Functional-diﬀerential equations of parabolic type with the involution operator
In this work, mathematical models important for applications of nonlinear optics are considered in the form of nonlinear functional diﬀerential equations of parabolic type with feedback and a transformation of spatial variables (which defines the involution operator). The property of the involution operator (rotation, reflection) allows us to reduce the original equation to a system of equations without transforming the spatial variables. The set of solutions of such equations is determined by two parameters: a small one - diﬀusion coeﬃcient and a large one - coeﬃcient of flow intensity. The equation is given on a ring domain with conditions of the third kind in the class of periodic functions. Important special cases of stationary and non-stationary solutions are investigated. For a stationary solution that depends only on the angular coordinate, the nature of the stationary points and their stability are studied in detail. The variety of solutions of particular equations is also inherited in the general case. The particular solutions found are used to construct asymptotic solutions of the original equations. The work cites corresponding references to publications of the authors