Nonlocal Complex Ginzburg-Landau Equation for Electrochemical Systems

Garcı́a-Morales, Vladimir; Krischer, Katharina

doi:10.1103/physrevlett.100.054101

Cited by 39 publications

(19 citation statements)

References 17 publications

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“…and ω(.) are given by (1.2) these equations are the normal form close to a standard supercritical Hopf bifurcation for any model with scalar nonlocal dispersal [29,36]. Consequently, results for (1.4) are significant for any ecological model with low amplitude populations cycles and with dispersal terms that are nonlocal with the same dispersal coefficient for each population.…”

Section: Introduction Many Natural Populations Exhibit Long-term Oscmentioning

confidence: 90%

Periodic Traveling Waves in Integrodifferential Equations for Nonlocal Dispersal

Sherratt¹

2014

SIAM J. Appl. Dyn. Syst.

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Periodic traveling waves (wavetrains) have been extensively studied for reaction-diffusion equations. One important motivation for this work has been the identification of periodic traveling wave patterns in spatiotemporal data sets in ecology. However, for many ecological populations, diffusion is no more than a rough phenomenological representation of dispersal, and spatial convolution with a dispersal kernel is more realistic. This paper concerns periodic traveling wave solutions of differential equations with nonlocal dispersal terms, and with local dynamics of lambda-omega form. These kinetics include the normal form near a standard supercritical Hopf bifurcation and are therefore significant for a wide range of applications. For general dispersal kernels, an explicit family of periodic traveling wave solutions is derived, as well as the condition for waves to be stable to perturbations of arbitrarily small wavenumber. Three specific kernels are then considered in detail: Laplace, Gaussian, and top hat. For Laplace and Gaussian kernels, it is shown that stability to perturbations of arbitrarily small wavenumber implies stability, a result that also applies for reaction-diffusion equations with lambda-omega kinetics. However, for the top hat kernel it is shown that periodic traveling waves may be stable to perturbations with small wavenumber but not to those with larger wavenumber. The wave family for the top hat kernel also shows significant qualitative differences from those for the Laplace and Gaussian kernels, and for reaction-diffusion equations with the same kinetics.Motivated by the form of PTW solutions of (1.1), I look for solutions of (2.1) with the form r = R, θ = αx + βt, where R ≥ 0, α ≥ 0, and β are constants, givingNote that (2.2) is only valid for even kernels K(.). Figure 1 shows various examples of such PTW solutions, when K(.) is the Laplace kernel (defined in (3.1) below). Figure 1. Examples of PTW solutions of (1.4) for the Laplace kernel (3.1) with a = 1, plotted as a function of space x. The functions λ(.) and ω(.) are given by (1.2) with λ1 = 1; the values of λ0 and α are as shown in the figure. Note that the values of ω0 and ω1 do not affect the PTW as a function of space. The left-and right-hand columns illustrate wave families for λ(0) < 1 and λ(0) > 1, respectively. In both cases, waves for small α have large wavelength and high amplitude, approaching a spatially homogeneous oscillation as α → 0 + . When λ(0) < 1, the wave amplitude is zero at a finite value of α, which is 2 for the value of λ0 used in the left-hand column of the figure. In contrast when λ(0) > 1 there is no upper bound on the values of α giving waves, and the wavelength → 0 with the amplitude remaining nonzero as α → ∞.Since λ(.) is strictly decreasing, (2.2) has a (unique) solution for R and β if and only if C 0 ∈ [1 − λ(0), 1]. Intuitive understanding of this is helped by considering the cases α = 0 and Downloaded 11/06/14 to 137.195.26.108. Redistribution subject to SIAM license or copyright; see PERIODIC WAVES...

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Section: Introduction Many Natural Populations Exhibit Long-term Oscmentioning

confidence: 90%

Periodic Traveling Waves in Integrodifferential Equations for Nonlocal Dispersal

Sherratt¹

2014

SIAM J. Appl. Dyn. Syst.

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“…[68] Migration coupling is long-range and tends to synchronize the double-layer potential at adjacent regions, although it can also promote migration-driven electrochemical turbulence. [69,70] Finally, the galvanostatic operation mode also introduces a coupling between the individual oscillators, and this interaction is global, that is, it acts on all oscillators simultaneously with the same strength or intensity, [71,72] where the term "with the same strength" indicates that the evolution rate of the state of an oscillator is independent of its position in space. The origin of the global coupling can be grasped intuitively.…”

Section: Effect Of Pt Nanostructurementioning

confidence: 99%

Oscillatory behaviour in Galvanostatic Formaldehyde Oxidation on Nanostructured Pt/Glassy Carbon Model Electrodes

et al. 2010

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The electrocatalytic oxidation of formaldehyde, which results in CO(2) and HCOOH formation, was investigated under galvanostatic conditions on nanostructured Pt/glassy carbon (GC) electrodes fabricated by employing colloidal lithography (CL). The measurements were performed on structurally well-defined model electrodes of different Pt surface coverages under different applied currents (current densities) and at constant electrolyte transport in a thin-layer flow cell connected to a differential electrochemical mass spectrometry (DEMS) setup to monitor the dynamic response of the reaction selectivity under these conditions. Periodic oscillations of the electrode potential and the CO(2) formation rate appear not only for a continuous Pt film, but also for the nanostructured Pt/GC electrodes when a critical current density is exceeded. The critical current density for achieving regular oscillation patterns increased with decreasing Pt nanodisk density. Lower oscillation frequencies of the electrode potential and lower CO(2) formation rate for nanostructured Pt/GC electrodes compared to continuous Pt film at similar applied current densities suggest that transport processes play an essential role. Moreover, from the simple periodic response of the nanostructured electrodes it follows that all individual Pt disks in the array oscillate in synchrony. This result is discussed in terms of the different modes of spatial coupling present in the system: global coupling, migration coupling and mass transport of the essential chemical species, and the coverage of corresponding adsorbates.

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“…Such prospects are supported by early 32 and more recent 33 investigation of non-local physics in superconductors, as well as research into pattern formation due to long-range non-locality in biological systems. [34][35][36] Acknowledgements. The computations were performed on resources at Chalmers Centre for Computational Science and Engineering (C3SE) provided by the Swedish National Infrastructure for Computing (SNIC).…”

mentioning

confidence: 99%

Phase crystals

Holmvall

Fogelström

Löfwander

et al. 2020

Phys. Rev. Research

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Superconducting order owes its properties to broken symmetry of the U (1) phase. In the presence of competing orders, or in pairbreaking environments, usual superconductivity may give way to modified condensates. Here we describe one such alternative, where the low-temperature ground state is more ordered due to formation of a periodic modulation of the phase. This state is manifested by spatially-periodic superflow patterns and circulating currents which break time-reversal symmetry. We demonstrate using microscopic theory how it can be realized near edges of superconductors that host flat bands of zero-energy Andreev bound states. More generally, we trace the origin of phase crystallization to non-local properties of the gradient energy, which implies existence of similar pattern-forming instabilities in many other systems. arXiv:1906.04793v1 [cond-mat.supr-con]

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Nonlocal Complex Ginzburg-Landau Equation for Electrochemical Systems

Cited by 39 publications

References 17 publications

Periodic Traveling Waves in Integrodifferential Equations for Nonlocal Dispersal

Periodic Traveling Waves in Integrodifferential Equations for Nonlocal Dispersal

Oscillatory behaviour in Galvanostatic Formaldehyde Oxidation on Nanostructured Pt/Glassy Carbon Model Electrodes

Phase crystals

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