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Barberi L, Kruse K. Localized States in Active Fluids. PHYSICAL REVIEW LETTERS 2023; 131:238401. [PMID: 38134762 DOI: 10.1103/physrevlett.131.238401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2022] [Accepted: 11/13/2023] [Indexed: 12/24/2023]
Abstract
Biological active matter is typically tightly coupled to chemical reaction networks affecting its assembly-disassembly dynamics and stress generation. We show that localized states can emerge spontaneously if assembly of active matter is regulated by chemical species that are advected with flows resulting from gradients in the active stress. The mechanochemical localized patterns form via a subcritical bifurcation and for parameter values for which patterns do not exist in absence of the advective coupling. Our work identifies a generic mechanism underlying localized cellular patterns.
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Affiliation(s)
- Luca Barberi
- Department of Biochemistry, University of Geneva, 1211 Geneva, Switzerland
- Department of Theoretical Physics, University of Geneva, 1211 Geneva, Switzerland
| | - Karsten Kruse
- Department of Biochemistry, University of Geneva, 1211 Geneva, Switzerland
- Department of Theoretical Physics, University of Geneva, 1211 Geneva, Switzerland
- NCCR for Chemical Biology, University of Geneva, 1211 Geneva, Switzerland
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2
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Aguilera-Rojas PJ, Clerc MG, Gonzalez-Cortes G, Jara-Schulz G. Localized standing waves induced by spatiotemporal forcing. Phys Rev E 2021; 104:044209. [PMID: 34781469 DOI: 10.1103/physreve.104.044209] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2021] [Accepted: 09/14/2021] [Indexed: 11/07/2022]
Abstract
Particle-type solutions are observed in out-of-equilibrium systems. These states can be motionless, oscillatory, or propagative depending on the injection and dissipation of energy. We investigate a family of localized standing waves based on a liquid-crystal light valve with spatiotemporal modulated optical feedback. These states are nonlinear waves in which energy concentrates in a localized and oscillatory manner. The organization of the family of solutions is characterized as a function of the applied voltage. Close to the reorientation transition, an amplitude equation allows us to elucidate the origin of these localized states and establish their bifurcation diagram. Theoretical findings are in qualitative agreement with experimental observations. Our results open the possibility of manipulating localized states induced by light, which can be used to expand and improve the storage and manipulation of information.
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Affiliation(s)
- P J Aguilera-Rojas
- Departamento de Física and Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - M G Clerc
- Departamento de Física and Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - G Gonzalez-Cortes
- Departamento de Física and Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - G Jara-Schulz
- Departamento de Física and Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile.,Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, Palaiseau 91120, France
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3
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Parra-Rivas P, Gelens L, Leo F. Localized structures in dispersive and doubly resonant optical parametric oscillators. Phys Rev E 2019; 100:032219. [PMID: 31639956 DOI: 10.1103/physreve.100.032219] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/21/2019] [Indexed: 06/10/2023]
Abstract
We study temporally localized structures in doubly resonant degenerate optical parametric oscillators in the absence of temporal walk-off. We focus on states formed through the locking of domain walls between the zero and a nonzero continuous-wave solution. We show that these states undergo collapsed snaking and we characterize their dynamics in the parameter space.
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Affiliation(s)
- P Parra-Rivas
- OPERA-photonics, Université libre de Bruxelles, 50 Avenue F. D. Roosevelt, CP 194/5, B-1050 Bruxelles, Belgium
- Laboratory of Dynamics in Biological Systems, KU Leuven Department of Cellular and Molecular Medicine, University of Leuven, B-3000 Leuven, Belgium
| | - L Gelens
- Laboratory of Dynamics in Biological Systems, KU Leuven Department of Cellular and Molecular Medicine, University of Leuven, B-3000 Leuven, Belgium
| | - F Leo
- OPERA-photonics, Université libre de Bruxelles, 50 Avenue F. D. Roosevelt, CP 194/5, B-1050 Bruxelles, Belgium
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4
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Garbin B, Javaloyes J, Barland S, Tissoni G. Interactions and collisions of topological solitons in a semiconductor laser with optical injection and feedback. CHAOS (WOODBURY, N.Y.) 2017; 27:114308. [PMID: 29195338 DOI: 10.1063/1.5006751] [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
We present experimental and numerical results about dynamical interactions of topological solitons in a semiconductor laser with coherent injection and feedback. We show different kind of interactions such as repulsion, annihilation, or formation of soliton bound states, depending on laser parameters. Collisions between single structures and bound states conserve momentum and charge.
