Periodic Differential Operators

Brown, B. M.; Eastham, M. S. P.; Schmidt, Karl Michael

doi:10.1007/978-3-0348-0528-5

Cited by 54 publications

(46 citation statements)

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“…where k is the wavenumber of the perturbation and a j are the derivatives relevant for determining stability against homogeneous perturbations (see (3)). The linear dispersion relation is given by dðk; k; p; cÞ :…”

Section: Striped Pattern Formationmentioning

confidence: 99%

Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes

Siero¹,

Doelman²,

Eppinga³

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

124

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For water-limited arid ecosystems, where water distribution and infiltration play a vital role, various models have been set up to explain vegetation patterning. On sloped terrains, vegetation aligned in bands has been observed ubiquitously. In this paper, we consider the appearance, stability, and bifurcations of 2D striped or banded patterns in an arid ecosystem model. We numerically show that the resilience of the vegetation bands is larger on steeper slopes by computing the stability regions (Busse balloons) of striped patterns with respect to 1D and transverse 2D perturbations. This is corroborated by numerical simulations with a slowly decreasing water input parameter. Here, long wavelength striped patterns are unstable against transverse perturbations, which we also rigorously prove on flat ground through an Evans function approach. In addition, we prove a "Squire theorem" for a class of two-component reaction-advection-diffusion systems that includes our model, showing that the onset of pattern formation in 2D is due to 1D instabilities in the direction of advection, which naturally leads to striped patterns. V C 2015 AIP Publishing LLC. This paper has been motivated by studies in one space dimension of a scaled phenomenological model for vegetation on possibly sloped planes in arid ecosystems. 53,60 One-dimensional patterns ideally represent striped patterns in two space dimensions by trivially extending them into a transversal direction. Such patterns are referred to as banded vegetation and have received considerable attention after reports of widespread observations. 8,58 Understanding the appearance and disappearance of vegetation bands may ultimately help prevent land degradation. The restriction to one space dimension may overestimate stability: patterns that are stable against 1D perturbations are not necessarily stable against all 2D perturbations. Natural questions to pose are:• Which of the 1D stable patterns extend to 2D stable striped patterns? • In case of destabilization by 2D perturbations, which mechanisms are responsible?In this paper, we answer these questions for the arid ecosystem model and determine the impact of slope induced advection of water. The influence of advection on striped pattern formation is studied in a more general setting. This approach provides a clear argumentation that is unobscured by model-specific details. Equally important, the results will be applicable to a wide range of models. Applicability to the arid ecosystem model is carefully checked though, assuring that the abstract requirements can in fact be met in practice.

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Section: Striped Pattern Formationmentioning

confidence: 99%

Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes

Siero¹,

Doelman²,

Eppinga³

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

124

View full text Add to dashboard Cite

show abstract

“…We prove that if all instability intervals are empty, then the matrix potential is diagonal with the two diagonal entries equal. The analogous results for Hill's equation can be found in , pages 94–111] and in which it is assumed that all finite instability intervals vanish. It should be noted that in the Hill's equation case, there is always an unbounded instability interval while for the first‐order systems considered in this paper all instability intervals are finite.…”

Section: Introductionmentioning

confidence: 64%

“…Here, the analytic structure of the discriminant played a central role. An up‐to‐date survey of this area as well as of periodic one‐dimensional Dirac problems can be found in , pages 1–29]. Classical results on the Hill's equation can be found in and on the one‐dimensional Dirac equation in .…”

Section: Introductionmentioning

confidence: 99%

First‐order systems in on with periodic matrix potentials and vanishing instability intervals

Currie

Roth

Watson

2014

Math Methods in App Sciences

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Canonical systems in R 2 with absolutely continuous real symmetric π-periodic potentials matrices are considered. A through analysis of the discriminant is given along with the indexing and interlacing of the eigenvalues of the periodic, anti-periodic and Dirichlettype boundary value problems on [0, π]. The periodic and anti-periodic eigenvalues are characterized in terms of Dirichlet type eigenvalues. It is shown that all instability intervals vanish if and only if the potential is the product of an absolutely continuous real valued function with the identity matrix.

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“…Self-adjoint systems have been studied extensively in the last century, see [3]- [6]. Periodic problems for self-adjoint systems with integrable potentials have received consistent attention, [38]. This is especially true recently for the Ambarzumyan and Borg uniqueness-type results, [7]- [12], [13] and [14].…”

Section: Introductionmentioning

confidence: 99%

Borg's Periodicity Theorems for First-Order Self-Adjoint Systems with Complex Potentials

Currie¹,

Roth²,

Watson³

2016

Proceedings of the Edinburgh Mathematical Society

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A self-adjoint first order system with Hermitian π-periodic potential Q(z), integrable on compact sets, is considered. It is shown that all zeros of ∆ + 2e −i π 0 ℑqdt are double zeros if and only if this self-adjoint system is unitarily equivalent to one in which Q(z) is π 2 -periodic. Furthermore, the zeros of ∆−2e −i π 0 ℑqdt are all double zeros if and only if the associated self-adjoint system is unitarily equivalent to one in which Q(z) = σ 2 Q(z)σ 2 . Here ∆ denotes the discriminant of the system and σ 0 , σ 2 are Pauli matrices. Finally, it is shown that all instability intervals vanish if and only if Q = rσ 0 + qσ 2 , for some real valued π-periodic functions r and q integrable on compact sets. * Keywords: Dirac system, inverse problems, spectral theory. MSC(2010): 34A55, 34L40, 34B05.

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Periodic Differential Operators

Cited by 54 publications

References 0 publications

Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes

Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes

First‐order systems in on with periodic matrix potentials and vanishing instability intervals

Borg's Periodicity Theorems for First-Order Self-Adjoint Systems with Complex Potentials

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