  Roll solutions at the zigzag boundary, typically selected by patterns and
defects in numerical simulations, are shown to be nonlinearly stable. This
result also serves as an example that linear decay weaker than the classical
diffusive decay, together with quadratic nonlinearity, still gives nonlinear
stability of spatially periodic patterns. Linear analysis reveals that, instead
of the classical $t^{-1}$ diffusive decay rate, small perturbations of roll
solutions at the zigzag boundary decay with a $t^{-3/4}$ rate along with time,
due to the degeneracy of the quadratic term of the continuation of the
translational mode of the linearized operator in the Bloch-Fourier spaces. The
nonlinear stability proof is based on a decomposition of the neutral
translational mode and the faster decaying modes in the Bloch-Fourier space,
and a fixed-point argument, demonstrating the irrelevancy of the nonlinear
terms.
