Theoretical and Mathematical Physics最新文献

We study bifurcations of nonlinear waves (spatially inhomogeneous solutions) emerging from homogeneous equilibrium states of an initial boundary value problem, arising in nonlinear optics, for a nonlinear parabolic equation on a disk with a spatial argument rescaling operator and with time delay. In the plane of the main parameters of the equation, we construct stability (instability) domains of homogeneous equilibrium states and study the dynamics of the stability domains depending on the rescaling coefficient. We investigate the mechanisms of stability loss by homogeneous equilibrium states, the possible bifurcations of spatially inhomogeneous self-oscillatory solutions, and their stability. We demonstrate the possibility of bifurcation of stable rotational and spiral waves.

We consider singularly perturbed operator differential transport equations of a special form in the case where the transport operator acts on space–time variables; a linear operator acting on an additional variable describes the interaction that “scrambles” the solution with respect to that variable. We construct a formal asymptotic expansion of the solution of the Cauchy problem for a singularly perturbed operator differential transport equation with small nonlinearity and weak diffusion in the case of several spatial variables. Under some conditions assumed for these problems, the leading term of the asymptotics is described by a quasilinear parabolic equation. The remainder term is estimated with respect to the residual under certain conditions.

We describe the geometric and probabilistic properties of a noncommutative (2)-torus in a magnetic field. We study the volume invariance, integrated scalar curvature, and the volume form by using the operator method of perturbation by an inner derivation of the magnetic Laplacian operator on the noncommutative (2)-torus. We then analyze the magnetic stochastic process describing the motion of a particle subject to a uniform magnetic field on the noncommutative (2)-torus, and discuss the related main properties.

The main result of this paper is a mathematical model that describes the dynamics of the motion of several cars in areas with different speed limits. As such areas, we can consider speed limit zones and speed bumps or uneven road surfaces. The model is a system of differential equations with a delayed argument. The dynamical properties of the model are studied by numerical methods. A computer program has been developed that uses the model to describe the motion of traffic flows in various road situations. The simulation results coincide with the observation data of real traffic flows.

We continue studies of the nonlocal erosion equation that is used as a mathematical model of the formation of a spatially inhomogeneous relief on semiconductor surfaces. We show that such a relief can form as a result of local bifurcations in the case where the stability of the spatially homogeneous equilibrium state changes. We consider a periodic boundary-value problem and study its codimension-(2) bifurcations. For solutions describing an inhomogeneous relief, we obtain asymptotic formulas and study their stability. The analysis of the mathematical problem is based on modern methods of the theory of dynamical systems with an infinite-dimensional phase space, in particular, on the method of integral manifolds and on the theory of normal forms.

We present the results of a study of nonlinearity in inverse problems of the orbital dynamics of Jupiter’s outer satellites, discovered in 2018–2022, and of potentially hazardous asteroids. The results show that for a more accurate study of orbital uncertainty, we must first find the minimum value of a nonlinearity indicator by varying the initial epoch within the measurable interval for different parametric spaces.

We consider a system of autonomous nonlinear ordinary differential equations for which the existence conditions for an invariant manifold are satisfied in the case where this manifold is central. It is well known that the theorem on the existence of a central invariant manifold cannot be supplemented with the statement of its uniqueness. We obtain sufficient conditions that guarantee the uniqueness of the central invariant manifold.

The existence of stationary solutions of singularly perturbed systems of reaction–diffusion–advection equations is studied in the case of fast and slow reaction–diffusion–advection equations with nonlinearities containing the gradient of the squared sought function (KPZ nonlinearities). The asymptotic method of differential inequalities is used to prove the existence theorems. The boundary layer asymptotics of solutions are constructed in the case of Neumann and Dirichlet boundary conditions. The case of quasimonotone sources and systems without the quasimonotonicity requirement is also considered.

We consider a periodic problem for a singularly perturbed parabolic reaction–diffusion–advection equation of the Burgers type with the modulus advection; it has a solution in the form of a moving front. We formulate conditions for the existence of such a solution and construct its asymptotic approximation. We pose a control problem where the required front propagation law is implemented by a specially chosen boundary condition. We construct an asymptotic solution of the boundary control problem. Using the asymptotic method of differential inequalities, we estimate the accuracy of the solution of the control problem. We propose an original numerical algorithm for solving singularly perturbed problems involving the modulus advection.

We obtain a contrast-structure type solution of a system of equations for the baretting effect that include a nonlinear singularly perturbed parabolic equation and an additional nonlocal integral relation. We prove the existence of the solution with an internal transition layer and construct the asymptotic approximation of this solution. We obtain estimates of the main physical model parameters, which coincide with experimental data and the estimates obtained previously by other methods.