Boundary value problem with an inclined derivative for a nonlinear equation with an involution in a rectangle

A. A. Kornuta

V. I. Vernadsky Crimean Federal University, Simferopol

Abstract: One of the important areas of research in applied nonlinear dynamics is the study of mathematical models of processes and phenomena of nonlinear optics that exhibit various modes of self-organization of the light field. It is the models of nonlinear optics by controlling the internal parameters of the system that make it possible to implement a wide range of changes in the light field in experiments. One of such systems is an optical device, which is a specially arranged external contour, which is called a “two-dimensional feedback loop” consists of various optical devices (lenses, prisms, etc.) and a thin layer of a nonlinear medium. As models describing the dynamics of nonlinear optical systems with feedback, ordinary differential equations, partial differential equations, functional differential equations with the transformation of spatial variables can be used, the type of which is determined by the device of the feedback loop, the choice of parameters, conditions at the boundary. The configuration of the area in which the corresponding model is considered is determined by the transverse aperture of the system. In the articles by A. V. Razgulin, E. P. Belan, one-dimensional problems in a circle with rotation transformation, problems on a segment with reflection transformation were considered, a two-dimensional problem in a circle with rotation transformation, as well as with a combination of rotation and radial compression, was investigated. The problem for the ring was considered in the works by A. A. Kornuta and V. A. Lukianenko [23]. Non-classical problems in partial differential equations that are close to the considered ones are investigated in the works by A. L. Skubachevsky and his students. The dynamics of solutions to the problems under consideration largely depends on the boundary conditions. As a rule, boundary value problems with Neumann conditions are investigated. However, it is known that boundary conditions with an oblique (oblique) derivative make it possible to model spiral waves that occur in nonlinear optical systems. The issues of the existence and stability of spiral waves in functional differential equation of diffusion with a delay describing the dynamics of nonlinear optical system with a feedback loop in a thin ring with boundary conditions with an oblique derivative are investigated in the works by A. V. Razgulin and S. S. Budzinsky. This paper discusses the existence and stability of solutions of a functional differential equation of parabolic type in a rectangle with a reflection transformation of a spatial variable and boundary conditions with an oblique derivative.

Keywords: functional differential equation, bifurcation, spectral problem, central manifold, boundary conditions with an oblique derivative.

UDC: 517.95

MSC: 35K10, 35K55