Theoretical and Mathematical Physics

We study the properties of the Gelfand-Shilov spaces \(S_{a_k}^{b_n}\) in the context of deformation quantization. Our main result is a characterization of their corresponding multiplier algebras with respect to a twisted convolution, which is given in terms of the inclusion relation between these algebras and the duals of the spaces of pointwise multipliers with an explicit description of these functional spaces. The proof of the inclusion theorem essentially uses the equality \(S_{a_k}^{b_n}=S^{b_n}\cap{S_{a_k}}\).

We determine the velocities and lengths of solitary envelope waves whose velocity is located in a left half-neighborhood of the phase velocity minimum in the dispersion relation for shallow basins under ice cover. The ice cover is modeled as an elastic Kirchhoff-Love ice plate. The Euler equation for the liquid layer (water) includes an additional pressure from the plate, which Boats freely on the liquid surface. We consider the case of weakly nonlinear waves in the limit of long wavelengths and small amplitudes where the initial dimensionless stress in the ice cover does not exceed one third. These waves are described by a fifth-order Kawahara equation. We then compare the obtained results with the parameters found using a strongly nonlinear description. The comparison yields very good results for shallow depths of the considered basin. This phenomenon is explained by the properties of the lowest nonlinearity coefficient in the equations describing the solitary envelope waves branching from the phase velocity minimum on the dispersion curve. We discuss possible applications of the obtained results to experimental wave measurements under an ice cover.

We construct a solution expressed in terms of Schur Q-functions of a strongly coupled B-type Kadomtsev-Petviashvili hierarchy. As a generalization of these functions, we introduce universal characters satisfying the bilinear equations of a new infinite-dimensional integrable system called the strongly coupled B-type universal character hierarchy.

We obtain solutions of the discrete nonlinear Schrödinger equation with an impurity center in two ways. In the first of them, we construct the wave function as a series in a certain parameter. In the second, approximate method, we obtain the wave function in the continuum limit. We compare the obtained solutions with numerical results, and the relative accuracy of the solution in the form of a series does not exceed 10⁻¹⁵in order of magnitude.

We study the gravitational frequency shift of light signals from the center of a spherically symmetric nonstatic matter distribution. In the case of oscillating scalar dark matter with a logarithmic potential, we obtain explicit formulas for the ratio of the emitted and received frequencies.