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Descalzi O, Carvalho MI, Facão M, Brand HR. Dissipative solitons stabilized by nonlinear gradient terms: Time-dependent behavior and generic properties. CHAOS (WOODBURY, N.Y.) 2022; 32:123107. [PMID: 36587340 DOI: 10.1063/5.0118348] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/04/2022] [Accepted: 11/07/2022] [Indexed: 06/17/2023]
Abstract
We study the time-dependent behavior of dissipative solitons (DSs) stabilized by nonlinear gradient terms. Two cases are investigated: first, the case of the presence of a Raman term, and second, the simultaneous presence of two nonlinear gradient terms, the Raman term and the dispersion of nonlinear gain. As possible types of time-dependence, we find a number of different possibilities including periodic behavior, quasi-periodic behavior, and also chaos. These different types of time-dependence are found to form quite frequently from a window structure of alternating behavior, for example, of periodic and quasi-periodic behaviors. To analyze the time dependence, we exploit extensively time series and Fourier transforms. We discuss in detail quantitatively the question whether all the DSs found for the cubic complex Ginzburg-Landau equation with nonlinear gradient terms are generic, meaning whether they are stable against structural perturbations, for example, to the additions of a small quintic perturbation as it arises naturally in an envelope equation framework. Finally, we examine to what extent it is possible to have different types of DSs for fixed parameter values in the equation by just varying the initial conditions, for example, by using narrow and high vs broad and low amplitudes. These results indicate an overlapping multi-basin structure in parameter space.
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Affiliation(s)
- Orazio Descalzi
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Av. Mons. Álvaro del Portillo 12.455, Las Condes, Santiago, Chile
| | - M I Carvalho
- DEEC/FEUP and INESC TEC. Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
| | - M Facão
- Departamento de Física, Universidade de Aveiro and I3N Campus Universitário de Santiago, 3810-193 Aveiro, Portugal
| | - Helmut R Brand
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
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Zemskov EP, Tsyganov MA, Kassner K, Horsthemke W. Nonlinear waves in a quintic FitzHugh-Nagumo model with cross diffusion: Fronts, pulses, and wave trains. CHAOS (WOODBURY, N.Y.) 2021; 31:033141. [PMID: 33810726 DOI: 10.1063/5.0043919] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/12/2021] [Accepted: 03/02/2021] [Indexed: 06/12/2023]
Abstract
We study a tristable piecewise-linear reaction-diffusion system, which approximates a quintic FitzHugh-Nagumo model, with linear cross-diffusion terms of opposite signs. Basic nonlinear waves with oscillatory tails, namely, fronts, pulses, and wave trains, are described. The analytical construction of these waves is based on the results for the bistable case [Zemskov et al., Phys. Rev. E 77, 036219 (2008) and Phys. Rev. E 95, 012203 (2017) for fronts and for pulses and wave trains, respectively]. In addition, these constructions allow us to describe novel waves that are specific to the tristable system. Most interesting is the pulse solution with a zigzag-shaped profile, the bright-dark pulse, in analogy with optical solitons of similar shapes. Numerical simulations indicate that this wave can be stable in the system with asymmetric thresholds; there are no stable bright-dark pulses when the thresholds are symmetric. In the latter case, the pulse splits up into a tristable front and a bistable one that propagate with different speeds. This phenomenon is related to a specific feature of the wave behavior in the tristable system, the multiwave regime of propagation, i.e., the coexistence of several waves with different profile shapes and propagation speeds at the same values of the model parameters.
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Affiliation(s)
- Evgeny P Zemskov
- Federal Research Center for Computer Science and Control, Russian Academy of Sciences, Vavilova 40, 119333 Moscow, Russia
| | - Mikhail A Tsyganov
- Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Institutskaya 3, 142290 Pushchino, Moscow Region, Russia
| | - Klaus Kassner
- Institut für Physik, Otto-von-Guericke Universität, Universitätsplatz 2, 39106 Magdeburg, Germany
| | - Werner Horsthemke
- Department of Chemistry, Southern Methodist University, Dallas, Texas 75275-0314, USA
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Descalzi O, Brand HR. Breaking of symmetry of interacting dissipative solitons can lead to partial annihilation. Phys Rev E 2020; 101:040201. [PMID: 32422738 DOI: 10.1103/physreve.101.040201] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/23/2019] [Accepted: 04/05/2020] [Indexed: 11/07/2022]
Abstract
We show that for a large range of approach velocities and over a large interval of stabilizing cubic cross-coupling between counterpropagating waves, a collision of stationary pulses leads to a partial annihilation of pulses via a spontaneous breaking of symmetry. This result arises for coupled cubic-quintic complex Ginzburg-Landau equations for traveling waves for sufficiently large values of the stabilizing cubic cross-coupling and for large enough approach velocities of the pulses. Briefly, we show in addition that the collision of counterpropagating pulses in a system of two coupled cubic Ginzburg-Landau equations with nonlinear gradients (Raman effect) might also lead to partial annihilation, indicating that this breaking of symmetry is generic. Systems of experimental interest include surface reactions, convective onset, biosolitons, and nonlinear optics.
