Finger dynamics in pattern forming systems

Macroscopic systems subjected to external forcing exhibit complex spatiotemporal behaviors as result of dissipative self-organization. We consider a paradigmatic Swift-Hohenberg equation in both variational and nonvariational forms in two-dimensions. We investigate in both equations the occurrence of a curvature instability generating spot multiplication, self-replication or fingering instability that affects the circular shape of two-dimensional localized structures . We show that when increasing the radius of localized structures, the first angular index m to become unstable is \(m=2\). This mode corresponds to an elliptical deformation of the circular shape of localized structures. We show also that for a fixed value of the radius of localized structures, the mode \(m=2\) becomes unstable for small values of the diffusion coefficient and higher modes become unstable for large angular index m. These result hold for both variational and nonvariational models. In addition, we analyze the stability of a single stripe localized structure and extended pattern for both variational and nonvariational Swift-Hohenberg equations.