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Kozyreff G. Speed of wave packets and the nonlinear Schrödinger equation. Phys Rev E 2023; 107:014219. [PMID: 36797873 DOI: 10.1103/physreve.107.014219] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/24/2022] [Accepted: 01/11/2023] [Indexed: 06/18/2023]
Abstract
The universal theory of weakly nonlinear wave packets given by the nonlinear Schrödinger equation is revisited. In the limit where the group and phase velocities are very close together, a multiple-scale analysis carried out beyond all orders reveals that a single soliton, bright or dark, can travel at a different speed than the group velocity. In an exponentially small but finite range of parameters, the envelope of the soliton is locked to the rapid oscillations of the carrier wave. Eventually, the dynamics is governed by an equation analogous to that of a pendulum, in which the center of mass of the soliton is subjected to a periodic potential. Consequently, the soliton speed is not constant and generally contains a periodic component. Furthermore, the interaction between two distant solitons can in principle be profoundly altered by the aforementioned effective periodic potential and we conjecture the existence of new bound states. These results are derived on a wide class of wave models and in such a general way that they are believed to be of universal validity.
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Affiliation(s)
- Gregory Kozyreff
- Optique Nonlinéaire Théorique, Université libre de Bruxelles (U.L.B.), CP 231, Belgium
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2
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Dolinina D, Yulin A. Dissipative switching waves and solitons in the systems with spontaneously broken symmetry. Phys Rev E 2021; 103:052207. [PMID: 34134290 DOI: 10.1103/physreve.103.052207] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2021] [Accepted: 04/15/2021] [Indexed: 11/07/2022]
Abstract
The paper addresses the bistability caused by spontaneous symmetry breaking bifurcation in a one-dimensional periodically corrugated nonlinear waveguide pumped by coherent light at normal incidence. The formation and the stability of the switching waves connecting the states of different symmetries are studied numerically. It is shown that the switching waves can form stable resting and moving bound states (dissipative solitons). The protocols of the creation of the discussed nonlinear localized waves are suggested and verified by numerical simulations.
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Affiliation(s)
- D Dolinina
- Faculty of Physics, ITMO University, Saint-Petersburg 197101, Russia
| | - A Yulin
- Faculty of Physics, ITMO University, Saint-Petersburg 197101, Russia
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3
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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] [What about the content of this article? (0)] [Affiliation(s)] [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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Meron E, Bennett JJR, Fernandez-oto C, Tzuk O, Zelnik YR, Grafi G. Continuum Modeling of Discrete Plant Communities: Why Does It Work and Why Is It Advantageous? Mathematics 2019; 7:987. [DOI: 10.3390/math7100987] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
Understanding ecosystem response to drier climates calls for modeling the dynamics of dryland plant populations, which are crucial determinants of ecosystem function, as they constitute the basal level of whole food webs. Two modeling approaches are widely used in population dynamics, individual (agent)-based models and continuum partial-differential-equation (PDE) models. The latter are advantageous in lending themselves to powerful methodologies of mathematical analysis, but the question of whether they are suitable to describe small discrete plant populations, as is often found in dryland ecosystems, has remained largely unaddressed. In this paper, we first draw attention to two aspects of plants that distinguish them from most other organisms—high phenotypic plasticity and dispersal of stress-tolerant seeds—and argue in favor of PDE modeling, where the state variables that describe population sizes are not discrete number densities, but rather continuous biomass densities. We then discuss a few examples that demonstrate the utility of PDE models in providing deep insights into landscape-scale behaviors, such as the onset of pattern forming instabilities, multiplicity of stable ecosystem states, regular and irregular, and the possible roles of front instabilities in reversing desertification. We briefly mention a few additional examples, and conclude by outlining the nature of the information we should and should not expect to gain from PDE model studies.
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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] [What about the content of this article? (0)] [Affiliation(s)] [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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6
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Abstract
The properties of a hinged floating elastic sheet of finite length under compression are considered. Numerical continuation is used to compute spatially localized buckled states with many spatially localized folds. Both symmetric and antisymmetric states are computed and the corresponding bifurcation diagrams determined. Weakly nonlinear analysis is used to analyze the transition from periodic wrinkles to singlefold and multifold states and to compute their energy. States with the same number of folds have energies that barely differ from each other and the energy gap decreases exponentially as localization increases. The stability of the different competing states is studied and the multifold solutions are all found to be unstable. However, the decay time into solutions with fewer folds can be so slow that multifolds may appear to be stable.
