Nonlinear instability of rolls in the 2-dimensional generalized Swift-Hohenberg equation

2025/11/30 by Chae, Myeongju, Jung, Soyeun
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.00764

Abstract

Within the framework developed in \citeGr, JLL, RT1, we rigorously establish the nonlinear instability of roll solutions to the two-dimensional generalized Swift-Hohenberg equation (gSHE). Our analysis is based on spectral information near the maximally unstable Bloch mode, combined with precise semigroup estimates. We construct a certain class of small initial perturbations that grow in time and cause the solution to deviate from the underlying roll solution within a finite time. This result provides a clear transition from spectral to nonlinear instability in a genuinely two-dimensional setting, where the Bloch parameter σ ranges over an unbounded domain.

Citations

• Long-wave instability of periodic shear flows for the 2D Navier-Stokes equations
• Asymptotic behaviour of solutions of linearized Navier Stokes equations in the long waves regime
• Nonlinear Instability of Periodic Traveling Waves
• Pointwise nonlinear stability of nonlocalized modulated periodic\n reaction-diffusion waves
• An overview of periodic elliptic operators
• Transverse instability of the line solitary water-waves
• Transverse nonlinear instability for two-dimensional dispersive models
• Pattern formation outside of equilibrium
• Singularity Analysis of Generating Functions

Related

• Nonlinear Stability at the Zigzag Boundary
• Localised states in an extended Swift-Hohenberg equation
• Geometric blow-up of a dynamic Turing instability in the Swift-Hohenberg equation
• Dynamic Transitions of the Swift-Hohenberg Equation with Third-Order Dispersion
• Non-linear Stability of Modulated Fronts for the Swift-Hohenberg\n Equation