Том 512, № 1 (2023)

The article discusses various modifications of the nonlinear Burgers equation with small parameter and degenerate in solution of the form

\(F(u,\varepsilon ) = {{u}_{t}} - {{u}_{{xx}}} + u{{u}_{x}} + \varepsilon {{u}^{2}} - f(x,t) = 0,\)            (1)

where \(F:\Omega \to C([0,\pi ] \times [0,T])\), \(T > 0\), \(\Omega = {{C}^{2}}([0,\pi ] \times [0,T]\,)\,\mathbb{R}\) and \(u(0,t) = u(\pi ,t) = 0\), \(u(x,0) = \varphi (x)\), \(f(x,t) \in C([0,\pi ] \times [0,T])\), \(\varphi (x) \in C[0,\pi ]\). We will be interested in the most important in applications case of a small parameter ε with oscillating initial conditions of the form \(\varphi (x) = k\sin x\), where k –some, generally speaking, constant depending on ε, and study the question of the existence of a solution in neighborhood of the trivial \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), which corresponds to \(k = k{\kern 1pt} * = 0\) and at what initial Under certain conditions on the values of k, it is possible to construct an analytical approximation of this solution for small ε.

We will look for a solution in the traditional way of separation of variables on a subspace of functions of the form \(u(x,t) = v(t)u(x)\), where \(v(t) = c{{e}^{{ - t}}}\), \(u(x) \in {{\mathcal{C}}^{2}}([0,\pi ])\). In this case, the problem under consideration is degenerate at the point \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), since \({\text{Im}}F_{u}^{'}(u{\kern 1pt} *,\varepsilon {\kern 1pt} *) \ne Z = \mathcal{C}([0,\pi ] \times [0,T])\). This follows from the Sturm-Liouville theory. To achieve our goals, we apply the apparatus of p-regularity theory [6, 7, 15, 16] and show that the mapping \(F(u,\varepsilon )\) is 3-regular at the point \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), т.е. p = 3.

A generalized method for finding optimal control of the amplitude of one-dimensional oscillations in the vicinity of the equilibrium position for a scleronomous multidimensional mechanical system with friction is proposed. The oscillatory degree of freedom of the system does not lend itself to direct control. Its movement is influenced by other, directly controlled degrees of freedom. They are being chosen as the control functions. The number of directly controlled coordinates can include both positional and cyclic coordinates. The method does not use conjugate variables in the sense of the Pontryagin’s maximum principle and does not increase the dimension of the original system of differential equations of motion. The effectiveness of the proposed method is demonstrated by the example of a specific oscillatory mechanical model with dry and viscous friction.

We investigate the spatial restricted circular three-body problem in the nonresonant case. Namely, we apply Gaussian averaging to obtain averaged equations in terms of osculating elements and then investigate them. Keplerian ellipse with a focus in the main body (the Sun) is taken as an unperturbed orbit assuming the semi-major axis of the ellipse to be less than the radius of the orbit of the outer planet (internal problem). Using the Parseval formula we have derived the twice-averaged perturbed force function of the problem in the form of an explicit analytical series with coefficients expressed in terms of the Gauss and Clausen hypergeometric functions. An investigation of averaged force function along it’s curves of non-analyticity showed that the series are asymptotic by Poincaré. For a reduced system with one degree of freedom, phase portraits of oscillations in the plane of the Keplerian elements \(e\), \(\omega \) are constructed in the second and fourth approximations. It is shown the topology of the phase portrait is more complicated in fourth approximation then in first and second approximations provided that the constant \({{c}_{1}}\) of the Lidov–Kozai integral belongs to the interval (0, 0.382).

A higher integrability of the gradient of a solution to the Zaremba problem in a bounded Lipschitz plane domain is proved for the inhomogeneous p-Laplace equation.

The paper considers the problem of choosing the initial approximation when using gradient optimization methods for solving the inverse problem of restoring the distribution of velocities in a heterogeneous continuous medium. A system of acoustic equations is used to describe the behavior of the medium, and a finite-difference scheme is used to solve the direct problem. L-BFGS-B is used as a gradient optimization method. Adjoint state method is used to calculate the gradient of the error functional with respect to the medium parameters. The initial approximation for the gradient method is obtained using a convolutional neural network. The network is trained to predict the distribution of velocities in the medium from the wave response from it. The paper shows that a neural network trained on responses from simple layered structures can be successfully used to solve the inverse problem for a complex Marmousi model.

Within the framework of the development of the theory of hidden oscillations, the problem of determining the boundary of global stability and revealing its hidden parts corresponding to the non-local birth of hidden oscillations is considered. For a phase-locked loop with a proportional-integrating filter and a piecewise-linear phase detector characteristic, effective methods for determination of bifurcations of the global stability loss, for obtaining analytical formulas of the bifurcation values, and for constructing trivial and hidden parts of the global stability boundary are suggested.

In this paper we show that the one–dimensional finite–gap Schrödinger operator can be obtained by passing to the limit from a second–order difference operator that commutes with some odd–order difference operator; the coefficients of these difference operators are functions defined on the line and depend on a small parameter. Moreover, the spectral curve of the difference operators does not depend on the small parameter and coincides with the spectral curve of the Schrödinger operator.

A method has been developed for the numerical solution of a nonlinear equation describing the diffusion transfer of radiation energy. The method is based on the introduction of the second time derivative with a small parameter into the parabolic equation and an explicit difference scheme. Explicit approximation of the initial equation makes it possible to implement on its basis an algorithm that is effectively adapted to the architecture of high-performance computing systems. The new scheme provides, in comparison with the original scheme, a larger time integration step and a sufficiently high resolution quality of the solution structure, providing the second order of accuracy. A heuristic algorithm for choosing the parameters of a three-layer difference scheme is proposed. A promising field of application of the method can be problems of plasma physics and astrophysics.