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Affiliation(s)
- Leonardo Gordillo
- Departamento de Física, Universidad de Santiago de Chile, Av. Ecuador 3493, Estación Central, Santiago, Chile
| | - Edgar Knobloch
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
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7
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Susanto H, Kusdiantara R, Li N, Kirikchi OB, Adzkiya D, Putri ERM, Asfihani T. Snakes and ghosts in a parity-time-symmetric chain of dimers. Phys Rev E 2018; 97:062204. [PMID: 30011512 DOI: 10.1103/physreve.97.062204] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2017] [Indexed: 11/07/2022]
Abstract
We consider linearly coupled discrete nonlinear Schrödinger equations with gain and loss terms and with a cubic-quintic nonlinearity. The system models a parity-time (PT)-symmetric coupler composed by a chain of dimers. We study uniform states and site-centered and bond-centered spatially localized solutions and present that each solution has a symmetric and antisymmetric configuration between the arms. The symmetric solutions can become unstable due to bifurcations of asymmetric ones, that are called ghost states, because they exist only when an otherwise real propagation constant is taken to be complex valued. When a parameter is varied, the resulting bifurcation diagrams for the existence of standing localized solutions have a snaking behavior. The critical gain and loss coefficient above which the PT symmetry is broken corresponds to the condition when bifurcation diagrams of symmetric and antisymmetric states merge. Past the symmetry breaking, the system no longer has time-independent states. Nevertheless, equilibrium solutions can be analytically continued by defining a dual equation that leads to ghost states associated with growth or decay, that are also identified and examined here. We show that ghost localized states also exhibit snaking bifurcation diagrams. We analyze the width of the snaking region and provide asymptotic approximations in the limit of strong and weak coupling where good agreement is obtained.
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Affiliation(s)
- H Susanto
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
| | - R Kusdiantara
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom.,Centre of Mathematical Modelling and Simulation, Institut Teknologi Bandung, 1st Floor, Labtek III, Jl. Ganesha No. 10, Bandung, 40132, Indonesia
| | - N Li
- School of Computer Science and Electronic Engineering, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
| | - O B Kirikchi
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
| | - D Adzkiya
- Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Sukolilo, Surabaya 60111, Indonesia
| | - E R M Putri
- Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Sukolilo, Surabaya 60111, Indonesia
| | - T Asfihani
- Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Sukolilo, Surabaya 60111, Indonesia
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Parra-Rivas P, Gomila D, Gelens L, Knobloch E. Bifurcation structure of localized states in the Lugiato-Lefever equation with anomalous dispersion. Phys Rev E 2018; 97:042204. [PMID: 29758631 DOI: 10.1103/physreve.97.042204] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2018] [Indexed: 06/08/2023]
Abstract
The origin, stability, and bifurcation structure of different types of bright localized structures described by the Lugiato-Lefever equation are studied. This mean field model describes the nonlinear dynamics of light circulating in fiber cavities and microresonators. In the case of anomalous group velocity dispersion and low values of the intracavity phase detuning these bright states are organized in a homoclinic snaking bifurcation structure. We describe how this bifurcation structure is destroyed when the detuning is increased across a critical value, and determine how a bifurcation structure known as foliated snaking emerges.
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Affiliation(s)
- P Parra-Rivas
- Laboratory of Dynamics in Biological Systems, KU Leuven Department of Cellular and Molecular Medicine, University of Leuven, B-3000 Leuven, Belgium
- Applied Physics Research Group, APHY, Vrije Universiteit Brussel, 1050 Brussels, Belgium
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
| | - D Gomila
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
| | - L Gelens
- Laboratory of Dynamics in Biological Systems, KU Leuven Department of Cellular and Molecular Medicine, University of Leuven, B-3000 Leuven, Belgium
- Applied Physics Research Group, APHY, Vrije Universiteit Brussel, 1050 Brussels, Belgium
| | - E Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
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Abstract
We consider the discrete Swift-Hohenberg equation with cubic and quintic nonlinearity, obtained from discretizing the spatial derivatives of the Swift-Hohenberg equation using central finite differences. We investigate the discretization effect on the bifurcation behavior, where we identify three regions of the coupling parameter, i.e., strong, weak, and intermediate coupling. Within the regions, the discrete Swift-Hohenberg equation behaves either similarly or differently from the continuum limit. In the intermediate coupling region, multiple Maxwell points can occur for the periodic solutions and may cause irregular snaking and isolas. Numerical continuation is used to obtain and analyze localized and periodic solutions for each case. Theoretical analysis for the snaking and stability of the corresponding solutions is provided in the weak coupling region.
