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Echeverría-Alar S, Clerc MG, Bordeu I. Emergence of disordered branching patterns in confined chiral nematic liquid crystals. Proc Natl Acad Sci U S A 2023; 120:e2221000120. [PMID: 37027428 PMCID: PMC10104489 DOI: 10.1073/pnas.2221000120] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2022] [Accepted: 03/06/2023] [Indexed: 04/08/2023] Open
Abstract
Spatial branching processes are ubiquitous in nature, yet the mechanisms that drive their growth may vary significantly from one system to another. In soft matter physics, chiral nematic liquid crystals provide a controlled setting to study the emergence and growth dynamic of disordered branching patterns. Via an appropriate forcing, a cholesteric phase may nucleate in a chiral nematic liquid crystal, which self-organizes into an extended branching pattern. It is known that branching events take place when the rounded tips of cholesteric fingers swell, become unstable, and split into two new cholesteric tips. The origin of this interfacial instability and the mechanisms that drive the large-scale spatial organization of these cholesteric patterns remain unclear. In this work, we investigate experimentally the spatial and temporal organization of thermally driven branching patterns in chiral nematic liquid crystal cells. We describe the observations through a mean-field model and find that chirality is responsible for the creation of fingers, regulates their interactions, and controls the tip-splitting process. Furthermore, we show that the complex dynamics of the cholesteric pattern behaves as a probabilistic process of branching and inhibition of chiral tips that drives the large-scale topological organization. Our theoretical findings are in good agreement with the experimental observations.
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Affiliation(s)
- Sebastián Echeverría-Alar
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, 837.0415Santiago, Chile
- Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, 837.0415Santiago, Chile
| | - Marcel G. Clerc
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, 837.0415Santiago, Chile
- Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, 837.0415Santiago, Chile
| | - Ignacio Bordeu
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, 837.0415Santiago, Chile
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2
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Parra-Rivas P, Fernandez-Oto C. Formation of localized states in dryland vegetation: Bifurcation structure and stability. Phys Rev E 2020; 101:052214. [PMID: 32575306 DOI: 10.1103/physreve.101.052214] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/24/2019] [Accepted: 04/27/2020] [Indexed: 11/07/2022]
Abstract
We study theoretically the emergence of localized states of vegetation close to the onset of desertification. These states are formed through the locking of vegetation fronts, connecting a uniform vegetation state with a bare soil state, which occurs nearby the Maxwell point of the system. To study these structures we consider a universal model of vegetation dynamics in drylands, which has been obtained as the normal form for different vegetation models. Close to the Maxwell point localized gaps and spots of vegetation exist and undergo collapsed snaking. The presence of gaps strongly suggest that the ecosystem may undergo a recovering process. In contrast, the presence of spots may indicate that the ecosystem is close to desertification.
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Affiliation(s)
- P Parra-Rivas
- Service OPERA-photonics, Universit libre de Bruxelles, 50 Avenue F. D. Roosevelt, CP 194/5, B-1050 Bruxelles, Belgium.,Laboratory of Dynamics in Biological Systems, Department of Cellular and Molecular Medicine, University of Leuven, B-3000 Leuven, Belgium
| | - C Fernandez-Oto
- Complex Systems Group, Facultad de Ingenieria y Ciencias Aplicadas, Universidad de los Andes, Av. Mon. Alvaro del Portillo 12455 Santiago, Chile
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3
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Castillero PM, Yochelis A. Comb-like Turing patterns embedded in Hopf oscillations: Spatially localized states outside the 2:1 frequency locked region. CHAOS (WOODBURY, N.Y.) 2017; 27:043110. [PMID: 28456181 DOI: 10.1063/1.4981394] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
A generic mechanism for the emergence of spatially localized states embedded in an oscillatory background is demonstrated by using a 2:1 frequency locking oscillatory system. The localization is of Turing type and appears in two space dimensions as a comb-like state in either π phase shifted Hopf oscillations or inside a spiral core. Specifically, the localized states appear in absence of the well known flip-flop dynamics (associated with collapsed homoclinic snaking) that is known to arise in the vicinity of Hopf-Turing bifurcation in one space dimension. Derivation and analysis of three Hopf-Turing amplitude equations in two space dimensions reveal a local dynamics pinning mechanism for Hopf fronts, which in turn allows the emergence of perpendicular (to the Hopf front) Turing states. The results are shown to agree well with the comb-like core size that forms inside spiral waves. In the context of 2:1 resonance, these localized states form outside the 2:1 resonance region and thus extend the frequency locking domain for spatially extended media, such as periodically driven Belousov-Zhabotinsky chemical reactions. Implications to chlorite-iodide-malonic-acid and shaken granular media are also addressed.
