Spinning solid flames, so named because reacting solid compounds produce solid reaction products, and spinning detonation waves may be phenomenologically described by the sixth-order evolution-type PDE (Strunin, IMA J. Appl. Math., 63, 1999, 163). The equation is formulated in terms of the location of combustion front defined as the surface separating cold fresh mixture from hot reaction products. The equation contains nonlinear energy source term, nonlinear term facilitating energy flux towards smaller scales, and linear dissipation term expressed by sixth-order derivative. The model yields stable kink-type solutions travelling along spiral trajectories on cylindrical sample. Only large initial distortions of plane front will develop into nontrivial regimes, namely the spinning waves; small distortions will decay. Qualitatively, such dynamics well addresses the real phenomena. In terms of derivative of the front profile with respect to transverse coordinate the kinks correspond to pulses, or auto-solitons. The auto-soliton manifests itself as fundamental solution of the equation. We demonstrate numerically that arbitrary initial condition eventually breaks up into a set of identical pulses. Stress that the travelling waves (pulses) here are produced by purely dissipative mechanism. We present and analyse new results of numerical simulations of the model, focusing on bifurcations of regimes with various wave numbers. Of particular interest is the question whether chaotic regimes occur in large domains. We give some preliminary evidences in favour of positive answer to the question. We also discuss mathematical links of the equation to the generalized nonlinear phase diffusion equation of Kuramoto.