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Affiliation(s)
- Orazio Descalzi
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Avenida Monseñor Álvaro del Portillo 12.455, Las Condes, Santiago, Chile.,Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
| | - Helmut R Brand
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
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Zemskov EP, Tsyganov MA, Horsthemke W. Oscillatory multipulsons: Dissipative soliton trains in bistable reaction-diffusion systems with cross diffusion of attractive-repulsive type. Phys Rev E 2020; 101:032208. [PMID: 32289978 DOI: 10.1103/physreve.101.032208] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2019] [Accepted: 02/24/2020] [Indexed: 06/11/2023]
Abstract
One-dimensional localized sequences of bound (coupled) traveling pulses, wave trains with a finite number of pulses, are described in a piecewise-linear reaction-diffusion system of the FitzHugh-Nagumo type with linear cross-diffusion terms of opposite signs. The simplest case of two bound pulses, the paired-pulse waves (pulse pairs), is solved analytically. The solutions contain oscillatory tails in the wave profiles so that the pulse pairs consist of a double-peak core and wavy edges. Several pulse pairs with different profile shapes and propagation speeds can appear for the same parameter values of the model when the cross diffusion is dominant. The more general case of many bound pulses, multipulse waves, is studied numerically. It is shown that, dependent on the values of the cross-diffusion coefficients, the multipulse waves upon collision can pass through one another with unchanged size and shape, exhibiting soliton behavior. Moreover, multipulse collisions with the system boundaries can generate a rich variety of wave transformations: the transition from the multipulse waves to pulse-front waves and further to simple fronts or to annihilation as well the transition to solitary pulses or to multipulse waves with lower numbers of pulses. Analytical and numerical results for the pulse pairs agree well with each other.
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Affiliation(s)
- Evgeny P Zemskov
- Federal Research Center for Computer Science and Control, Russian Academy of Sciences, Vavilova 40, 119333 Moscow, Russia
| | - Mikhail A Tsyganov
- Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Institutskaya 3, 142290 Pushchino, Moscow Region, Russia
| | - Werner Horsthemke
- Department of Chemistry, Southern Methodist University, Dallas, Texas 75275-0314, USA
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Zadorin AS, Rondelez Y, Gines G, Dilhas V, Urtel G, Zambrano A, Galas JC, Estevez-Torres A. Synthesis and materialization of a reaction-diffusion French flag pattern. Nat Chem 2017; 9:990-996. [PMID: 28937677 DOI: 10.1038/nchem.2770] [Citation(s) in RCA: 67] [Impact Index Per Article: 9.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2016] [Accepted: 03/16/2017] [Indexed: 12/25/2022]
Abstract
During embryo development, patterns of protein concentration appear in response to morphogen gradients. These patterns provide spatial and chemical information that directs the fate of the underlying cells. Here, we emulate this process within non-living matter and demonstrate the autonomous structuration of a synthetic material. First, we use DNA-based reaction networks to synthesize a French flag, an archetypal pattern composed of three chemically distinct zones with sharp borders whose synthetic analogue has remained elusive. A bistable network within a shallow concentration gradient creates an immobile, sharp and long-lasting concentration front through a reaction-diffusion mechanism. The combination of two bistable circuits generates a French flag pattern whose 'phenotype' can be reprogrammed by network mutation. Second, these concentration patterns control the macroscopic organization of DNA-decorated particles, inducing a French flag pattern of colloidal aggregation. This experimental framework could be used to test reaction-diffusion models and fabricate soft materials following an autonomous developmental programme.