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Affiliation(s)
| | - Arik Yochelis
- Department of Solar Energy and Environmental Physics, Swiss Institute for Dryland Environmental and Energy Research, Blaustein Institutes for Desert Research (BIDR), Ben-Gurion University of the Negev, Sede Boqer Campus, 8499000 Midreshet Ben-Gurion, Israel
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4
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Odent V, Louvergneaux E, Clerc MG, Andrade-Silva I. Optical wall dynamics induced by coexistence of monostable and bistable spatial regions. Phys Rev E 2016; 94:052220. [PMID: 27967167 DOI: 10.1103/physreve.94.052220] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2016] [Indexed: 11/06/2022]
Abstract
When nonequilibrium extended homogeneous systems exhibit multistability, it leads to the presence of domain walls between the existing equilibria. Depending on the stability of the steady states, the dynamics differs. Here, we consider the interface dynamics in the case of a spatially inhomogeneous system, namely, an optical system where the control parameter is spatially Gaussian. Then interfaces connect the monostable and the bistable nonuniform states that are associated with two distinct spatial regions. The coexistence of these two regions of different stability induces relaxation dynamics and the propagation of a wall with a time-dependent speed. We emphasize analytically these two dynamical behaviors using a generic bistable model. Experimentally, an inhomogeneous Gaussian light beam traveling through either a dye-doped liquid crystal cell or a Kerr cavity depicts these behaviors, in agreement with the theoretical predictions.
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Affiliation(s)
- V Odent
- Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59000 Lille, France.,Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - E Louvergneaux
- Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59000 Lille, France
| | - M G Clerc
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - I Andrade-Silva
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
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5
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Contreras DAC, Clerc MG. Internal noise and system size effects induce nondiffusive kink dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032922. [PMID: 25871187 DOI: 10.1103/physreve.91.032922] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/12/2015] [Indexed: 06/04/2023]
Abstract
We investigate the effects of inherent fluctuations and system size in the dynamics of domain between uniform symmetric states. In the case of monotonous kinks, this dynamics is characterized by exhibiting nonsymmetric random walks, being attracted to the system borders. For nonmonotonous interface, the dynamics is replaced by a hopping dynamic. Based on bistable universal models, we characterize the origin of these unexpected dynamics through use of the stochastic kinematic laws for the interface position and the survival probability. Numerical simulations show a quite good agreement with the theoretical predictions.
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Affiliation(s)
- Diego A C Contreras
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - Marcel G Clerc
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
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6
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Gomila D, Colet P, Walgraef D. Theory for the spatiotemporal dynamics of domain walls close to a nonequilibrium Ising-Bloch transition. PHYSICAL REVIEW LETTERS 2015; 114:084101. [PMID: 25768763 DOI: 10.1103/physrevlett.114.084101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/17/2014] [Indexed: 06/04/2023]
Abstract
We derive a generic model for the interaction of domain walls close to a nonequilibrium-Bloch transition. The universal scenario predicted by the model includes stationary Ising and Bloch localized structures (dissipative solitons), as well as drifting and oscillating Bloch structures. Our theory also explains the behavior of Bloch walls during a collision. The results are confirmed by numerical simulations of the Ginzburg-Landau equation forced at twice its natural frequency and are in agreement with previous observations in several physical systems.