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Affiliation(s)
- B Garbin
- The Dodd-Walls Centre for Photonic and Quantum Technologies, and Physics Department, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
| | - J Javaloyes
- Departament de Física, Universitat de les Illes Balears, Cra. De Valldemossa, km 7.5, E-07122 Palma, Spain
| | - S Barland
- Université Côte d'Azur, CNRS, Institut de Physique de Nice, 1361 Route des Lucioles, 06560 Valbonne, France
| | - G Tissoni
- Université Côte d'Azur, CNRS, Institut de Physique de Nice, 1361 Route des Lucioles, 06560 Valbonne, France
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Javaloyes J, Marconi M, Giudici M. Nonlocality Induces Chains of Nested Dissipative Solitons. PHYSICAL REVIEW LETTERS 2017; 119:033904. [PMID: 28777628 DOI: 10.1103/physrevlett.119.033904] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2016] [Indexed: 06/07/2023]
Abstract
Dissipative solitons often behave as quasiparticles, and they may form molecules characterized by well-defined bond distances. We show that pointwise nonlocality may lead to a new kind of molecule where bonds are not rigid. The elements of this molecule can shift mutually one with respect to the others while remaining linked together, in a manner similar to interlaced rings in a chain. We report experimental observations of these chains of nested dissipative solitons in a time-delayed laser system.
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Affiliation(s)
- J Javaloyes
- Departament de Física, Universitat de les Illes Baleares, C/Valldemossa km 7.5, 07122 Mallorca, Spain
| | - M Marconi
- Université Côte d'Azur, CNRS, Institut de Physique de Nice, F-06560 Valbonne, France
| | - M Giudici
- Université Côte d'Azur, CNRS, Institut de Physique de Nice, F-06560 Valbonne, France
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6
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Yulin AV, Aladyshkina A, Shalin AS. Motion of dissipative optical fronts under the action of an oscillating pump. Phys Rev E 2016; 94:022205. [PMID: 27627296 DOI: 10.1103/physreve.94.022205] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2016] [Indexed: 06/06/2023]
Abstract
The dynamics of domain walls in optical bistable systems with pump and loss is considered. It is shown that an oscillating component of the pump affects the average drift velocity of the domain walls. The cases of harmonic and biharmonic pumps are considered. It is demonstrated that in the case of biharmonic pulse the velocity of the domain wall can be controlled by the mutual phase of the harmonics. The analogy between this phenomenon and the ratchet effect is drawn. Synchronization of the moving domain walls by the oscillating pump in discrete systems is studied and discussed.
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Affiliation(s)
- A V Yulin
- ITMO University 197101, Kronverksky pr. 49, St. Petersburg, Russian Federation
| | - A Aladyshkina
- National Research University Higher School of Economics, Bolshaya Pecherskaya 603155, 25/12, Nizhny Novgorod, Russian Federation
| | - A S Shalin
- ITMO University 197101, Kronverksky pr. 49, St. Petersburg, Russian Federation
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7
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Clerc MG, Coulibaly S, Ferré MA, García-Ñustes MA, Rojas RG. Chimera-type states induced by local coupling. Phys Rev E 2016; 93:052204. [PMID: 27300877 DOI: 10.1103/physreve.93.052204] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/15/2015] [Indexed: 06/06/2023]
Abstract
Coupled oscillators can exhibit complex self-organization behavior such as phase turbulence, spatiotemporal intermittency, and chimera states. The latter corresponds to a coexistence of coherent and incoherent states apparently promoted by nonlocal or global coupling. Here we investigate the existence, stability properties, and bifurcation diagram of chimera-type states in a system with local coupling without different time scales. Based on a model of a chain of nonlinear oscillators coupled to adjacent neighbors, we identify the required attributes to observe these states: local coupling and bistability between a stationary and an oscillatory state close to a homoclinic bifurcation. The local coupling prevents the incoherent state from invading the coherent one, allowing concurrently the existence of a family of chimera states, which are organized by a homoclinic snaking bifurcation diagram.
