Localized states in the generalized Swift-Hohenberg equation.

The Swift-Hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states. The presence of these states is related to a phenomenon called homoclinic snaking. Numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states. The evolution of the localized states once they lose stability is illustrated using direct simulations in time.