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Affiliation(s)
- Damià Gomila
- IFISC (CSIC-UIB), Instituto de Física Interdisciplinar y Sistemas Complejos, E-07122 Palma de Mallorca, Spain
| | - Pere Colet
- IFISC (CSIC-UIB), Instituto de Física Interdisciplinar y Sistemas Complejos, E-07122 Palma de Mallorca, Spain
| | - Daniel Walgraef
- IFISC (CSIC-UIB), Instituto de Física Interdisciplinar y Sistemas Complejos, E-07122 Palma de Mallorca, Spain
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7
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Escaff D, Fernandez-Oto C, Clerc MG, Tlidi M. Localized vegetation patterns, fairy circles, and localized patches in arid landscapes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:022924. [PMID: 25768586 DOI: 10.1103/physreve.91.022924] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/25/2014] [Indexed: 06/04/2023]
Abstract
We investigate the formation of localized structures with varying widths in one- and two-dimensional systems. The mechanism of stabilization is attributed to strongly nonlocal coupling mediated by a Lorentzian type of kernel. We show that, in addition to stable dips found recently [see, e.g. Fernandez-Oto et al., Phys. Rev. Lett. 110, 174101 (2013)], there are stable localized peaks which appear as a result of strongly nonlocal coupling. We applied this mechanism to arid ecosystems by considering a prototype model of a Nagumo type. In one dimension, we study the front connecting the stable uniformly vegetated state to the bare one under the effect of strongly nonlocal coupling. We show that strongly nonlocal coupling stabilizes both-dip and peak-localized structures. We show analytically and numerically that the width of a localized structure, which we interpret as a fairy circle, increases strongly with the aridity parameter. This prediction is in agreement with published observations. In addition, we predict that the width of localized patch decreases with the degree of aridity. Numerical results are in close agreement with analytical predictions.
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Affiliation(s)
- D Escaff
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Monseñor Alvaro del Portillo 12.455, Las Condes, Santiago, Chile
| | - C Fernandez-Oto
- Faculté des Sciences, Université Libre de Bruxelles (U.L.B.), C.P. 231, Campus Plaine, B-1050 Bruxelles, Belgium
| | - M G Clerc
- Departamento de Física, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile
| | - M Tlidi
- Faculté des Sciences, Université Libre de Bruxelles (U.L.B.), C.P. 231, Campus Plaine, B-1050 Bruxelles, Belgium
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8
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de Valcárcel GJ, Martínez-Quesada M, Staliunas K. Phase-bistable pattern formation in oscillatory systems via rocking: application to nonlinear optical systems. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2014; 372:rsta.2014.0008. [PMID: 25246675 DOI: 10.1098/rsta.2014.0008] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We present a review, together with new results, of a universal forcing of oscillatory systems, termed 'rocking', which leads to the emergence of a phase bistability and to the kind of pattern formation associated with it, characterized by the presence of phase domains, phase spatial solitons and phase-bistable extended patterns. The effects of rocking are thus similar to those observed in the classic 2 : 1 resonance (the parametric resonance) of spatially extended systems of oscillators, which occurs under a spatially uniform, time-periodic forcing at twice the oscillations' frequency. The rocking, however, has a frequency close to that of the oscillations (it is a 1 : 1 resonant forcing) and hence is a good alternative to the parametric forcing when the latter is inefficient (e.g. in optics). The key ingredient is that the rocking amplitude is modulated either in time or in space, such that its sign alternates (exhibits π-phase jumps). We present new results concerning a paradigmatic nonlinear optical system (the two-level laser) and show that phase domains and dark-ring (phase) solitons replace the ubiquitous vortices that characterize the emission of free-running, broad area lasers.