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Affiliation(s)
- M G Clerc
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - S Coulibaly
- Laboratoire de Physique des Lasers, Atomes et Molécules, CNRS UMR 8523, Université de Lille 1 Sciences et Technologies, 59655 Villeneuve d'Ascq Cedex, France
| | - M A Ferré
- Instituto de Física, Pontificia Universidad Católica de Valparaíso, casilla 4059, Valparaíso, Chile
| | - M A García-Ñustes
- Instituto de Física, Pontificia Universidad Católica de Valparaíso, casilla 4059, Valparaíso, Chile
| | - R G Rojas
- Instituto de Física, Pontificia Universidad Católica de Valparaíso, casilla 4059, Valparaíso, Chile
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Yochelis A, Knobloch E, Köpf MH. Origin of finite pulse trains: Homoclinic snaking in excitable media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032924. [PMID: 25871189 DOI: 10.1103/physreve.91.032924] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/23/2013] [Indexed: 06/04/2023]
Abstract
Many physical, chemical, and biological systems exhibit traveling waves as a result of either an oscillatory instability or excitability. In the latter case a large multiplicity of stable spatially localized wavetrains consisting of different numbers of traveling pulses may be present. The existence of these states is related here to the presence of homoclinic snaking in the vicinity of a subcritical, finite wavenumber Hopf bifurcation. The pulses are organized in a slanted snaking structure resulting from the presence of a heteroclinic cycle between small and large amplitude traveling waves. Connections of this type require a multivalued dispersion relation. This dispersion relation is computed numerically and used to interpret the profile of the pulse group. The different spatially localized pulse trains can be accessed by appropriately customized initial stimuli, thereby blurring the traditional distinction between oscillatory and excitable systems. The results reveal a new class of phenomena relevant to spatiotemporal dynamics of excitable media, particularly in chemical and biological systems with multiple activators and inhibitors.
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Affiliation(s)
- Arik Yochelis
- Department of Solar Energy and Environmental Physics, Swiss Institute for Dryland Environmental and Energy Research, Jacob Blaustein Institutes for Desert Research (BIDR), Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 84990, Israel
| | - Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Michael H Köpf
- Département de Physique, École Normale Supérieure, 24 rue Lhomond, 75005 Paris, France
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9
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Gandhi P, Knobloch E, Beaume C. Localized states in periodically forced systems. PHYSICAL REVIEW LETTERS 2015; 114:034102. [PMID: 25659000 DOI: 10.1103/physrevlett.114.034102] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/07/2014] [Indexed: 06/04/2023]
Abstract
The theory of stationary spatially localized patterns in dissipative systems driven by time-independent forcing is well developed. With time-periodic forcing, related but time-dependent structures may result. These may consist of breathing localized patterns, or states that grow for part of the cycle via nucleation of new wavelengths of the pattern followed by wavelength annihilation during the remainder of the cycle. These two competing processes lead to a complex phase diagram whose structure is a consequence of a series of resonances between the nucleation time and the forcing period. The resulting diagram is computed for the periodically forced quadratic-cubic Swift-Hohenberg equation, and its details are interpreted in terms of the properties of the depinning transition for the fronts bounding the localized state on either side. The results are expected to shed light on localized states in a large variety of periodically driven systems.
