2023/10/18 by Abhijit Chowdhary, Mason Haberle, Chowdhary, Abhijit +5
Computer Science · Mathematics · #35B10 #35B35 #35B36 #35B40 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Dynamics and Pattern Formation #Pattern Formation and Solitons (nlin.PS) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2310.12365
openalex publication_date 2023/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.