Front propagation in weakly subcritical pattern-forming systems

Ponedel, Benjamin C.; Kao, Hsuan-Yun; Knobloch, Edgar

doi:10.1103/physreve.96.032208

Cited by 6 publications

(3 citation statements)

References 40 publications

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“…Rescaling (4.3) yields the Ginzburg-Landau equation analysed by Kao & Knobloch [26] and Ponedel et al [50]:…”

Section: Parallel Stripe Frontsmentioning

confidence: 99%

“…We note that in the cubic-quintic case a = 0. The depinning fronts we are interested in correspond to nonlinear travelling fronts of the form [50]…”

Section: Parallel Stripe Frontsmentioning

confidence: 99%

“…This is the main aim of the paper. Furthermore, the weakly nonlinear analysis (see for instance Ponedel et al [50]) suggests there is pattern selection mechanism for both invading and retreating fronts where they select a unique propagation speed c and far-field wavenumber k x but the semi-analytical perturbation analysis does not say anything about a selection mechanism. Hence, we wish to also clarify when such a pattern selection mechanism occurs or not.…”

Section: Introductionmentioning

confidence: 99%

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Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation

Lloyd¹

2019

Preprint

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It is well-known that stationary localised patterns involving a periodic stripe core can undergo a process that is known as 'homoclinic snaking' where patterns are added to the stripe core as a bifurcation parameter is varied. The parameter region where homoclinic snaking takes place usually occupies a small region in the bistability region between the stripes and quiescent state. Outside the homoclinic snaking region, the localised patterns invade or retreat where stripes are either added or removed from the core forming depinning fronts. It remains an open problem to carry out a numerical bifurcation analysis of depinning fronts. In this paper, we carry out a numerical bifurcation analysis of depinning of fronts near the homoclinic snaking region, involving a spatial stripe cellular pattern embedded in a quiescent state, in the two-dimensional Swift-Hohenberg equation with either a quadratic-cubic or cubic-quintic nonlinearity. We focus on depinning fronts involving stripes that are orientated either parallel, oblique and perpendicular to the front interface, and almost planar depinning fronts. We show that invading parallel depinning fronts select both a far-field wavenumber and a propagation wavespeed whereas retreating parallel depinning fronts come in families where the wavespeed is a function of the far-field wavenumber. Employing a far-field core decomposition, we propose a boundary value problem for the invading depinning fronts which we numerically solve and use path-following routines to trace out bifurcation diagrams. We then carry out a thorough numerical investigation of the parallel, oblique, perpendicular stripe, and almost planar invasion fronts. We find that almost planar invasion fronts in the cubic-quintic Swift-Hohenberg equation bifurcate off parallel invasion fronts and co-exist close to the homoclinic snaking region. Sufficiently far from the 1D homoclinic snaking region, no almost planar invasion fronts exist and we find that parallel invasion stripe fronts may regain transverse stability if they propagate above a critical speed. Finally, we show that depinning fronts shed light on the time simulations of fully localised patches of stripes on the plane. The numerical algorithms detailed have wider application to general modulated fronts and reaction-diffusion systems.

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“…Rescaling (4.3) yields the Ginzburg-Landau equation analysed by Kao & Knobloch [26] and Ponedel et al [50]:…”

Section: Parallel Stripe Frontsmentioning

confidence: 99%

“…We note that in the cubic-quintic case a = 0. The depinning fronts we are interested in correspond to nonlinear travelling fronts of the form [50]…”

Section: Parallel Stripe Frontsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation

Lloyd¹

2019

Preprint

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On the speed of propagation in Turing patterns for reaction–diffusion systems

Klika,

Gaffney,

Maini

2024

Physica D: Nonlinear Phenomena

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Defects of Bénard cell on a propagating front

Duan

Kang

2020

Physics of Fluids

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Bénard-Marangoni convection can be used to self-organize hexagonal convective cells, but defects easily emerge in the hexagonal pattern, which hinders its application in industry. The dynamics of front propagation and defect generation are studied in this paper. We focus especially on the onset process of a local disturbance of a hexagonal pattern, named the “nucleus.” The front propagation of the nucleus has been researched through numerical simulations of a model equation and experiments. In the numerical simulations, a single nucleus can evolve into a perfect hexagon pattern under critical or subcritical conditions, and a random disturbance can generate multiple nuclei which evolve into grain boundaries. In addition, under supercritical conditions, defects also emerge as a single nucleus grows. The instability of front propagation is considered to be the mechanism for the generation of irregular patterns. The curvature effect makes the protrusion of the front have a larger velocity in supercriticality, which results in a wavy front, and defects are generated in the concave portion of the front. Also, because of the curvature effect, the front of an irregular pattern has a larger velocity than that of the regular pattern since the protrusion of the front in the irregular pattern increases the average velocity. Experiments have also been carried out by using an infrared camera to analyze front propagation. The results are qualitatively in agreement with the results of numerical simulations. Through the study of defect generation in front propagation, we put forward a method for generating a hexagon pattern which greatly reduces the number of defects.

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Front propagation in weakly subcritical pattern-forming systems

Cited by 6 publications

References 40 publications

Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation

Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation

On the speed of propagation in Turing patterns for reaction–diffusion systems

Defects of Bénard cell on a propagating front

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