13 results on '"MODULATIONAL INSTABILITIES"'
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2. Vector solitons in an extended coupled Schrödinger equations with modulated nonlinearities.
- Author
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Zakeri, Gholam-Ali and Yomba, Emmanuel
- Subjects
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VECTOR analysis , *SOLITONS , *SCHRODINGER equation , *NONLINEAR theories , *QUINTIC equations , *COEFFICIENTS (Statistics) - Abstract
We developed a method to investigate the vector solitons in an extended non-autonomous coupled Schrödinger equations with cubic and quintic nonlinearities in which the external potentials of the model depend in space and time, and all the remaining of its coefficients are time dependent. We mapped this system into an autonomous one under a specific set of constraint conditions on the variable coefficients of the model and then analyzed the regions with modulational instabilities. Front, bright and dark vector solitons associated with the original system are obtained. Some applications and a numerical simulation based on a Fourier split-step method are discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
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3. A population-competition model for analyzing transverse optical patterns including optical control and structural anisotropy
- Author
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Y C Tse, Chris K P Chan, M H Luk, N H Kwong, P T Leung, R Binder, and Stefan Schumacher
- Subjects
modulational instabilities ,pattern competitions ,switching control and anisotropy ,steady state phase diagrams ,semiconductor microcavities ,Science ,Physics ,QC1-999 - Abstract
We present a detailed study of a low-dimensional population-competition (PC) model suitable for analysis of the dynamics of certain modulational instability patterns in extended systems. The model is applied to analyze the transverse optical exciton–polariton patterns in semiconductor quantum well microcavities. It is shown that, despite its simplicity, the PC model describes quite well the competitions among various two-spot and hexagonal patterns when four physical parameters, representing density saturation, hexagon stabilization, anisotropy, and switching beam intensity, are varied. The combined effects of the last three parameters are given detailed considerations here. Although the model is developed in the context of semiconductor polariton patterns, its equations have more general applicability, and the results obtained here may benefit the investigation of other pattern-forming systems. The simplicity of the PC model allows us to organize all steady state solutions in a parameter space ‘phase diagram’. Each region in the phase diagram is characterized by the number and type of solutions. The main numerical task is to compute inter-region boundary surfaces, where some steady states either appear, disappear, or change their stability status. The singularity types of the boundary points, given by Catastrophe theory, are shown to provide a simple geometric overview of the boundary surfaces. With all stable and unstable steady states and the phase boundaries delimited and characterized, we have attained a comprehensive understanding of the structure of the four-parameter phase diagram. We analyze this rich structure in detail and show that it provides a transparent and organized interpretation of competitions among various patterns built on the hexagonal state space.
- Published
- 2015
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4. Locking of domain walls and quadratic frequency combs in doubly resonant optical parametric oscillators
- Abstract
The formation of frequency combs (FCs) in high-Q microresonators with Kerr type of nonlinearity has attracted a lot of attention in the past decade [1]. Recently it has been shown that FCs can be also generated in dissipative dispersive cavities with quadratic nonlinearities [2,3], opening a new possibility of generating combs in previously unattainable spectral regions. Previous work has shown that modulational instability (MI) induces pattern and FC formation in degenerate optical parametric oscillators (OPOs) [4]. However, the existence of dissipative solitons or localized structures (LSs) is still unclear., info:eu-repo/semantics/published
- Published
- 2019
5. Bifurcation structure of localized patterns and spikes in dispersive Kerr cavities
- Abstract
In the past ten years externally driven dispersive cavities with Kerr type of nonlinearity received a lot of attention as sources for optical frequency comb generation. These combs consist in very sharp equidistant spectral lines that can be used to measure light frequencies with extremely high accuracy, finding applications in broad areas of science and technology [1]. These combs correspond to the frequency spectra of the light patterns circulating inside the cavity, and therefore it is possible to understand the dynamics and stability of the former by studying the latter. Since these cavities are dissipative, light patterns are commonly called dissipative structures (DSs). The dynamics of the DSs are well described by the Lugiato-Lefever (LL) equation [2], that in its normalized form can be written as, info:eu-repo/semantics/published
- Published
- 2019
6. Theory of pattern forming systems under traveling-wave forcing
- Author
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Rüdiger, Sten, Nicola, Ernesto M., Casademunt, Jaume, and Kramer, Lorenz
- Subjects
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FORCING (Model theory) , *PROPERTIES of matter , *SOLUTION (Chemistry) , *EQUATIONS - Abstract
