Wavenumber selection via spatial parameter jump

Scheel, Arnd; Weinburd, Jasper

doi:10.1098/rsta.2017.0191

Cited by 11 publications

(23 citation statements)

References 45 publications

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“…This project was highly inspired by the techniques and the perspective in the memoir [FLTW17]. Our results are corroborated by the results in [SW18], which puts the ideas we advocate for in a safe ground for comparison with other mathematical tools. Some of the questions we address below are suggestively related to well established mathematical techniques ( §8.1- §8.3), while others ( §8.4- §8.8) have a pure speculative nature; regardless of their plausibility, they should be read with caution.…”

Section: Open Problems and Further Commentssupporting

confidence: 77%

The Swift–Hohenberg Equation under Directional-Quenching: Finding Heteroclinic Connections Using Spatial and Spectral Decompositions

Monteiro¹,

Yoshinaga

2019

Arch Rational Mech Anal

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We study the existence of patterns (nontrivial, stationary solutions) in the one-dimensional Swift-Hohenberg Equation in a directional quenching scenario, that is, on x ≤ 0 the energy potential associated to the equation is bistable, whereas on x ≥ 0 it is monostable. This heterogeneity in the medium induces a symmetry break that makes the existence of heteroclinic orbits of the type point-to-periodic not only plausible but, as we prove here, true. In this search, we use an interesting result of [FLTW17] in order to understand the multiscale structure of the problem, namely, how multiple scales -fast/slow-interact with each other. In passing, we advocate for a new approach in finding connecting orbits, using what we call "far/near decompositions", relying both on information about the spatial behavior of the solutions and on Fourier analysis. Our method is functional analytic and PDE based, relying minimally on dynamical system techniques and making no use of comparison principles whatsoever.

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Section: Open Problems and Further Commentssupporting

confidence: 77%

The Swift–Hohenberg Equation under Directional-Quenching: Finding Heteroclinic Connections Using Spatial and Spectral Decompositions

Monteiro¹,

Yoshinaga

2019

Arch Rational Mech Anal

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“…Pattern forming systems with a step-like (also called "jump-type") heterogeneity have a rich history in the mathematical literature (see e.g. [29][30][31][32][33][34][35][36][37][38]) where, they have been studied in the context of front-pinning [36], pulse localization [37], and wavenumber selection [38], to name a few recent examples. These studies predominantly focussed on excitable media and the models studied are not mass-conserving.…”

Section: B Local Equilibria Theorymentioning

confidence: 99%

Pattern localization to a domain edge

et al. 2020

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The formation of protein patterns inside cells is generically described by reaction-diffusion models.The study of such systems goes back to Turing, who showed how patterns can emerge from a homogenous steady state when two reactive components have different diffusivities (e.g. membranebound and cytosolic states). However, in nature, systems typically develop in a heterogeneous environment, where upstream protein patterns affect the formation of protein patterns downstream. Examples for this are the polarization of Cdc42 adjacent to the previous bud-site in budding yeast, and the formation of an actin-recruiter ring that forms around a PIP3 domain in macropinocytosis. This suggests that previously established protein patterns can serve as a template for downstream proteins and that these downstream proteins can 'sense' the edge of the template. A mechanism for how this edge sensing may work remains elusive.Here we demonstrate and analyze a generic and robust edge-sensing mechanism, based on a twocomponent mass-conserving reaction-diffusion (McRD) model. Our analysis is rooted in a recently developed theoretical framework for McRD systems, termed local equilibria theory. We extend this framework to capture the spatially heterogeneous reaction kinetics due to the template. This enables us to graphically construct the stationary patterns in the phase space of the reaction kinetics. Furthermore, we show that the protein template can trigger a regional mass-redistribution instability near the template edge, leading to the accumulation of protein mass, which eventually results in a stationary peak at the template edge. We show that simple geometric criteria on the reactive nullcline's shape predict when this edge-sensing mechanism is operational. Thus, our results provide guidance for future studies of biological systems, and for the design of synthetic pattern forming systems. * These three authors contributed equally † frey@lmu.de 1 GTPases are hydrolase enzymes that can bind and hydrolyze guanosine triphosphate (GTP). Ras is a subfamily of small GTPases.

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“…Already linear stability of the selected stripe solution with respect to such perturbations is not evident. As demonstrated in [35], relying on [13], parallel stripes selected by slowly moving triggers are unstable against transverse perturbations due to a zigzag instability. The instability may however be propagating at a slower speed than the quenching, such that unstable patterns can be observed in a large region in the wake of the quenching line.…”

Section: Stability Modulation and Selectionmentioning

confidence: 99%

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

Goh

Scheel

2018

J. London Math. Soc.

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We study the effect of domain growth on the orientation of striped phases in a Swift–Hohenberg equation. Domain growth is encoded in a step‐like parameter dependence that allows stripe formation in a half plane, and suppresses patterns in the complement, while the boundary of the pattern‐forming region is propagating with fixed normal velocity. We construct front solutions that leave behind stripes in the pattern‐forming region that are parallel to or at a small oblique angle to the boundary. Technically, the construction of stripe formation parallel to the boundary relies on ill‐posed, infinite‐dimensional spatial dynamics. Stripes forming at a small oblique angle are constructed using a functional‐analytic, perturbative approach. Here, the main difficulties are the presence of continuous spectrum and the fact that small oblique angles appear as a singular perturbation in a traveling‐wave problem. We resolve the former difficulty using a farfield‐core decomposition and Fredholm theory in weighted spaces. The singular perturbation problem is resolved using preconditioners and boot‐strapping.

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Wavenumber selection via spatial parameter jump

Cited by 11 publications

References 45 publications

The Swift–Hohenberg Equation under Directional-Quenching: Finding Heteroclinic Connections Using Spatial and Spectral Decompositions

The Swift–Hohenberg Equation under Directional-Quenching: Finding Heteroclinic Connections Using Spatial and Spectral Decompositions

Pattern localization to a domain edge

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

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