ^d of marked trajectories in each of which the distance between the centers of any pair of these balls is greater than r₀. We describe a method for computing the asymptotic form of the probability density W_n(r) of the distance r between the centers of the initial and final balls of a trajectory consisting of n individual moves of a particle of the ensemble. The number n, the trajectory modulus, is a random variable in this model in addition to the distance r. This makes it necessary to determine the distribution of n, for which we use the canonical distribution obtained from the most probable distribution of particles in the ensemble over the moduli of their trajectories. Averaging the density W_n(r) over the canonical distribution of the modulus n allows finding the asymptotic behavior of the probability density of the distance r between the ends of the paths of the canonical ensemble of particles in a self-avoiding random walk in ℝ^d for 2 ≤ d < 4.

We study the linear stability of a three-layer flow of immiscible liquids located in a periodic normal electric field. We consider certain porous media assumed to be uniform, homogeneous, and isotropic. We analytically and numerically simulate the system of linear evolution equations of such a medium. The linearized problem leads to a system of two Mathieu equations with complex coefficients of the damping terms. We study the effects of the streaming velocity, permeability of the porous medium, and the electrical properties of the flow of a thin layer (film) of liquid on the flow instability. We consider several special cases of such systems. As a special case, we consider a uniform electric field and solve the transition curve equations up to the second order in a small dimensionless parameter. We show that the dielectric constant ratio and also the electric field play a destabilizing role in the stability criteria, while the porosity has a dual effect on the wave motion. In the case of an alternating electric field and a periodic velocity, we use the method of multiple time scales to calculate approximate solutions and analyze the stability criteria in the nonresonance and resonance cases; we also obtain transition curves in these cases. We show that an increase in the velocity and the electric field promote oscillations and hence have a destabilizing effect.

Conformal Killing vectors (CKVs) in static plane symmetric space–times were recently studied by Saifullah and Yazdan, who concluded by remarking that static plane symmetric space–times do not admit any proper CKV except in the case where these space–times are conformally flat. We present some non-conformally flat static plane symmetric space–time metrics admitting proper CKVs. For these space–times, we also investigate a special type of CKVs, known as inheriting CKVs.