Nonlinear Stability at the Zigzag Boundary

TL;DR

This paper analyzes the nonlinear stability of roll solutions at the zigzag boundary of the Swift-Hohenberg equation, revealing an algebraic decay rate of perturbations due to degeneracy effects.

Contribution

It provides a rigorous analysis of the decay rates at the zigzag boundary, highlighting the algebraic decay and the role of degeneracy in the nonlinear stability.

Findings

Perturbations decay at a rate of t^{-1/4} instead of t^{-1/2}.

Degeneracy of the quadratic term affects the decay rate.

Nonlinear terms are shown to be irrelevant in the stability analysis.

Abstract

We investigate the dynamics of roll solutions at the zigzag boundary of the planar Swift-Hohenberg equation. Linear analysis shows an algebraic decay of small perturbation with a $t^{- 1/4}$ rate, instead of the classical $t^{- 1/2}$ diffusive decay rate, 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 proof is based on a decomposition of the neutral mode and the faster decaying modes in the Bloch-Fourier space, and a fixed-point argument, demonstrating the irrelevancy of the nonlinear terms.