1
|
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.
Collapse
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
| |
Collapse
|
2
|
Descalzi O, Cartes C, Brand HR. Oscillatory dissipative solitons stabilized by nonlinear gradient terms: The transition to localized spatiotemporal disorder. Phys Rev E 2022; 105:L062201. [PMID: 35854493 DOI: 10.1103/physreve.105.l062201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2022] [Accepted: 05/24/2022] [Indexed: 11/07/2022]
Abstract
We investigate properties of oscillatory dissipative solitons (DSs) in a cubic complex Ginzburg-Landau equation stabilized by nonlinear gradient terms. As a main result we find a transition to dissipative solitons with spatiotemporal disorder as a function of the diffusion coefficient. This transition proceeds via quasiperiodicity and shows incommensurate satellites next to the fundamental frequency and its harmonics indicating a possible route to localized spatiotemporal chaos. The transition back to oscillatory DSs follows a similar scenario.
Collapse
Affiliation(s)
- Orazio Descalzi
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, 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, Chile
| | - Helmut R Brand
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
| |
Collapse
|
3
|
Descalzi O, Cartes C, Brand HR. Multiplicative noise can induce a velocity change of propagating dissipative solitons. Phys Rev E 2021; 103:L050201. [PMID: 34134314 DOI: 10.1103/physreve.103.l050201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/12/2020] [Accepted: 04/13/2021] [Indexed: 11/07/2022]
Abstract
We investigate the influence of spatially homogeneous multiplicative noise on propagating dissipative solitons (DSs) of the cubic complex Ginzburg-Landau equation stabilized by nonlinear gradient terms. Here we focus on the nonlinear gradient terms, in particular on the influence of the Raman term and the delayed nonlinear gain. We show that a fairly small amount of multiplicative noise can lead to a change in the mean velocity for such systems. This effect is exclusively due to the presence of the stabilizing nonlinear gradient terms. For a range of parameters we find a velocity change proportional to the noise intensity for the Raman term and for delayed nonlinear gain. We note that the dissipative soliton decreases the modulus of its velocity when only one type of nonlinear gradient is present. We present a straightforward mean field analysis to capture this simple scaling law. At sufficiently high noise strength the nonlinear gradient stabilized DSs collapse.
Collapse
Affiliation(s)
- Orazio Descalzi
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, 7620001, 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, 7620001, Santiago, Chile
| | - Helmut R Brand
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
| |
Collapse
|
4
|
Descalzi O, Cartes C, Brand HR. Interaction of dissipative solitons stabilized by nonlinear gradient terms. Phys Rev E 2021; 103:042215. [PMID: 34005884 DOI: 10.1103/physreve.103.042215] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/06/2020] [Accepted: 04/02/2021] [Indexed: 06/12/2023]
Abstract
We study the interaction of stable dissipative solitons of the cubic complex Ginzburg-Landau equation which are stabilized only by nonlinear gradient terms. In this paper we focus for the interactions in particular on the influence of the nonlinear gradient term associated with the Raman effect. Depending on its magnitude, we find up to seven possible outcomes of theses collisions: Stationary bound states, oscillatory bound states, meandering oscillatory bound states, bound states with large-amplitude oscillations, partial annihilation, complete annihilation, and interpenetration. Detailed results and their analysis are presented for one value of the corresponding nonlinear gradient term, while the results for two other values are just mentioned briefly. We compare our results with those obtained for coupled cubic-quintic complex Ginzburg-Landau equations and with the cubic-quintic complex Swift-Hohenberg equation. It turns out that both meandering oscillatory bound states as well as bound states with large-amplitude oscillations appear to be specific for coupled cubic complex Ginzburg-Landau equations with a stabilizing cubic nonlinear gradient term. Remarkably, we find for the large-amplitude oscillations a linear relationship between oscillation amplitude and period.
Collapse
Affiliation(s)
- Orazio Descalzi
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Santiago 7620001, 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, Santiago 7620001, Chile
| | - Helmut R Brand
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
| |
Collapse
|
5
|
Clerc MG, Coullet P, Rojas RG, Tlidi M. Introduction to Focus Issue: Instabilities and nonequilibrium structures. CHAOS (WOODBURY, N.Y.) 2020; 30:110401. [PMID: 33261348 DOI: 10.1063/5.0033273] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/14/2020] [Accepted: 10/15/2020] [Indexed: 06/12/2023]
Abstract
This Focus Issue on instabilities and nonequilibrium structures includes invited contributions from leading researchers across many different fields. The issue was inspired in part by the "VII Instabilities and Nonequilibrium Structures 2019" conference that took place at the Pontifica Universidad Católica de Valparaiso, Chile in December 2019. The conference, which is devoted to nonlinear science, is one of the oldest conferences in South America (since December 1985). This session has an exceptional character since it coincides with the 80th anniversary of Professor Enrique Tirapegui. We take this opportunity to highlight Tirapegui's groundbreaking contributions in the field of random perturbations experienced by macroscopic systems and in the formation of spatiotemporal structures in such systems operating far from thermodynamic equilibrium. This issue addresses a cross-disciplinary area of research as can be witnessed by the diversity of systems considered from inert matter such as photonics, chemistry, and fluid dynamics, to biology.
Collapse
Affiliation(s)
- Marcel G Clerc
- Departamento de Física and Millennium Institute for Research in Optics, FCFM, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - Pierre Coullet
- Institut de Physique de Nice, CNRS, Université Côte d'Azur, Nice, France
| | - Rene G Rojas
- Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile
| | - Mustapha Tlidi
- Faculté des Sciences, Université Libre de Bruxelles (U.L.B.), CP 231, Campus Plaine, B-1050 Bruxelles, Belgium
| |
Collapse
|
6
|
Clerc MG, Coulibaly S, Parra-Rivas P, Tlidi M. Nonlocal Raman response in Kerr resonators: Moving temporal localized structures and bifurcation structure. CHAOS (WOODBURY, N.Y.) 2020; 30:083111. [PMID: 32872794 DOI: 10.1063/5.0007350] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/26/2020] [Accepted: 07/13/2020] [Indexed: 06/11/2023]
Abstract
A ring resonator made of a silica-based optical fiber is a paradigmatic system for the generation of dissipative localized structures or dissipative solitons. We analyze the effect of the non-instantaneous nonlinear response of the fused silica or the Raman response on the formation of localized structures. After reducing the generalized Lugiato-Lefever to a simple and generic bistable model with a nonlocal Raman effect, we investigate analytically the formation of moving temporal localized structures. This reduction is valid close to the nascent bistability regime, where the system undergoes a second-order critical point marking the onset of a hysteresis loop. The interaction between fronts allows for the stabilization of temporal localized structures. Without the Raman effect, moving temporal localized structures do not exist, as shown in M. G. Clerc, S. Coulibaly, and M. Tlidi, Phys. Rev. Res. 2, 013024 (2020). The detailed derivation of the speed and the width associated with these structures is presented. We characterize numerically in detail the bifurcation structure and stability associated with the moving temporal localized states. The numerical results of the governing equations are in close agreement with analytical predictions.
Collapse
Affiliation(s)
- 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
| | - S Coulibaly
- Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59000 Lille, France
| | - P Parra-Rivas
- OPERA-photonique, Université libre de Bruxelles, 50 Avenue F. D. Roosevelt, CP 194/5, B-1050 Bruxelles, Belgium
| | - M Tlidi
- Faculté des Sciences, Université Libre de Bruxelles (U.L.B), CP 231, Campus Plaine, B-1050 Bruxelles, Belgium
| |
Collapse
|