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Affiliation(s)
- R Kusdiantara
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom.,Centre of Mathematical Modelling and Simulation, Institut Teknologi Bandung, 1st Floor, Labtek III, Jl. Ganesha No. 10, Bandung 40132, Indonesia
| | - H Susanto
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
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10
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Zelnik YR, Uecker H, Feudel U, Meron E. Desertification by front propagation? J Theor Biol 2017; 418:27-35. [DOI: 10.1016/j.jtbi.2017.01.029] [Citation(s) in RCA: 33] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/19/2016] [Revised: 11/26/2016] [Accepted: 01/19/2017] [Indexed: 11/19/2022]
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Abstract
We report on the emergence of robust multiclustered chimera states in a dissipative-driven system of symmetrically and locally coupled identical superconducting quantum interference device (SQUID) oscillators. The "snakelike" resonance curve of the single SQUID is the key to the formation of the chimera states and is responsible for the extreme multistability exhibited by the coupled system that leads to attractor crowding at the geometrical resonance (inductive-capacitive) frequency. Until now, chimera states were mostly believed to exist for nonlocal coupling. Our findings provide theoretical evidence that nearest-neighbor interactions are indeed capable of supporting such states in a wide parameter range. SQUID metamaterials are the subject of intense experimental investigations, and we are highly confident that the complex dynamics demonstrated in this paper can be confirmed in the laboratory.
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Affiliation(s)
- J Hizanidis
- Department of Physics, Crete Center for Quantum Complexity and Nanotechnology, University of Crete, P.O. Box 2208, 71003 Heraklion, Greece; Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, P.O. Box 1527, 71110 Heraklion, Greece; and National University of Science and Technology MISiS, Leninsky Prospekt 4, Moscow 119049, Russia
| | - N Lazarides
- Department of Physics, Crete Center for Quantum Complexity and Nanotechnology, University of Crete, P.O. Box 2208, 71003 Heraklion, Greece; Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, P.O. Box 1527, 71110 Heraklion, Greece; and National University of Science and Technology MISiS, Leninsky Prospekt 4, Moscow 119049, Russia
| | - G P Tsironis
- Department of Physics, Crete Center for Quantum Complexity and Nanotechnology, University of Crete, P.O. Box 2208, 71003 Heraklion, Greece; Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, P.O. Box 1527, 71110 Heraklion, Greece; and National University of Science and Technology MISiS, Leninsky Prospekt 4, Moscow 119049, Russia
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12
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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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13
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Breña-Medina V, Champneys A. Subcritical Turing bifurcation and the morphogenesis of localized patterns. Phys Rev E Stat Nonlin Soft Matter Phys 2014; 90:032923. [PMID: 25314520 DOI: 10.1103/physreve.90.032923] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2014] [Indexed: 05/03/2023]
Abstract
Subcritical Turing bifurcations of reaction-diffusion systems in large domains lead to spontaneous onset of well-developed localized patterns via the homoclinic snaking mechanism. This phenomenon is shown to occur naturally when balancing source and loss effects are included in a typical reaction-diffusion system, leading to a super- to subcritical transition. Implications are discussed [corrected]for a range of physical problems, arguing that subcriticality leads to naturally robust phase transitions to localized patterns.
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Affiliation(s)
- Víctor Breña-Medina
- Departamento de Nanotecnología, Centro de Física Aplicada y Tecnología Avanzada, Universidad Nacional Autónoma de México, Juriquilla No. 3001, Querétaro 76230, Mexico and Department of Engineering Mathematics, University of Bristol, Queen's Building, University Walk, Bristol BS8 1TR, United Kingdom
| | - Alan Champneys
- Department of Engineering Mathematics, University of Bristol, Queen's Building, University Walk, Bristol BS8 1TR, United Kingdom
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Kozyreff G, Chapman SJ. Analytical results for front pinning between an hexagonal pattern and a uniform state in pattern-formation systems. Phys Rev Lett 2013; 111:054501. [PMID: 23952408 DOI: 10.1103/physrevlett.111.054501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/05/2013] [Indexed: 06/02/2023]
Abstract
In pattern-forming systems, localized patterns are states of intermediate complexity between fully extended ordered patterns and completely irregular patterns. They are formed by stationary fronts enclosing an ordered pattern inside an homogeneous background. In two dimensions, the ordered pattern is most often hexagonal and the conditions for fronts to stabilize are still unknown. In this Letter, we show how the locking of these fronts depends on their orientation relative to the pattern. The theory rests on general asymptotic arguments valid when the spatial scale of the front is slow compared to that of the hexagonal pattern. Our analytical results are confirmed by numerical simulations with the Swift-Hohenberg equation, relevant to hydrodynamical and buckling instabilities, and a nonlinear optical cavity model.
