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arXiv · 2012.04154

Nonlinear Stability at the Zigzag Boundary

Mason Haberle·Abhijit Chowdhary·Qiliang Wu
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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.

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Mason Haberle, Abhijit Chowdhary, Qiliang Wu. 2020-12-08. Nonlinear Stability at the Zigzag Boundary. https://arxiv.org/abs/2012.04154

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