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

We study bifurcations of nonlinear waves (spatially inhomogeneous solutions) emerging from homogeneous equilibrium states of an initial boundary value problem, arising in nonlinear optics, for a nonlinear parabolic equation on a disk with a spatial argument rescaling operator and with time delay. In the plane of the main parameters of the equation, we construct stability (instability) domains of homogeneous equilibrium states and study the dynamics of the stability domains depending on the rescaling coefficient. We investigate the mechanisms of stability loss by homogeneous equilibrium states, the possible bifurcations of spatially inhomogeneous self-oscillatory solutions, and their stability. We demonstrate the possibility of bifurcation of stable rotational and spiral waves.