Spatial Wavenumber Selection in Recurrent Precipitation

Goh, Ryan; Mesuro, Samantha; Scheel, Arnd

doi:10.1137/100793086

Cited by 24 publications

(49 citation statements)

References 34 publications

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“…It is worth noting that our perturbation analysis is insensitive to these instabilities, as the main non‐degeneracy assumption only relies on coperiodic stability of the periodic patterns. Other instabilities are non‐resonant with the mode of propagation and therefore do not change the Fredholm index; see [, § 4] for the notion of non‐resonance in this context.…”

Section: Applications and Discussionmentioning

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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Section: Applications and Discussionmentioning

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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“…Many examples of this triggered pattern formation arise in systems with mass-conserving properties. Typical model equations for such systems are the Cahn-Hilliard equation [15,29,30,41], the Keller-Segel model for chemotaxis [1,33], reaction-diffusion systems [32], or phase-field systems [14,18,19]. Other examples arise in ion-bombardment studies [17] and are modeled by Kuramoto-Shivashinsky-type models, while still others arise when studying general osciliatory instabilities using real and complex Ginzburg-Landau models [4,6,7].…”

Section: Introductionmentioning

confidence: 99%

Pattern formation in the wake of triggered pushed fronts

Goh

Scheel²

2016

Nonlinearity

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Pattern-forming fronts are often controlled by an external stimulus which progresses through a stable medium at a fixed speed, rendering it unstable in its wake. By controlling the speed of excitation, such stimuli, or "triggers," can mediate pattern forming fronts which freely invade an unstable equilibrium and control which pattern is selected. In this work, we analytically and numerically study when the trigger perturbs an oscillatory pushed free front. In such a situation, the resulting patterned front, which we call a pushed trigger front, exhibits a variety of interesting phenomenon, including snaking, non-monotonic wavenumber selection, and hysteresis. Assuming the existence of a generic oscillatory pushed free front, we use heteroclinic bifurcation techniques to prove the existence of trigger fronts in an abstract setting motivated by the spatial dynamics approach. We then derive a leading order expansion for the selected wavenumber in terms of the trigger speed. Furthermore, we show that such a bifurcation curve is governed by the difference of certain strong-stable and weakly-stable spatial eigenvalues associated with the decay of the free pushed front. We also study prototypical examples of these phenomena in the cubic-quintic complex Ginzburg Landau equation and a modified Cahn-Hilliard equation.Running head: Triggered pushed fronts

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“…Invasion fronts do however exhibit predictable wavenumber selection mechanisms [7,18,32], yielding phenomena similar to Figure 1.3. It is worth noticing that linear stability information is qualitatively contained in information on the geometry of the curves of equilibria, in analogy to Remark 2.1 on the sign of group velocities, here; see [7].…”

Section: Discussionmentioning

confidence: 68%

Advection and Autocatalysis as Organizing Principles for Banded Vegetation Patterns

et al. 2018

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We motivate and analyze a simple model for the formation of banded vegetation patterns. The model incorporates a minimal number of ingredients for vegetation growth in semi-arid landscapes. It allows for comprehensive analysis and sheds new light onto phenomena such as the migration of vegetation bands and the interplay between their upper and lower edges. The key ingredient is the formulation as a closed reaction-diffusion system, thus introducing a conservation law that both allows for analysis and provides ready intuition and understanding through analogies with characteristic speeds of propagation and shock waves.

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Spatial Wavenumber Selection in Recurrent Precipitation

Cited by 24 publications

References 34 publications

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

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

Pattern formation in the wake of triggered pushed fronts

Advection and Autocatalysis as Organizing Principles for Banded Vegetation Patterns

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