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Affiliation(s)
- G Kozyreff
- Optique Nonlinéaire Théorique, Université Libre de Bruxelles, CP 231, Campus Plaine, B-1050 Bruxelles, Belgium
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15
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Abstract
Localized states are found in many pattern forming systems. The aim of this paper is to investigate the occurrence of oscillatory localized states in two-dimensional Boussinesq magnetoconvection. Initially considering an idealized model, in which the vertical structure of the system has been simplified by a projection onto a small number of Fourier modes, we find that these states are restricted to the low ζ regime (where ζ represents the ratio of the magnetic to thermal diffusivities). These states always exhibit bistability with another nontrivial solution branch; in other words, they show no evidence of subcritical behavior. This is due to the weak flux expulsion that is exhibited by these time-dependent solutions. Using the results of this parameter survey, we locate corresponding states in a fully resolved two-dimensional system, although the mode of oscillation is more complex in this case. This is the first time that a localized oscillatory state, of this kind, has been found in a fully resolved magnetoconvection simulation.
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Affiliation(s)
- Matthew C Buckley
- School of Mathematics and Statistics, Newcastle University, Newcastle Upon Tyne, United Kingdom
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16
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Faye G, Rankin J, Chossat P. Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis. J Math Biol 2013; 66:1303-38. [DOI: 10.1007/s00285-012-0532-y] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2012] [Revised: 04/06/2012] [Indexed: 11/25/2022]
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Haudin F, Rojas RG, Bortolozzo U, Residori S, Clerc MG. Homoclinic snaking of localized patterns in a spatially forced system. Phys Rev Lett 2011; 107:264101. [PMID: 22243157 DOI: 10.1103/physrevlett.107.264101] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/29/2011] [Indexed: 05/31/2023]
Abstract
Dissipative localized structures exhibit intricate bifurcation diagrams. An adequate theory has been developed in one space dimension; however, discrepancies arise with the experiments. Based on an optical feedback with spatially modulated input beam, we set up a 1D forced configuration in a nematic liquid crystal layer. We characterize experimentally and theoretically the homoclinic snaking diagram of localized patterns, providing a reconciliation between theory and experiments.
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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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19
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Matthews PC, Susanto H. Variational approximations to homoclinic snaking in continuous and discrete systems. Phys Rev E Stat Nonlin Soft Matter Phys 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] [What about the content of this article? (0)] [Affiliation(s)] [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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20
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Susanto H, Matthews PC. Variational approximations to homoclinic snaking. Phys Rev E Stat Nonlin Soft Matter Phys 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] [What about the content of this article? (0)] [Affiliation(s)] [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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22
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Abstract
We show that spiral vortices in oscillatory systems can lose stability to secondary modes to form dual-mode spiral vortices. The secondary modes grow at the vortex core where the oscillation amplitude vanishes but are nonlinearly damped by the oscillatory mode away from the core. Gradients of the oscillation phase, induced by the hosted secondary mode, can lead to additional hosting events that culminate in periodic core oscillations or in a novel form of spatiotemporal chaos. The results of this study apply to physical, chemical, and biological systems that go through cusp-Hopf, fold-Hopf, and Hopf-Turing bifurcations.
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Affiliation(s)
- Yair Mau
- Physics Department, Ben-Gurion University, Beer-Sheva, Israel
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23
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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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24
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Houghton SM, Knobloch E. Homoclinic snaking in bounded domains. Phys Rev E Stat Nonlin Soft Matter Phys 2009; 80:026210. [PMID: 19792234 DOI: 10.1103/physreve.80.026210] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2009] [Revised: 05/27/2009] [Indexed: 05/28/2023]
Abstract
Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. On the real line this process continues forever. In finite domains snaking terminates once the domain is filled but the details of how this occurs depend critically on the choice of boundary conditions. With periodic boundary conditions the snaking branches terminate on a branch of spatially periodic states. However, with non-Neumann boundary conditions they turn continuously into a large amplitude filling state that replaces the periodic state. This behavior, shown here in detail for the Swift-Hohenberg equation, explains the phenomenon of "snaking without bistability," recently observed in simulations of binary fluid convection by Mercader et al. Phys. Rev. E 80, 025201 (2009).
