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

The parabolic functional differential equation
$$ \frac{\partial u}{\partial t}=D\frac{\partial^2u}{\partial x^2}u+K\left(1+\gamma\cos u(x+\theta,t-T)\right) $$
is considered on the circle $[0,2\pi]$. Here, $D>0$, $T>0$, $K>0$, and $\gamma\in(0,1)$. Such equations arise in the modeling of nonlinear optical systems with a time delay $T>0$ and a spatial argument rotated by an angle $\theta\in[0,2\pi)$ 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.