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Affiliation(s)
- Anton S Zadorin
- Laboratoire Jean Perrin, Université Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France.,CNRS, UMR 8237, 75005 Paris, France
| | - Yannick Rondelez
- LIMMS/CNRS-IIS, University of Tokyo, Komaba 4-6-2 Meguro-ku, Tokyo, Japan.,Ecole supérieure de physique et chimie industrielles, Laboratoire Gulliver, 10 rue Vauquelin, 75005, Paris, France
| | - Guillaume Gines
- LIMMS/CNRS-IIS, University of Tokyo, Komaba 4-6-2 Meguro-ku, Tokyo, Japan
| | - Vadim Dilhas
- Laboratoire Jean Perrin, Université Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France.,CNRS, UMR 8237, 75005 Paris, France
| | - Georg Urtel
- Ludwig-Maximilians-Universität München, Fakultät für Physik, Amalienstraße 54, 80799 Munich, Germany
| | - Adrian Zambrano
- Laboratoire Jean Perrin, Université Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France.,CNRS, UMR 8237, 75005 Paris, France
| | - Jean-Christophe Galas
- Laboratoire Jean Perrin, Université Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France.,CNRS, UMR 8237, 75005 Paris, France
| | - André Estevez-Torres
- Laboratoire Jean Perrin, Université Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France.,CNRS, UMR 8237, 75005 Paris, France
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Descalzi O, Cartes C, Brand HR. Multiplicative noise can lead to the collapse of dissipative solitons. Phys Rev E 2016; 94:012219. [PMID: 27575135 DOI: 10.1103/physreve.94.012219] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2016] [Indexed: 06/06/2023]
Abstract
We investigate the influence of spatially homogeneous multiplicative noise on the formation of localized patterns in the framework of the cubic-quintic complex Ginzburg-Landau equation. We find that for sufficiently large multiplicative noise the formation of stationary and temporally periodic dissipative solitons is suppressed. This result is characterized by a linear relation between the bifurcation parameter and the noise amplitude required for suppression. For the regime associated with exploding dissipative solitons we find a reduction in the number of explosions for larger noise strength as well as a conversion to other types of dissipative solitons or to filling-in and eventually a collapse to the zero solution.
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Affiliation(s)
- Orazio Descalzi
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Avenida Monseñor Álvaro del Portillo 12.455, Las Condes, Santiago, Chile
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
| | - Carlos Cartes
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Avenida Monseñor Álvaro del Portillo 12.455, Las Condes, Santiago, Chile
| | - Helmut R Brand
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
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Descalzi O, Brand HR. Non-unique results of collisions of quasi-one-dimensional dissipative solitons. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2015; 373:rsta.2015.0115. [PMID: 26527813 DOI: 10.1098/rsta.2015.0115] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 06/10/2015] [Indexed: 06/05/2023]
Abstract
We investigate collisions of quasi-one-dimensional dissipative solitons (DSs) for a large class of initial conditions, which are not temporally asymptotic quasi-one-dimensional DSs. For the case of sufficiently small approach velocity and sufficiently large values of the dissipative cross-coupling between the counter-propagating DSs, we find non-unique results for the outcome of collisions. We demonstrate that these non-unique results are intrinsically related to a modulation instability along the crest of the quasi-one-dimensional objects. As a model, we use coupled cubic-quintic complex Ginzburg-Landau equations. Among the final results found are stationary and oscillatory compound states as well as more complex assemblies consisting of quasi-one-dimensional and localized states. We analyse to what extent the final results can be described by the solutions of one cubic-quintic complex Ginzburg-Landau equation with effective parameters.
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Affiliation(s)
- Orazio Descalzi
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Av. Mons. Álvaro del Portillo 12.455, Las Condes, Santiago, Chile Department of Physics, University of Bayreuth, Bayreuth 95440, Germany
| | - Helmut R Brand
- Department of Physics, University of Bayreuth, Bayreuth 95440, Germany
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Descalzi O, Cartes C, Brand HR. Noisy localized structures induced by large noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:020901. [PMID: 25768449 DOI: 10.1103/physreve.91.020901] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2014] [Indexed: 06/04/2023]
Abstract
We investigate the influence of large noise on the formation of localized patterns in the framework of the cubic-quintic complex Ginzburg-Landau equation. The interaction of localization and noise can lead to filling in or noisy localized structures for fixed noise strength. To focus on the interaction between noise and localization we cover a region in parameter space, in particular, subcriticality, for which stationary stable deterministic pulses do not exist. Possible experimental tests of the work presented for autocatalytic chemical reactions and bioinspired systems are outlined.
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Affiliation(s)
- Orazio Descalzi
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Av. Mons. Álvaro del Portillo 12.455, Las Condes, Santiago, Chile
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
| | - Carlos Cartes
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Av. Mons. Álvaro del Portillo 12.455, Las Condes, Santiago, Chile
| | - Helmut R Brand
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
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