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Affiliation(s)
- S M Houghton
- School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom.
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25
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Mercader I, Batiste O, Alonso A, Knobloch E. Localized pinning states in closed containers: Homoclinic snaking without bistability. Phys Rev E Stat Nonlin Soft Matter Phys 2009; 80:025201. [PMID: 19792185 DOI: 10.1103/physreve.80.025201] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2008] [Revised: 06/25/2009] [Indexed: 05/28/2023]
Abstract
Binary mixtures with a negative separation ratio are known to exhibit time-independent spatially localized convection when heated from below. Numerical continuation of such states in a closed two-dimensional container with experimental boundary conditions and parameter values reveals the presence of a pinning region in Rayleigh number with multiple stable localized states but no bistability between the conduction state and an independent container-filling state. An explanation for this unusual behavior is offered.
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Affiliation(s)
- Isabel Mercader
- Departament de Física Aplicada, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
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Kozyreff G, Tlidi M, Mussot A, Louvergneaux E, Taki M, Vladimirov AG. Localized beating between dynamically generated frequencies. Phys Rev Lett 2009; 102:043905. [PMID: 19257421 DOI: 10.1103/physrevlett.102.043905] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/16/2008] [Indexed: 05/27/2023]
Abstract
We analyze the beating between intrinsic frequencies that are simultaneously generated by a modulation (Turing) instability in a nonlinear extended system. The model studied is that of a coherently driven photonic crystal fiber cavity. Beating in the form of a slow modulation of fast intensity oscillations is found to be stable for a wide range of parameters. We find that such beating can also be localized and contain only a finite number of slow modulations. These structures consist of dips in the amplitude of the fast intensity oscillations, which can either be isolated or regularly spaced. An asymptotic analysis close to the modulation instability threshold allows us to explain this phenomenon as a manifestation of homoclinic snaking for dissipative localized structures.
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Affiliation(s)
- G Kozyreff
- Optique Nonlinéaire Théorique, Université libre de Bruxelles, C.P. 231, Campus Plaine, B-1050 Bruxelles, Belgium
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Barbay S, Hachair X, Elsass T, Sagnes I, Kuszelewicz R. Homoclinic snaking in a semiconductor-based optical system. Phys Rev Lett 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] [What about the content of this article? (0)] [Affiliation(s)] [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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Bergeon A, Burke J, Knobloch E, Mercader I. Eckhaus instability and homoclinic snaking. Phys Rev E Stat Nonlin Soft Matter Phys 2008; 78:046201. [PMID: 18999502 DOI: 10.1103/physreve.78.046201] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2008] [Indexed: 05/27/2023]
Abstract
Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable, spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. This behavior is simplest to understand within the subcritical Swift-Hohenberg equation, but is also present in the subcritical regime of doubly diffusive convection driven by horizontal gradients. In systems that are unbounded in one spatial direction homoclinic snaking continues indefinitely as the localized structure grows to resemble a spatially periodic state of infinite extent. In finite domains or in periodic domains with finite spatial period the process must terminate. In this paper we show that the snaking branches in general turn over once the length of the localized state becomes comparable to the domain, and examine the factors that determine the location of the termination point or points, and their relation to the Eckhaus instability of the spatially periodic state.
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Affiliation(s)
- A Bergeon
- IMFT UMR CNRS 5502-UPS UFR MIG, 31062 Toulouse Cedex, France
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Clerc MG, Falcon C, Tirapegui E. Comment on "asymptotics of large bound States of localized structures". Phys Rev Lett 2008; 100:049401-049402. [PMID: 18352342 DOI: 10.1103/physrevlett.100.049401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/30/2007] [Indexed: 05/26/2023]
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Abstract
We investigate the dynamics of pattern-forming systems in large domains near a codimension-two point corresponding to a ‘strong spatial resonance’ where competing instabilities with wavenumbers in the ratio 1 : 2 or 1 : 3 occur. We supplement the standard amplitude equations for such a mode interaction with Ginzburg–Landau-type modulational terms, appropriate to pattern formation in a large domain. In cases where the coefficients of these new diffusive terms differ substantially from each other, we show that spatially periodic solutions found near onset may be unstable to two long-wavelength modulational instabilities. Moreover, these instabilities generically occur near the codimension-two point only in the 1 : 2 and 1 : 3 cases, and not when weaker spatial resonances arise. The first instability is ‘amplitude-driven’ and is the analogue of the well-known Turing instability of reaction–diffusion systems. The second is a phase instability for which the subsequent nonlinear development is described, at leading order, by the Cahn–Hilliard equation.