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Affiliation(s)
- Germán J de Valcárcel
- Departament d'Òptica, Universitat de València, Dr. Moliner 50, Burjassot 46100, Spain
| | | | - Kestutis Staliunas
- ICREA and Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Colom 11, Terrassa 08222, Spain
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9
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Tlidi M, Staliunas K, Panajotov K, Vladimirov AG, Clerc MG. Localized structures in dissipative media: from optics to plant ecology. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2014; 372:20140101. [PMID: 25246688 PMCID: PMC4186218 DOI: 10.1098/rsta.2014.0101] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
Localized structures (LSs) in dissipative media appear in various fields of natural science such as biology, chemistry, plant ecology, optics and laser physics. The proposal for this Theme Issue was to gather specialists from various fields of nonlinear science towards a cross-fertilization among active areas of research. This is a cross-disciplinary area of research dominated by nonlinear optics due to potential applications for all-optical control of light, optical storage and information processing. This Theme Issue contains contributions from 18 active groups involved in the LS field and have all made significant contributions in recent years.
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Affiliation(s)
- M Tlidi
- Départment de Physique, Université Libre de Bruxelles (ULB), CP 231, Campus Plaine, Bruxelles 1050, Belgium
| | - K Staliunas
- Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Colom 11, Terrassa 08222 (Barcelona), Spain
| | - K Panajotov
- Brussels Photonics Team, Department of Applied Physics and Photonics (B-PHOT TONA), Vrije Unversiteit Brussels, Pleinlaan 2, Brussels 1050, Belgium Institute of Solid State Physics, 72 Tzarigradsko Chaussee Boulevard, Sofia 1784, Bulgaria
| | - A G Vladimirov
- Weierstrass Institute, Mohrenstrasse 39, Berlin 10117, Germany
| | - M G Clerc
- Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile
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10
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Paulau PV, Gomila D, Colet P, Malomed BA, Firth WJ. From one- to two-dimensional solitons in the Ginzburg-Landau model of lasers with frequency-selective feedback. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:036213. [PMID: 22060481 DOI: 10.1103/physreve.84.036213] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2011] [Indexed: 05/31/2023]
Abstract
We use the cubic complex Ginzburg-Landau equation linearly coupled to a dissipative linear equation as a model for lasers with an external frequency-selective feedback. This system may also serve as a general pattern-formation model in media driven by an intrinsic gain and selective feedback. While, strictly speaking, the approximation of the laser nonlinearity by a cubic term is only valid for small field intensities, it qualitatively reproduces results for dissipative solitons obtained in models with a more complex nonlinearity in the whole parameter region where the solitons exist. The analysis is focused on two-dimensional stripe-shaped and vortex solitons. An analytical expression for the stripe solitons is obtained from the known one-dimensional soliton solution, and its relation with vortex solitons is highlighted. The radius of the vortices increases linearly with their topological charge m, therefore the stripe-shaped soliton may be interpreted as the vortex with m=∞, and, conversely, vortex solitons can be realized as unstable stripes bent into stable rings. The results for the vortices are applicable for a broad class of physical systems.
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Affiliation(s)
- P V Paulau
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, E-07071 Palma de Mallorca, Spain
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11
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Woolley TE, Baker RE, Maini PK, Aragón JL, Barrio RA. Analysis of stationary droplets in a generic Turing reaction-diffusion system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:051929. [PMID: 21230522 DOI: 10.1103/physreve.82.051929] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2010] [Revised: 10/06/2010] [Indexed: 05/30/2023]
Abstract
Solitonlike structures called "droplets" are found to exist within a paradigm reaction-diffusion model that can be used to describe patterning in a number of biological systems, for example, on the skin of various fish species. They have also been found in many other systems that can be modeled with a complex Ginzburg-Landau system. These droplets can be analyzed in the biological paradigm model because the system has two nonzero stable steady states that are symmetric; however, the asymmetric case is more challenging. We first review the properties of the paradigm system and then extend a recently developed perturbation technique [D. Gomila, J. Opt. B: Quantum Semiclassical Opt. 6, S265 (2004)] to investigate the weakly asymmetric case. We compare the results of our mathematical analysis with numerical simulations and show good agreement in the region where the assumptions hold.
