This study presents the description of the parameterization of sound speed distribution in the Sea of Japan in the presence of a synoptic eddy. An analytical representation of the background sound speed profile (SSP) on its periphery is proposed. The perturbation of sound speed directly associated with the presence of an eddy is investigated. The proposed parameterization of the background SSP leads to a Sturm–Liouville problem for normal mode computation, which is equivalent to the eigenvalue problem for the Schrödinger equation with the Morse potential. This equivalence leads to simple analytical formulae for normal modes and their respective horizontal wavenumbers. It is shown that in the presence of an eddy causing moderate variations in sound speed, the standard perturbation theory for acoustic modes can be applied to describe the variability in horizontal wavenumbers across the area in which the eddy is localized. The proposed parameterization can be applied to the sound propagation modeling in the Sea of Japan.

This paper selects a set of reference points in the form of an arithmetic progression for planning an experiment to evaluate the parameters of systems of differential equations. This choice makes it possible to construct estimates of the parameters of a system of first-order differential equations based on the reversibility of the observation matrix, as well as estimates of the parameters of a system of second-order differential equations describing vibrations in a mechanical system by switching to a system of first-order differential equations. In turn, the reversibility of the observation matrix used in parameter estimation is established using the Vandermonde formula. A volumetric computational experiment has been carried out showing how, with an increase in the number of observations in the vicinity of reference points and with a decrease in the step of arithmetic progression, the accuracy of estimates of the parameters of the analyzed system increases. Among the estimated parameters, the most important are the oscillation frequencies of a conservative mechanical system, which establish its proximity to resonance, and therefore, determine the stability and reliability of the system.

The eigenvalue problem for a dynamic system describing in Schrödinger coordinates the oscillations of a homogeneous harmonic chain with dissipation at the boundaries is considered. The combinatorial Lagrange formula is used to obtain a uniform approximation of the eigenvalues for a sufficiently large number of particles in the chain.

In this paper, the problems of estimating the parameters of partial differential equations from numerous observations in the vicinity of some reference points are considered. The paper is devoted to estimating the diffusion coefficient in the diffusion equation and the parameters of one-soliton solutions of nonlinear partial differential equations. When estimating the diffusion coefficient, it was necessary to construct an estimate of the second derivative based on inaccurate observations of the solution of the diffusion equation. This procedure required consideration of two reference points when determining the first and second partial derivatives of the solution of the diffusion equation. To analyse one-soliton solutions of partial differential equations, a series of techniques have been developed that allow one to estimate the parameters of the solution itself, but not its equation. These techniques are used to estimate the parameters of one-soliton solutions of the equations kdv, mkdv, Sine–Gordon, Burgers and nonlinear Schrodinger. All the considered estimates were tested during computational experiments.

We consider a spectral problem for a dynamical system describing, in the Schrödinger variables, the motion in a finite homogeneous chain of coupled harmonic oscillators with boundary conditions that admit dissipation and energy pumping in the system. The solution of this problem is given for arbitrary parameter values in the boundary conditions and for any sufficiently large number of oscillators in the chain.