Russian Journal of Mathematical Physics最新文献

Investigations of the local dynamics of singularly perturbed equations with two delays are carried out for the casein which both the delays are asymptotically large and, at the same time, of the same order (proportional). Critical cases were identified, and it was shown that all of them have infinite dimension. To study the behavior of solutions in cases close to critical, special nonlinear equations (quasinormal forms) were constructed whose solutions provide asymptotic approximations of the solutions to the original problem.

The paper considers a system of two reaction-diffusion equations with diffusion coefficients differing by several orders of magnitude and discontinuous reactive terms. The existence, local uniqueness and Lyapunov asymptotic stability of a solution with an internal transition layer are established. The analysis is carried out using asymptotic method of differential inequalities. The obtained conditions for the existence of a stable solution determine the limits of applicability of model problems based on such systems of equations, in particular, for studying chemical transformations in the pore spaces of geological formations.

This paper investigates stationary boundary-layer solutions of singularly perturbed systems of Tikhonov-type equations arising in drift-diffusion models of sub-Debye semiconductors. Boundary-layer asymptotics for solutions are constructed for Neumann and Dirichlet boundary conditions. The asymptotic method of differential inequalities is used to prove the existence and asymptotic stability theorems for these solutions. A physical interpretation of the results thus obtained is provided.

The present paper provides a survey of the achievements of the research group led by Professor N. N. Nefedov in the field of asymptotic analysis of moving fronts for nonlinear singularly perturbed reaction–diffusion–advection equations. The primary research tool is the asymptotic method of differential inequalities, which allows one to carry out a rigorous justification of the proposed asymptotic approximations. The paper systematizes results for parabolic initial-boundary value problems and illustrates key approaches to their analysis.

This paper examines the global time solvability of the Cauchy problem for a nonlinear system of elliptic and parabolic equations. A large-time asymptotic solution to this Cauchy problem is obtained.

We consider a second order semilinear elliptic equation on a star graph with finitely many edges, some of which are supposed to have small lengths proportional to a small parameter (varepsilon.) At the common vertex, a general boundary condition is imposed, while at the boundary vertices, we impose the Dirichlet or Robin or Neumann condition. The nonlinearity in the equation depends only on the unknown function and variable and is supposed to satisfy a Lipschitz condition as well as a certain monotonicity condition. In addition, the nonlinearity on the small edges is multiplied by (varepsilon^{-1}.) The main results of the paper establish the unique solvability of the considered problem and describe the convergence and asymptotic expansions for the solution as (varepsilon) goes to zero. The limiting problem is considered on the graph without small edges and involves a certain limiting boundary condition at the common vertex. This limiting boundary condition depends in a nontrivial nonlinear way on the nonlinearities in the original equation on the small edges. The convergence is established uniformly in (L_2)-norm of the right-hand side in the equation. We also construct the asymptotic expansion for the solution of perturbed problem up to an arbitrary power of (varepsilon,) and estimate the remainders in the asymptotics uniformly in (L_2)-norm of the right-hand side in the equation.

For a hyperbolic semilinear partial differential equation, the problem of achieving a travelling wave is studied depending on the initial conditions. Numerical experiments show that the travelling wave structure depends on the rate of stabilization of the initial function at the leading edge.

Results on solarity of Deutsch boundedly compact sets are obtained. As application, we show that the set of extended exponential sums in (C[a, b]) is a sun.

In a neighborhood of a cuspidal caustic, the Maslov canonical operator can be represented as a linear combination of the composite Pearcey function and its derivatives with coefficients that are smooth functions independent of the small parameter of the asymptotics. This representation plays an important role when constructing asymptotic solutions of various equations describing wave propagation (such as the wave equation, Maxwell equations, and linearized shallow water equations). The paper provides a method for writing computationally efficient formulas for these coefficients, thus extending the preceding research in this direction.

In this paper, using the example of a problem for the hydrogen atom, an approach to constructing the asymptotics of the eigenfunctions of the Schrödinger operator corresponding to Lagrangian manifolds noncompact in momenta is discussed. The advantage of this approach is that it also enables one to describe functions corresponding to lower energy levels.