Abstract: The response of pattern forming systems to external forcing either spatial or temporal has received much attention for several decades. Combined spatio-temporal forcing has only been introduced recently, in particular in the form of a spatially resonant traveling-wave forcing [S. Rüdiger, D.G. Míguez, A.P. Muñuzuri, F. Sagués, J. Casademunt, Phys. Rev. Lett. 90 (2003) 128301]. Since then, both a series of experiments, in the context of Turing patterns in reaction–diffusion systems, and the development of the corresponding generic theory, have unveiled a wealth of new and unexpected phenomena. In this article we review these phenomena, we provide a unified and comprehensive description of them, and extend the theoretical analysis to new situations. We formulate the generic amplitude equations for different orders of spatial resonance for 1d and 2d patterns (stripes and hexagons). We identify and describe in detail the autonomous dynamical system which underlies the phenomenon of traveling-stripe resonance. For 1d we focus on localized solutions (kinks and pulses), their dynamics and their interaction for both 1:1 and 2:1 resonance. Specifically, we discuss the effect of wave-number mismatch (inexact resonance) combined with the non-gradient dynamics induced by the motion of the forcing, resulting in non-trivial interactions and complex spatio-temporal dynamics. Analytical results in the phase approximation are obtained, while numerical techniques are used to study the complete problem. We show that defect interaction is oscillatory with distance, allowing for the existence of locked chaotic wave trains. In 2d we discuss the modulational instabilities of striped patterns and focus mostly on the emergence of hexagons induced by traveling stripe forcing, for exact 1:1 resonance. In this case, we examine in detail the complex bifurcation scenario beyond primary instabilities. We finally discuss the problem of traveling-wave forcing in the context of a specific system, the photosensitive CDIMA reaction, where most experiments have been carried out. We compare experiments and theoretical predictions and propose other experimental systems where the study could be extended. Finally, we review related work by other authors and discuss possible further developments and open questions which hold the promise of new interesting findings. [Copyright &y& Elsevier]
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- 2007
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7. Modulational instability in the nonlocal -model
- Author
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Wyller, John, Królikowski, Wiesław Z., Bang, Ole, Petersen, Dan Erik, and Rasmussen, Jens Juul
- Subjects
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SOLITONS , *GEOMETRIC connections , *NONLINEAR theories , *THEORY of wave motion - Abstract
Abstract: We investigate in detail the linear regime of the modulational instability (MI) properties of the plane waves of the nonlocal model for -media formulated in Nikolov et al. [N.I. Nikolov, D. Neshev, O. Bang, W.Z. Królikowski, Quadratic solitons as nonlocal solitons, Phys. Rev. E 68 (2003) 036614; I.V. Shadrivov, A.A. Zharov, Dynamics of optical spatial solitons near the interface between two quadratically nonlinear media, J. Opt. Soc. Amer. B 19 (2002) 596–602]. It is shown that the MI is of the oscillatory type and of finite bandwidth. Moreover, it is possible to identify regions in the parameter space for which a fundamental gain band exists, and regions for which higher order gain bands and modulational stability exist. We also show that the MI analysis for the nonlocal model is applicable in the finite walk-off case. Finally, we show that the plane waves of the nonlocal -model are recovered as the asymptotic limit of one of the branches of the plane waves (i.e. the adiabatic branch or the acoustic branch) of the full -model by means of a singular perturbational approach. It is also proven that the stability results for the adiabatic branch continuously approach those of the nonlocal -model, by using the singular perturbational approach. The other branch of the plane waves (i.e. the nonadiabatic branch or the optical branch) is always modulationally unstable. We compare the MI results for the adiabatic branch with the predictions obtained from the full -model in the non-walk-off limit. It is concluded that for most physical relevant parameter regimes it suffices to use the nonlocal model in order to determine the MI properties. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
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8. Modulation of electron-acoustic waves
- Author
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Shukla, P.K., Hellberg, M.A., and Stenflo, L.
- Subjects
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ION acoustic waves , *IONOSPHERE - Abstract
FAST observations have indicated signatures of large amplitude solitary waves in the auroral zone of the earth's ionosphere. Our objective here is to propose a model for the generation of density cavities by the ponderomotive force of electron-acoustic waves. For this purpose, we derive a nonlinear Schro¨dinger equation for the electron-acoustic wave envelope as well as a driven (by the electron-acoustic wave ponderomotive force) ion-acoustic wave equation. Possible stationary solutions of our coupled equations are obtained. [Copyright &y& Elsevier]
- Published
- 2003
- Full Text
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9. Propagation of partially incoherent light in nonlinear media via the Wigner transform method.
- Author
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Helczynski, L., Anderson, D., Fedele, R., Hall, B., and Lisak, M.