The normal forms for strong spatial resonances are also well known to permit uniformly travelling wave solutions. We also show that these travelling waves may be similarly unstable.
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Affiliation(s)
- J.H.P Dawes
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of CambridgeWilberforce Road, Cambridge CB3 0WA, UK
| | - M.R.E Proctor
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of CambridgeWilberforce Road, Cambridge CB3 0WA, UK
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Firth WJ, Columbo L, Scroggie AJ. Proposed resolution of theory-experiment discrepancy in homoclinic snaking. Phys Rev Lett 2007; 99:104503. [PMID: 17930391 DOI: 10.1103/physrevlett.99.104503] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/18/2007] [Indexed: 05/25/2023]
Abstract
In spatially extended Turing-unstable systems, parameter variation should, in theory, produce only fully developed patterns. In experiment, however, localized patterns or solitons sitting on a smooth background often appear. Addition of a nonlocal nonlinearity can resolve this discrepancy by tilting the "snaking" bifurcation diagram characteristic of such problems.
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Affiliation(s)
- W J Firth
- SUPA and Department of Physics, University of Strathclyde, Glasgow, Scotland, United Kingdom
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Abstract
The bistable Swift-Hohenberg equation exhibits multiple stable and unstable spatially localized states of arbitrary length in the vicinity of the Maxwell point between spatially homogeneous and periodic states. These states are organized in a characteristic snakes-and-ladders structure. The origin of this structure in one spatial dimension is reviewed, and the stability properties of the resulting states with respect to perturbations in both one and two dimensions are described. The relevance of the results to several different physical systems is discussed.
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Affiliation(s)
- John Burke
- Department of Physics, University of California, Berkeley, California 94720, USA.
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Abstract
We derive asymptotically an order parameter equation in the limit where nascent bistability and long-wavelength modulation instabilities coalesce. This equation is a variant of the Swift-Hohenberg equation that generally contains nonvariational terms of the form psinabla(2)psi and /nablapsi/(2). We briefly review some of the properties already derived for this equation and derive it on three examples taken from chemical, biological, and optical contexts. Finally, we derive the equation on a general class of partial differential systems.
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Affiliation(s)
- G Kozyreff
- Optique Nonlinéaire Théorique, Université Libre de Bruxelles (U.L.B.), Campus de la Plaine C.P. 231, Boulevard du Triomphe, B-1050 Bruxelles, Belgium
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Abstract
The existence of localized structures, including so-called cavity solitons, in driven optical systems is discussed. In theory, they should exist only below the threshold of a subcritical modulational instability, but in experiment they often appear spontaneously on parameter variation. The addition of a nonlocal nonlinearity may resolve this discrepancy by tilting the "snaking" bifurcation diagram characteristic of such problems.
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Affiliation(s)
- W J Firth
- SUPA and Department of Physics, University of Strathclyde, 107 Rottenrow, G4 0NG Glasgow, Scotland
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Yochelis A, Burke J, Knobloch E. Reciprocal oscillons and nonmonotonic fronts in forced nonequilibrium systems. Phys Rev Lett 2006; 97:254501. [PMID: 17280359 DOI: 10.1103/physrevlett.97.254501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/27/2006] [Indexed: 05/13/2023]
Abstract
The formation of oscillons in a synchronously oscillating background is studied in the context of both damped and self-exciting oscillatory media. Using the forced complex Ginzburg-Landau equation we show that such states bifurcate from finite amplitude homogenous states near the 2:1 resonance boundary. In each case we identify a region in parameter space containing a finite multiplicity of coexisting stable oscillons with different structure. Stable time-periodic monotonic and nonmonotonic frontlike states are present in an overlapping region. Both types of structure are related to the presence of a Maxwell point between the zero and finite amplitude homogeneous states.
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Affiliation(s)
- Arik Yochelis
- Department of Physics, University of California, Berkeley, California 94720, USA
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