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Affiliation(s)
- Punit Gandhi
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Cédric Beaume
- Department of Aeronautics, Imperial College London, London SW7 2AZ, United Kingdom
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Colet P, Matías MA, Gelens L, Gomila D. Formation of localized structures in bistable systems through nonlocal spatial coupling. I. General framework. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012914. [PMID: 24580304 DOI: 10.1103/physreve.89.012914] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/02/2013] [Indexed: 05/23/2023]
Abstract
The present work studies the influence of nonlocal spatial coupling on the existence of localized structures in one-dimensional extended systems. We consider systems described by a real field with a nonlocal coupling that has a linear dependence on the field. Leveraging spatial dynamics we provide a general framework to understand the effect of the nonlocality on the shape of the fronts connecting two stable states. In particular we show that nonlocal terms can induce spatial oscillations in the front tails, allowing for the creation of localized structures, that emerge from pinning between two fronts. In parameter space the region where fronts are oscillatory is limited by three transitions: the modulational instability of the homogeneous state, the Belyakov-Devaney transition in which monotonic fronts acquire spatial oscillations with infinite wavelength, and a crossover in which monotonically decaying fronts develop spatial oscillations with a finite wavelength. We show how these transitions are organized by codimension 2 and 3 points and illustrate how by changing the parameters of the nonlocal coupling it is possible to bring the system into the region where localized structures can be formed.
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Affiliation(s)
- Pere Colet
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Manuel A Matías
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Lendert Gelens
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain and Applied Physics Research Group (APHY), Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium
| | - Damià Gomila
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
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11
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Kao HC, Beaume C, Knobloch E. Spatial localization in heterogeneous systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012903. [PMID: 24580293 DOI: 10.1103/physreve.89.012903] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/18/2013] [Indexed: 06/03/2023]
Abstract
We study spatial localization in the generalized Swift-Hohenberg equation with either quadratic-cubic or cubic-quintic nonlinearity subject to spatially heterogeneous forcing. Different types of forcing (sinusoidal or Gaussian) with different spatial scales are considered and the corresponding localized snaking structures are computed. The results indicate that spatial heterogeneity exerts a significant influence on the location of spatially localized structures in both parameter space and physical space, and on their stability properties. The results are expected to assist in the interpretation of experiments on localized structures where departures from spatial homogeneity are generally unavoidable.
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Affiliation(s)
| | - Cédric Beaume
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
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Thiele U, Archer AJ, Robbins MJ, Gomez H, Knobloch E. Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:042915. [PMID: 23679497 DOI: 10.1103/physreve.87.042915] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2013] [Revised: 03/21/2013] [Indexed: 05/11/2023]
Abstract
The conserved Swift-Hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state. The resulting phase field crystal model describes a variety of spatially localized structures, in addition to different spatially extended periodic structures. The location of these structures in the temperature versus mean order parameter plane is determined using a combination of numerical continuation in one dimension and direct numerical simulation in two and three dimensions. Localized states are found in the region of thermodynamic coexistence between the homogeneous and structured phases, and may lie outside of the binodal for these states. The results are related to the phenomenon of slanted snaking but take the form of standard homoclinic snaking when the mean order parameter is plotted as a function of the chemical potential, and are expected to carry over to related models with a conserved order parameter.
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Affiliation(s)
- Uwe Thiele
- Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom.
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Matthews PC, Susanto H. Variational approximations to homoclinic snaking in continuous and discrete systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:066207. [PMID: 22304178 DOI: 10.1103/physreve.84.066207] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/26/2011] [Revised: 10/04/2011] [Indexed: 05/31/2023]
Abstract
Localized structures appear in a wide variety of systems, arising from a pinning mechanism due to the presence of a small-scale pattern or an imposed grid. When there is a separation of length scales, the width of the pinning region is exponentially small and beyond the reach of standard asymptotic methods. We show how this behavior can be obtained using a variational method, for two systems. In the case of the quadratic-cubic Swift-Hohenberg equation, this gives results that are in agreement with recent work using exponential asymptotics. In addition, the method is applied to a discrete system with cubic-quintic nonlinearity, giving results that agree well with numerical simulations.
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Affiliation(s)
- P C Matthews
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
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Susanto H, Matthews PC. Variational approximations to homoclinic snaking. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:035201. [PMID: 21517550 DOI: 10.1103/physreve.83.035201] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/01/2010] [Revised: 02/01/2011] [Indexed: 05/30/2023]
Abstract
We investigate the snaking of localized patterns, seen in numerous physical applications, using a variational approximation. This method naturally introduces the exponentially small terms responsible for the snaking structure, which are not accessible via standard multiple-scales asymptotic techniques. We obtain the symmetric snaking solutions and the asymmetric "ladder" states, and also predict the stability of the localized states. The resulting approximate formulas for the width of the snaking region show good agreement with numerical results.