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Affiliation(s)
- Thomas E Woolley
- Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford OX1 3LB, United Kingdom.
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12
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Haudin F, Elías RG, Rojas RG, Bortolozzo U, Clerc MG, Residori S. Front dynamics and pinning-depinning phenomenon in spatially periodic media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:056203. [PMID: 20866303 DOI: 10.1103/physreve.81.056203] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/05/2010] [Indexed: 05/29/2023]
Abstract
Front propagation in one- and two-dimensional spatially modulated media is studied both experimentally and theoretically. The pinning-depinning phenomenon, long ago predicted by Pomeau [Physica D 23, 3 (1986)], is obtained and verified experimentally in a nematic liquid-crystal cell under various configurations of optical forcing. The front dynamics is characterized with respect to the different forcing parameters and the observations are compared with numerical simulations of a full model for the tilt angle of the liquid crystals under optical feedback. A spatially forced dissipative ϕ4 model is derived near the points of nascent bistability. From this model we derive analytical results that account qualitatively for the observed front dynamics and pinning range. Localized structures of different sizes and shapes are found to exist inside the pinning range and experimentally proved to be stable states of the spatially forced system.
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Affiliation(s)
- F Haudin
- INLN, Université de Nice-Sophia Antipolis, CNRS, 1361 route des Lucioles, 06560 Valbonne, France
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13
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Gelens L, Gomila D, Van der Sande G, Matías MA, Colet P. Nonlocality-induced front-interaction enhancement. PHYSICAL REVIEW LETTERS 2010; 104:154101. [PMID: 20481992 DOI: 10.1103/physrevlett.104.154101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2009] [Indexed: 05/23/2023]
Abstract
We demonstrate that nonlocal coupling strongly influences the dynamics of fronts connecting two equivalent states. In two prototype models we observe a large amplification in the interaction strength between two opposite fronts increasing front velocities several orders of magnitude. By analyzing the spatial dynamics we prove that way beyond quantitative effects, nonlocal terms can also change the overall qualitative picture by inducing oscillations in the front profile. This leads to a mechanism for the formation of localized structures not present for local interactions. Finally, nonlocal coupling can induce a steep broadening of localized structures, eventually annihilating them.
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Affiliation(s)
- L Gelens
- Department of Applied Physics and Photonics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
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14
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Tassin P, Van der Sande G, Veretennicoff I, Kockaert P, Tlidi M. Pattern formation without diffraction matching in optical parametric oscillators with a metamaterial. OPTICS EXPRESS 2009; 17:9428-9433. [PMID: 19466195 DOI: 10.1364/oe.17.009428] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
We consider a degenerate optical parametric oscillator containing a left-handed material. We show that the inclusion of a left-handed material layer allows for controlling the strength and sign of the diffraction coefficient at either the pump or the signal frequency. Subsequently, we demonstrate the existence of stable dissipative structures without diffraction matching, i.e., without the usual relationship between the diffraction coefficients of the signal and pump fields. Finally, we investigate the size scaling of these light structures with decreasing diffraction strength.
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Affiliation(s)
- Philippe Tassin
- Department of Applied Physics and Photonics, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium.
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15
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Barrio RA, Baker RE, Vaughan B, Tribuzy K, de Carvalho MR, Bassanezi R, Maini PK. Modeling the skin pattern of fishes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:031908. [PMID: 19391972 DOI: 10.1103/physreve.79.031908] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/01/2008] [Revised: 02/11/2009] [Indexed: 05/27/2023]
Abstract
Complicated patterns showing various spatial scales have been obtained in the past by coupling Turing systems in such a way that the scales of the independent systems resonate. This produces superimposed patterns with different length scales. Here we propose a model consisting of two identical reaction-diffusion systems coupled together in such a way that one of them produces a simple Turing pattern of spots or stripes, and the other traveling wave fronts that eventually become stationary. The basic idea is to assume that one of the systems becomes fixed after some time and serves as a source of morphogens for the other system. This mechanism produces patterns very similar to the pigmentation patterns observed in different species of stingrays and other fishes. The biological mechanisms that support the realization of this model are discussed.
