Localized radial solutions of the Swift-Hohenberg equation

Stationary localized solutions of the planar Swift-Hohenberg equation are investigated in the parameter region where the trivial solution is stable. In the parameter region where rolls bifurcate subcritically, localized radial ring-like pulses are shown to bifurcate from the trivial solution. Furthermore, radial spot-like pulses are shown to bifurcate from the trivial state, regardless of the criticality of roll patterns. These theoretical results apply also to general reaction-diffusion systems near Turing instabilities. Numerical computations show that planar radial pulses 'snake' near the Maxwell point where, by definition, the one-dimensional roll patterns have the same energy as the trivial state. These computations also reveal that spots, which are stable in a certain parameter region, become unstable with respect to hexagonal perturbations, leading to fully localized hexagon patterns.