A new resonance mechanism in the Swift--Hohenberg equation with time-periodic forcing

TL;DR

This paper investigates a new resonance mechanism in the Swift--Hohenberg equation with periodic forcing, revealing how the forcing interacts with pattern nucleation and annihilation, supported by asymptotic theory and numerical validation.

Contribution

It introduces a novel resonance phenomenon in the Swift--Hohenberg equation with time-periodic forcing, linking forcing period to pattern nucleation dynamics and developing an asymptotic theory for it.

Findings

Identification of a new resonance phenomenon between forcing period and pattern nucleation.

Development of an asymptotic theory predicting resonance regions in parameter space.

Qualitative and some quantitative agreement between theory and numerical simulations.

Abstract

The generalized Swift--Hohenberg equation with a quadratic-cubic nonlinearity is used to study the persistence and decay of localized patterns in the presence of time-periodic parametric forcing. A novel resonance phenomenon between the forcing period and the time required to nucleate one wavelength of the pattern outside the pinning region is identified. The resonance generates distinct regions in parameter space characterized by the net number of wavelengths gained or lost in one forcing cycle. These regions are well described by an asymptotic theory based on the wavelength nucleation/annihilation time near the boundaries of the pinning region. The resulting theory leads to predictions that are qualitatively correct and, in some cases, provide quantitative agreement with numerical simulations.