Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes

Goh, Ryan; Scheel, Arnd

doi:10.1112/jlms.12122

Cited by 24 publications

(38 citation statements)

References 41 publications

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“…The boundary sink operator on the rectangular domain Ω bdy was discretized similarly using fourthorder centered finite differences with N s grid points in the spatial direction and a Fourier spectral method with N t grid points in the periodic temporal direction. Following the methods and terminology in [15,22], the boundary sink is computed numerically by decomposing the domain into a "far-field" region in which the asymptotic wave train is translated in time and space and a "core" region where the Neumann boundary condition has an effect on the wave shape. Note that in this case the core refers to the area near the boundary.…”

Section: Methodsmentioning

confidence: 99%

Determining the Source of Period-Doubling Instabilities in Spiral Waves

Dodson¹,

Sandstede²

2019

SIAM J. Appl. Dyn. Syst.

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Spiral wave patterns observed in models of cardiac arrhythmias and chemical oscillations develop alternans and stationary line defects, which can both be thought of as period-doubling instabilities. These instabilities are observed on bounded domains, and may be caused by the spiral core, far-field asymptotics, or boundary conditions. Here, we introduce a methodology to disentangle the impacts of each region on the instabilities by analyzing spectral properties of spiral waves and boundary sinks on bounded domains with appropriate boundary conditions. We apply our techniques to spirals formed in reaction-diffusion systems to investigate how and why alternans and line defects develop. Our results indicate that the mechanisms driving these instabilities are quite different; alternans are driven by the spiral core, whereas line defects appear from boundary effects. Moreover, we find that the shape of the alternans eigenfunction is due to the interaction of a point eigenvalue with curves of continuous spectra.

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Section: Methodsmentioning

confidence: 99%

Determining the Source of Period-Doubling Instabilities in Spiral Waves

Dodson¹,

Sandstede²

2019

SIAM J. Appl. Dyn. Syst.

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“…Comparing with the results in [], one would wish to extend the results here to situations periodic in y , or to more general equations such as the Cahn‐Hilliard equation. Results on such periodic configurations, with two‐dimensional structure have recently been obtained in [] for the Swift‐Hohenberg equation, again in a perturbative setting.…”

Section: Applications and Discussionmentioning

confidence: 99%

Contact angle selection for interfaces in growing domains

Monteiro

Scheel

2018

Z Angew Math Mech

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We study interfaces in an Allen‐Cahn equation, separating two metastable states. Our focus is on a directional quenching scenario, where a parameter renders the system bistable in a half plane and monostable in its complement, with the region of bistability expanding at a fixed speed. We show that the growth mechanism selects a contact angle between the boundary of the region of bistability and the interface separating the two metastable states. Technically, we focus on a perturbative setting in a vicinity of a symmetric situation with perpendicular contact. The main difficulty stems from the lack of Fredholm properties for the linearization in translation invariant norms. We overcome those difficulties establishing Fredholm properties in weighted spaces and farfield‐core decompositions to compensate for negative Fredholm indices.

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“…Beyond the transition from perpendicular to oblique stripes, we have analyzed detachment [13,14], k y ∼ 0 [12], and k y = 0, c x ∼ 0 [11] in prior work. From this "completist" perspective, the major challenge appears to be a description of the moduli space in a vicinity of k y , c x = 0.…”

Section: Detaching All Stripesmentioning

confidence: 99%

Growing Stripes, with and without Wrinkles

Avery¹,

Goh²,

Goodloe³

et al. 2019

SIAM J. Appl. Dyn. Syst.

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We present results on stripe formation in the Swift-Hohenberg equation with a directional quenching term. Stripes are "grown" in the wake of a moving parameter step line, and we analyze how the orientation of stripes changes depending on the speed of the quenching line and on a lateral aspect ratio. We observe stripes perpendicular to the quenching line, but also stripes created at oblique angles, as well as periodic wrinkles created in an otherwise oblique stripe pattern. Technically, we study stripe formation as traveling-wave solutions in the Swift-Hohenberg equation and in reduced Cahn-Hilliard and Newell-Whitehead-Segel models, analytically, through numerical continuation, and in direct simulations.

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Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes

Cited by 24 publications

References 41 publications

Determining the Source of Period-Doubling Instabilities in Spiral Waves

Determining the Source of Period-Doubling Instabilities in Spiral Waves

Contact angle selection for interfaces in growing domains

Growing Stripes, with and without Wrinkles

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