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Affiliation(s)
- H Susanto
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG72RD, United Kingdom
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Dawes JHP. The emergence of a coherent structure for coherent structures: localized states in nonlinear systems. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2010; 368:3519-3534. [PMID: 20603365 DOI: 10.1098/rsta.2010.0057] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
Coherent structures emerge from the dynamics of many kinds of dissipative, externally driven, nonlinear systems, and continue to provoke new questions that challenge our physical and mathematical understanding. In one specific subclass of such problems, in which a pattern-forming, or 'Turing', instability occurs, rapid progress has been made recently in our understanding of the formation of localized states: patches of regular pattern surrounded by the unpatterned homogeneous background state. This short review article surveys the progress that has been made for localized states and proposes three areas of application for these ideas that would take the theory in new directions and ultimately be of substantial benefit to areas of applied science. Finally, I offer speculations for future work, based on localized states, that may help researchers to understand coherent structures more generally.
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Affiliation(s)
- J H P Dawes
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.
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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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18
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Kozyreff G, Assemat P, Chapman SJ. Influence of boundaries on localized patterns. PHYSICAL REVIEW LETTERS 2009; 103:164501. [PMID: 19905698 DOI: 10.1103/physrevlett.103.164501] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2009] [Indexed: 05/28/2023]
Abstract
We analytically study the influence of boundaries on distant localized patterns generated by a Turing instability. To this end, we use the Swift-Hohenberg model with arbitrary boundary conditions. We find that the bifurcation diagram of these localized structures generally involves four homoclinic snaking branches, rather than two for infinite or periodic domains. Second, steady localized patterns only exist at discrete locations, and only at the center of the domain if their size exceeds a critical value. Third, reducing the domain size increases the pinning range.
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Affiliation(s)
- G Kozyreff
- Optique Nonlinéaire Théorique, Université Libre de Bruxelles (U.L.B.), CP 231, Belgium
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Barbay S, Hachair X, Elsass T, Sagnes I, Kuszelewicz R. Homoclinic snaking in a semiconductor-based optical system. PHYSICAL REVIEW LETTERS 2008; 101:253902. [PMID: 19113709 DOI: 10.1103/physrevlett.101.253902] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2008] [Indexed: 05/27/2023]
Abstract
We report on experimental observations of homoclinic snaking in a vertical-cavity semiconductor optical amplifier. Our observations in a quasi-one-dimensional and two-dimensional configurations agree qualitatively well with what is expected from recent theoretical and numerical studies. In particular, we show the bifurcation sequence leading to a snaking bifurcation diagram linking single localized states to "localized patterns" or clusters of localized states and demonstrate a parameter region where cluster states are inhibited.
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Affiliation(s)
- S Barbay
- Laboratoire de Photonique et de Nanostructures, CNRS, Route de Nozay, 91460 Marcoussis, France.
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20
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Bortolozzo U, Clerc MG, Residori S. Local theory of the slanted homoclinic snaking bifurcation diagram. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:036214. [PMID: 18851128 DOI: 10.1103/physreve.78.036214] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2008] [Revised: 08/24/2008] [Indexed: 05/26/2023]
Abstract
Localized states in out of equilibrium one-dimensional systems are described by the homoclinic snaking associated with the infinite sequence of multibump localized solutions of the corresponding time reversible dynamical system. We show that when the pattern undergoes a saddle-node bifurcation the homoclinic snaking bifurcation diagram becomes slanted and a finite set of localized states continue to exist outside the region of bistability. This generic behavior offers a local theory resolution of the discrepancy between models and experiments.
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Affiliation(s)
- U Bortolozzo
- INLN, Université de Nice Sophia-Antipolis, CNRS, 1361 route des Lucioles 06560 Valbonne, France
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