- Abstract
The propagation of partially incoherent light in nonlinear media is analyzed using the Wigner transform method. The power and versatility of this approach is illustrated by several examples which clearly demonstrate how partial incoherence tends to suppress coherent instabilities by weakening the nonlinearity. In particular, it is found that the effect of partial incoherence on modulational instabilities can be described in terms of a Landau-like damping effect, which counteracts the coherent growth rate of the instability. Similarly, in the case of the self-focusing collapse instability, the nonlinear focusing effect becomes successively smaller as the coherence length of the light decreases and eventually no collapse phenomenon occurs [ABSTRACT FROM PUBLISHER]
- Published
- 2002
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10. Swift-Hohenberg equation with third-order dispersion for optical fiber resonators
- Author
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A. Hariz, L. Cherbi, B. Kostet, L. Bahloul, Marcel G. Clerc, M. A. Ferre, Krassimir Panajotov, Etienne Averlant, Mustapha Tlidi, Brussels Photonics Team, and Applied Physics and Photonics
- Subjects
Physics ,Optical fiber ,Bistability ,SUPERCONTINUUM GENERATION ,MODULATIONAL INSTABILITIES ,LOCALIZED STRUCTURES ,TRANSVERSE PATTERNS ,SOLITONS ,DYNAMICS ,Physics::Optics ,01 natural sciences ,010305 fluids & plasmas ,law.invention ,Swift–Hohenberg equation ,Nonlinear system ,Modulational instability ,Resonator ,Dissipative Solitons ,law ,Quantum electrodynamics ,0103 physical sciences ,Dissipative system ,Nonlinear Cavity ,010306 general physics ,Swift-Hohenberg equation ,Photonic-crystal fiber - Abstract
We investigate the dynamics of a ring cavity made of photonic crystal fiber and driven by a coherent beam working near to the resonant frequency of the cavity. By means of a multiple-scale reduction of the Lugiato-Lefever equation with high-order dispersion, we show that the dynamics of this optical device, when operating close to the critical point associated with bistability, is captured by a real order parameter equation in the form of a generalized Swift-Hohenberg equation. A Swift-Hohenberg equation has been derived for several areas of nonlinear science such as chemistry, biology, ecology, optics, and laser physics. However, the peculiarity of the obtained generalized Swift-Hohenberg equation for photonic crystal fiber resonators is that it possesses a third-order dispersion. Based on a weakly nonlinear analysis in the vicinity of the modulational instability threshold, we characterize the motion of dissipative structures by estimating their propagation speed. Finally, we numerically investigate the formation of moving temporal localized structures often called cavity solitons.
- Published
- 2019
11. Stable and unstable time quasi periodic solutions for a system of coupled NLS equations
- Author
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Grébert, Benoît, Vilaça Da Rocha, Victor, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Basque Center for Applied Mathematics (BCAM), Basque Center for Applied Mathematics, ANR-15-CE40-0001,BEKAM,Au-delà de la théorie KAM(2015), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), European Project: 669689,H2020,ERC-2014-ADG,HADE(2015), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
EDP non linéaires ,modulational instabilities ,Small amplitude solutions ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Solutions de faible amplitude ,KAM theory ,nonlinear PDE ,Instabilitié modulationnelle ,Physics::Plasma Physics ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,Théorie KAM ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Hamiltonian systems ,Systèmes hamiltoniens - Abstract
International audience; We prove that a system of coupled nonlinear Schrödinger equations on the torus exhibits both stable and unstable small KAM tori. In particular the unstable tori are related to a beating phenomena which has been proved recently in [6]. This is the first example of unstable tori for a 1d PDE.; Nous prouvons qu'un système d'équations couplées de Schrödinger sur le tore exhibe à la fois des tores KAM stables et instables. En particulier, les tores instables sont reliés au phénomène de battement qui a été récemment prouvé dans [6]. C'est le premier exemple de tore instable pour une EDP en dimension 1.
- Published
- 2018
12. Origin and stability of dark pulse Kerr frequency combs in normal dispersion microresonators
- Abstract
We theoretically analyze dark pulse Kerr frequency combs in normal dispersion microresonators. A wide range of dark pulses of different widths are found to coexist, and can be described as interlocked switching waves., info:eu-repo/semantics/published
- Published
- 2016
13. On homoclinic snaking in optical systems
- Author
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William J. Firth, Lorenzo Columbo, and Tommaso Maggipinto
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Physics ,LOCALIZED STRUCTURES ,Applied Mathematics ,DYNAMICAL PROPERTIES ,General Physics and Astronomy ,Pattern formation ,Statistical and Nonlinear Physics ,MODULATIONAL INSTABILITIES ,Bifurcation diagram ,Nonlinear system ,Modulational instability ,Classical mechanics ,Quantum mechanics ,Homoclinic orbit ,BULK SEMICONDUCTOR MICROCAVITIES ,Nonlinear Sciences::Pattern Formation and Solitons ,CAVITY SOLITONS ,Mathematical Physics ,Bifurcation - 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. (c) 2007 American Institute of Physics.
- Published
- 2007
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