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Affiliation(s)
- Rafael A Barrio
- Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford OX1 3LB, United Kingdom
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16
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Pesch M, Lange W, Gomila D, Ackemann T, Firth WJ, Oppo GL. Two-dimensional front dynamics and spatial solitons in a nonlinear optical system. PHYSICAL REVIEW LETTERS 2007; 99:153902. [PMID: 17995167 DOI: 10.1103/physrevlett.99.153902] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2007] [Indexed: 05/25/2023]
Abstract
Two-dimensional fronts and coarsening dynamics with a t{1/2} power law are analyzed experimentally and theoretically in a nonlinear optical system of a sodium vapor cell with single-mirror feedback. Modifications of the t{1/2} power law are observed in the vicinity of a modulational instability leading to the formation of spatial solitons of different sizes. The experimental and numerical observations give direct evidence for the locking of fronts as the mechanism of soliton formation. A phenomenological equation for the dynamics of the domain radius explains the observed behavior.
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Affiliation(s)
- M Pesch
- Institut für Angewandte Physik, Westfälische Wilhelms-Universität Münster, Corrensstr. 2/4, D-48149 Münster, Germany
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17
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Kolokolnikov T, Tlidi M. Spot deformation and replication in the two-dimensional Belousov-Zhabotinski reaction in a water-in-oil microemulsion. PHYSICAL REVIEW LETTERS 2007; 98:188303. [PMID: 17501615 DOI: 10.1103/physrevlett.98.188303] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2006] [Indexed: 05/15/2023]
Abstract
In the limit of a large duffusivity ratio, spotlike solutions in the two-dimensional Belousov-Zhabotinski reaction in water-in-oil microemulsion are studied. It is shown analytically that such spots undergo an instability as the diffusivity ratio is decreased. An instability threshold is derived. For spots of small radius, it is shown that this instability leads to a spot splitting into precisely two spots. For larger spots, it leads to deformation, fingering patterns, and space-filling curves. Numerical simulations are shown to be in close agreement with the analytical predictions.
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18
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Marts B, Hagberg A, Meron E, Lin AL. Resonant and nonresonant patterns in forced oscillators. CHAOS (WOODBURY, N.Y.) 2006; 16:037113. [PMID: 17014247 DOI: 10.1063/1.2346153] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
Uniform oscillations in spatially extended systems resonate with temporal periodic forcing within the Arnold tongues of single forced oscillators. The Arnold tongues are wedge-like domains in the parameter space spanned by the forcing amplitude and frequency, within which the oscillator's frequency is locked to a fraction of the forcing frequency. Spatial patterning can modify these domains. We describe here two pattern formation mechanisms affecting frequency locking at half the forcing frequency. The mechanisms are associated with phase-front instabilities and a Turing-like instability of the rest state. Our studies combine experiments on the ruthenium catalyzed light-sensitive Belousov-Zhabotinsky reaction forced by periodic illumination, and numerical and analytical studies of two model systems, the FitzHugh-Nagumo model and the complex Ginzburg-Landau equation, with additional terms describing periodic forcing.
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Affiliation(s)
- Bradley Marts
- Department of Physics, Duke University, Durham, North Carolina 27708, USA
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Michinel H, Paz-Alonso MJ, Pérez-García VM. Turning light into a liquid via atomic coherence. PHYSICAL REVIEW LETTERS 2006; 96:023903. [PMID: 16486577 DOI: 10.1103/physrevlett.96.023903] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/23/2005] [Indexed: 05/06/2023]
Abstract
We study a four-level atomic system with electromagnetically induced transparency with giant chi(3) and chi(5) susceptibilities of opposite signs. This system will allow us to obtain multidimensional solitons and light condensates with surface tension properties analogous to those of usual liquids.
