Modeling the multifractal dynamics of COVID-19 pandemic.

To describe the COVID-19 pandemic, we propose to use a mathematical model of multifractal dynamics, which is alternative to other models and free of their shortcomings. It is based on the fractal properties of pandemics only and allows describing their time behavior using no hypotheses and assumptions about the structure of the disease process. The model is applied to describe the dynamics of the COVID-19 pandemic from day 1 to day 699 from the beginning of the pandemic. The calculated parameters of the model accurately determine the parameters of the trend and the large jump in daily diseases in this time interval. Within the framework of this model and finite-difference parametric nonlinear equations of the reduced SIR (Susceptible-Infected-Removed) model, the fractal dimensions of various segments of daily incidence in the world and variations in the main reproduction number of COVID-19 were calculated based on the data of COVID-19 world statistics.

Two-dimensional oscillator in time-dependent fields: comparison of some exact and approximate calculations.

Operator-difference multilayer schemes for solving the time-dependent Schrödinger equation up to sixth order of accuracy in the time step are presented. Reduced schemes for solving a set of coupled time-dependent Schrödinger equations with respect to the hyper-radial variable are devised using expansion of a wave packet over the set of appropriate basis angular functions. Further discretization of the resulting problem is realized by means of the finite-element method. The convergence of the expansion with respect to the number of basis functions and the efficiency of the numerical schemes are demonstrated in the exactly solvable model of an electric-field-driven two-dimensional oscillator (or a charged particle in a constant uniform magnetic field), in which we explicitly observed an effect of the periodical focusing and defocusing of the probability density flux.

Dynamics of a misaligned astigmatic twisted Gaussian beam in a Kerr-nonlinear parabolic waveguide.

Using the Galerkin criterion in the basis of flexible generalized Gaussian modes, the equations of motion are derived for the parameters of a misaligned astigmatic twisted Gaussian beam in an axially symmetric nonlinear medium. Nontrivial features of the beam dynamics (e.g., phase locking, cycle generation, nonlinear symmetry change) in a parabolic waveguide with Kerr nonlinearity are revealed.