Stability of Structures and Asymptotics of Nonlinear Parabolic Type Equations Solutions with Transformation of Arguments

The paper considers a parabolic-type functional differential equation with rotation and radial compression transformations on a ring, a circle and a circumference, which is a mathematical model of the dynamics of an optical system consisting of a thin Kerr type medium layer and a feedback loop that controls this dynamics. Using the method of central manifolds and the Galerkin method, we study the problem of constructing spatially inhomogeneous stationary solutions and solutions of the traveling wave type, which bifurcate from a spatially homogeneous solution. The analysis of the stability of the obtained solutions is carried out.