Fronts
                    <i>vs</i>
                    . Solitary Waves in Nonequilibrium Systems

Hakim, Vincent; Jakobsen, Per Kristen; Pomeau, Yves

doi:10.1209/0295-5075/11/1/004

Cited by 150 publications

(45 citation statements)

References 7 publications

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“…The continuous transition of these solutions from the conservative limit to the gradient limit in the parameter space of the CGLE has been studied in Ref. [16]. Although the dynamical properties of these pulselike solutions, their collisions and interactions are different from those of solitons of integrable systems, they have been called "solitons" in a number of works.…”

mentioning

confidence: 99%

Multisoliton Solutions of the Complex Ginzburg-Landau Equation

1997

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We present novel stable solutions which are soliton pairs and trains of the 1D complex GinzburgLandau equation (CGLE), and analyze them. We propose that the distance between the pulses and the phase difference between them is defined by energy and momentum balance equations. We present a two-dimensional phase plane ("interaction plane") for analyzing the stability properties and general dynamics of two-soliton solutions of the CGLE. [S0031-9007(97) [3], and spatially extended nonequilibrium systems [4,5]. In optics, it is useful in analyzing optical transmission lines [6,7], passively mode-locked fiber lasers [8,9], and spatial optical solitons [10]. In each case, the problem of the interaction of two, individually stable, juxtaposed elementary coherent structures (i.e., solitons) is crucial for understanding the general behavior of the system [11,12].Stable pulse-like solutions of the quintic CGLE have been found by Thual and Fauve [13]. Minimal requirements for their stability have been obtained in [14]. In the conservative limit, these solutions can be considered as perturbations of the nonlinear Schrödinger equation (NLSE) solitons [15]. The continuous transition of these solutions from the conservative limit to the gradient limit in the parameter space of the CGLE has been studied in Ref. [16]. Although the dynamical properties of these pulselike solutions, their collisions and interactions are different from those of solitons of integrable systems, they have been called "solitons" in a number of works. We follow this tradition, and also call them "solitons" or "soliton solutions."For the nonlinear Schrödinger equation, two solitons have zero binding energy. Hence, any nonlinear superposition of two solitons is neutrally stable, and can be made unstable with a very small perturbation. On the other hand, for the NLSE, there is no stationary solution in the form of two solitons with equal amplitudes and velocities and with a fixed separation. Frequently, real systems are not described by integrable equations (e.g., the NLSE), but by Hamiltonian generalizations of the NLSE. For these systems, the interaction between the pulses becomes inelastic, so that two-soliton solutions of the perturbed NLSE (when they exist) are unstable due to the energy exchange between the pulses [17]. The situation changes completely for nonconservative systems. Each soliton then has its own internal balance of energy which maintains its constant amplitude. Fixing the amplitudes effectively reduces the number of degrees of freedom in the system of two solitons and can make it stable. Bound states of two solitons in these systems were first analyzed by Malomed [18]. Using standard perturbation analysis for soliton interaction, he showed that stationary solutions in the form of bound states of two solitons, which are in-phase or out-of-phase, may exist. We also confirm that they do exist. However, careful numerical 0031-9007͞97͞79(21)͞4047(5)$10.00

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confidence: 99%

Multisoliton Solutions of the Complex Ginzburg-Landau Equation

1997

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“…The nonlinear interaction between the local amplitude and frequency seems to be the essential localization mechanism in this approximation. Indeed, one could find localized solutions of increasing length up to the limit of an infinitely long two-front state [59,74,75]. But some basical problems remained: Within the CGLE all pulses drift with the same velocity.…”

Section: E Comparison With Ltw Modelsmentioning

confidence: 99%

“…Furthermore, coexisting small stable and wide unstable LTWs were never seen in the CGLE but found in experiments by Kolodner [7]. Instead, stable broad pulse solutions are found in the model to arise in a saddle node bifurcation together with an unstable branch of smaller 'critical droplets' near the basic state [74]. Finally, numerical solutions of the field equations show the existence of stable LTWs even below the lowest TW saddle node [14].…”

Section: E Comparison With Ltw Modelsmentioning

confidence: 99%

Traveling wave fronts and localized traveling wave convection in binary fluid mixtures

Jung

Lücke

2005

Phys. Rev. E

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Nonlinear fronts between spatially extended traveling wave convection (TW) and quiescent fluid and spatially localized traveling waves (LTWs) are investigated in quantitative detail in the bistable regime of binary fluid mixtures heated from below. A finite-difference method is used to solve the full hydrodynamic field equations in a vertical cross section of the layer perpendicular to the convection roll axes. Results are presented for ethanol-water parameters with several strongly negative separation ratios where TW solutions bifurcate subcritically. Fronts and LTWs are compared with each other and similarities and differences are elucidated. Phase propagation out of the quiescent fluid into the convective structure entails a unique selection of the latter while fronts and interfaces where the phase moves into the quiescent state behave differently. Interpretations of various experimental observations are suggested.

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“…The GL equation is a ubiquitous model in many physical problems [56], and in different forms it appears as the simplest model for describing dissipative solitons [57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76], clusters of localized states rotating around a central vortex core [77], and laser patterns in cavities [78,79]. In appli- cation to waveguide arrays, a periodic index modulation can be modeled by a discrete GL equation that describes the presence of gain and loss due to optical amplifiers and saturable absorbers [42].…”

Section: Introductionmentioning

confidence: 99%

Collisions between discrete spatiotemporal dissipative Ginzburg-Landau solitons in two-dimensional photonic lattices

2009

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Abstract:Systematic results of collisions between discrete spatiotemporal dissipative Ginzburg-Landau solitons in two-dimensional photonic lattices are reported. The generic outcomes are identified for (i) the collision of two identical solitons located in the corner, at the edge, and in the center of the photonic lattice, and for (ii) the collision of two non-identical corner and edge solitons located at different distances from the boundaries of the photonic lattice. Depending on the values of the kick (collision momentum) and of the nonlinear (cubic) gain, the collision scenarios include soliton merging, creation of an extra soliton, soliton bouncing, soliton spreading, and quasi-elastic (symmetric) interactions. 05.45.Yv, 42.65.Tg, 42.65.Sf, 47.20 PACS (2008):

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Fronts vs . Solitary Waves in Nonequilibrium Systems

Abstract: Ginzburg-Landau amplitude equations are used here as a model for the nonlinear (') We refrain from using the word <

Cited by 150 publications

References 7 publications

Multisoliton Solutions of the Complex Ginzburg-Landau Equation

Multisoliton Solutions of the Complex Ginzburg-Landau Equation

Traveling wave fronts and localized traveling wave convection in binary fluid mixtures

Collisions between discrete spatiotemporal dissipative Ginzburg-Landau solitons in two-dimensional photonic lattices

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