A. V. Razgulin

Rotating waves in parabolic functional differential equations with rotation of spatial argument and time delay

AbstractThe parabolic functional differential equation

Finite-dimensional dynamics of distributed optical systems with delayed feedback

\frac{{\partial u}} {{\partial t}} = D\frac{{\partial ^2 u}} {{\partial x^2 }} - u + K(1 + \gamma \cos u(x + \theta ,t - T))

The attractor of the delayed functional-differential diffusion equation

is considered on the circle [0, 2π]. Here, D > 0, T > 0, K > 0, and γ ∈ (0, 1). Such equations arise in the modeling of nonlinear optical systems with a time delay T > 0 and a spatial argument rotated by an angle θ ∈ [0, 2π) in the nonlocal feedback loop in the approximation of a thin circular layer. The goal of this study is to describe spatially inhomogeneous rotating-wave solutions bifurcating from a homogeneous stationary solution in the case of a Andronov-Hopf bifurcation. The existence of such waves is proved by passing to a moving coordinate system, which makes it possible to reduce the problem to the construction of a nontrivial solution to a periodic boundary value problem for a stationary delay differential equation. The existence of rotating waves in an annulus resulting from a Andronov-Hopf bifurcation is proved, and the leading coefficients in the expansion of the solution in powers of a small parameter are obtained. The conditions for the stability of waves are derived by constructing a normal form for the Andronov-Hopf bifurcation for the functional differential equation under study.

Weighted estimate for the convergence rate of a projection difference scheme for a parabolic equation and its application to the approximation of the initial-data control problem

Abstract A mathematical approach for evaluation the complexity of spatio-temporal dynamics for delayed feedback optical systems is suggested. It is based on the concept of global attractor for discrete semigroups. Both the retarded differential-difference and retarded diffusion models of feedback optical systems are similarly investigated. Upper estimates for the number of determining modes and Hausdorff dimension of attractors are obtained.

Bifurcational light structures in nonlinear optical system with nonlocal interactions

We consider a class of functional-differential diffusion equations with delay and spatial rotation that arise in nonlinear optics. The existence of a compact global attractor is established for the corresponding discrete dynamic systems.

Modeling of distortion suppression in a nonlinear optical system with a delayed feedback loop

AbstractA new technique is proposed for analyzing the convergence of a projection difference scheme as applied to the initial value problem for a linear parabolic operator-differential equation. The technique is based on discrete analogues of weighted estimates reflecting the smoothing property of solutions to the differential problem for t > 0. Under certain conditions on the right-hand side, a new convergence rate estimate of order O(

Difference methods in problems of the optimal control of the steady selfaction of light beams

\sqrt \tau

On the matrix Fourier filtering problem for a class of models of nonlinear optical systems with a feedback

+ h) is obtained in a weighted energy norm without making any a priori assumptions on the additional smoothness of weak solutions. The technique leads to a natural projection difference approximation of the problem of controlling nonsmooth initial data. The convergence rate estimate obtained for the approximating control problems is of the same order O(