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Affiliation(s)
- Humberto Michinel
- Area de Optica, Facultade de Ciencias de Ourense, Universidade de Vigo, As Lagoas s/n, Ourense, 32004 Spain
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Pesch M, Grosse Westhoff E, Ackemann T, Lange W. Observation of a discrete family of dissipative solitons in a nonlinear optical system. PHYSICAL REVIEW LETTERS 2005; 95:143906. [PMID: 16241658 DOI: 10.1103/physrevlett.95.143906] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2005] [Indexed: 05/05/2023]
Abstract
We report on the observation of a discrete family of spatial dissipative solitons in a simple optical pattern forming system, which is based on a modified single-mirror feedback arrangement. After a pitchfork bifurcation the system possesses two (nearly) equivalent coexisting states of different polarizations. The spatial solitons correspond to excursions from one of the two states serving as a background state towards the other one. The members of the soliton family differ in the number of high-amplitude radial oscillations. The observations are in good agreement with numerical simulations and general expectations for dissipative solitons.
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Affiliation(s)
- M Pesch
- Institut für Angewandte Physik, Westfälische Wilhelms-Universität Münster, Corrensstrasse 2/4, D-48149 Münster, Federal Republic of Germany
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Scroggie AJ, Jeffers J, McCartney G, Oppo GL. Reversible soliton motion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:046602. [PMID: 15903801 DOI: 10.1103/physreve.71.046602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2004] [Revised: 09/07/2004] [Indexed: 05/02/2023]
Abstract
We show that spatial solitons on either phase- or amplitude-modulated backgrounds can change their direction of motion according to the modulation frequency. A soliton may, therefore, move up or down phase gradients or remain motionless regardless of where it is in relation to the background modulation. The general theory is in good agreement with numerical results in a variety of nonlinear systems.
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Affiliation(s)
- A J Scroggie
- Department of Physics, University of Strathclyde, 107 Rottenrow, Glasgow G4 ONG, Scotland, United Kingdom.
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Barashenkov IV, Woodford SR. Complexes of stationary domain walls in the resonantly forced Ginsburg-Landau equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:026613. [PMID: 15783447 DOI: 10.1103/physreve.71.026613] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/06/2004] [Revised: 11/15/2004] [Indexed: 05/24/2023]
Abstract
The parametrically driven Ginsburg-Landau equation has well-known stationary solutions-the so-called Bloch and Ne el, or Ising, walls. In this paper, we construct an explicit stationary solution describing a bound state of two walls. We also demonstrate that stationary complexes of more than two walls do not exist.
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Affiliation(s)
- I V Barashenkov
- Department of Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa
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Marts B, Martinez K, Lin AL. Front dynamics in an oscillatory bistable Belousov-Zhabotinsky chemical reaction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:056223. [PMID: 15600744 DOI: 10.1103/physreve.70.056223] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2004] [Indexed: 05/24/2023]
Abstract
We observe breathing front dynamics which select three distinct types of bistable patterns in the 2:1 resonance regime of the periodically forced oscillatory Belousov-Zhabotinsky reaction. We measure the curvature-driven shrinking of a circular domain R approximately t(1/2) at forcing frequencies below a specific value, and show that the fast time scale front oscillations (breathing) drive this slow time scale shrinking. Above a specific frequency, we observe fronts of higher curvature grow instead of shrink and labyrinth patterns form. Just below the transition frequency is a relatively narrow range of frequencies where the curvature-driven coarsening is balanced by a competing front interaction, which leads to a pattern of localized structures. The length scale of the localized structure and labyrinth patterns is set by the front interactions.
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Affiliation(s)
- Bradley Marts
- Center for Nonlinear and Complex Systems and Department of Physics, Duke University, Durham, North Carolina 27708, USA
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Gomila D, Colet P, Oppo GL, Miguel MS. Stable droplets and nucleation in asymmetric bistable nonlinear optical systems. ACTA ACUST UNITED AC 2004. [DOI: 10.1088/1464-4266/6/5/014] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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