Nonlinear waves in a parabolic equation with a spatial argument rescaling operator and with time delay

Bifurcations of nonlinear waves (spatially inhomogeneous solutions) from homogeneous equilibrium states of an initial boundary value problem in a circle for a nonlinear parabolic equation with a spatial argument stretching operator and a time delay arising in nonlinear optics are studied. In the plane of the basic parameters of the equation, the regions of stability (instability) of homogeneous equilibrium states are constructed, the dynamics of stability regions depending on the magnitude of the delay is studied. The mechanisms of loss of stability by homogeneous equilibrium states, possible bifurcations of spatially inhomogeneous self-oscillatory solutions and their stability are investigated. The possibility of bifurcation of stable rotational and spiral waves is shown.