Stochastic dissipative solitons (original) (raw)
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The European Physical …, 2010
We study the effects of noise on excitable DS found on nonlinear Kerr cavities, showing that the system exhibits coherence resonance, characterized by a maximum degree of regularity for intermediate noise intensities. This behavior is observed for two different ways of applying noise: an additive white uncorrelated spatio-temporal noise and including fluctuations in the intensity of an addressing beam.
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Physics Letters A, 2000
The conditions for trapping of Schrodinger solitons in a random potential are analysed. A stochastic ODE is derived for the position of the soliton centre and the behaviour of its solution is studied for different types of stochasticity. The problem of trapping or propagation of the solitons in the potential is connected with the calculation of the Lyapunov exponents for stochastic ODEs and rigorous criteria for trapping are provided. Possible applications to the problem of strong dispersion management are discussed. q
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A brief introduction is given to the concept of the soliton management, i.e., stable motion of localized pulses in media with strong periodic (or, sometimes, random) inhomogeneity, or conditions for the survival of solitons in models with strong time-periodic modulation of linear or nonlinear coefficients. It is demonstrated that a class of systems can be identified, in which solitons remain robust inherently coherent objects in seemingly "hostile" environments. Most physical models belonging to this class originate in nonlinear optics and Bose-Einstein condensation, although other examples are known too (in particular, in hydrodynamics). In this paper, the complexity of the soliton-management systems, and the robustness of solitons in them are illustrated using a recently explored fiber-optic setting combining a periodic concatenation of nonlinear and dispersive segments (the split-step model) for bimodal optical signals (i.e., ones with two polarizations of light), which includes the polarization mode dispersion, i.e., random linear mixing of the two polarization components at junctions between the fiber segment.
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The European Physical Journal Special Topics, 2013
We study the properties of the probability density function (PDF) of a bistable system driven by heavy tailed white symmetric Lévy noise. The shape of the stationary PDF is found analytically for the particular case of the Lévy index α = 1 (Cauchy noise). For an arbitrary Lévy index we employ numerical methods based on the solution of the stochastic Langevin equation and space fractional kinetic equation. In contrast with the bistable system driven by Gaussian noise, in the Lévy case the positions of maxima of the stationary PDF do not coincide with the positions of minima of the bistable potential. We provide a detailed study of the distance between the maxima and the minima as a function of the potential's depth and Lévy noise parameters.
Solitons in media with random dispersive perturbations
Physica D-nonlinear Phenomena, 1999
A statistical approach of the propagation of solitons in media with spatially random dispersive perturbations is developed. Applying the inverse scattering transform several regimes are put into evidence which are determined by the mass and the velocity of the incoming soliton and also by the correlation length of the perturbation. Namely, the mass of the soliton is almost conserved if it is initially large. If the initial mass is too small, then the mass decays with the length of the system. The decay rate is exponential in case of a white noise perturbation, but the mass will decrease as the inverse of the square root of the length if the central wave number of the soliton lies in the tail of the spectrum of the perturbation.
Solitons in regular and random split-step systems
Journal of the Optical Society of America B, 2003
ABSTRACT Fundamental properties of solitons in the recently introduced split-step model (SSM) are investigated. The SSM is a system that consists of periodically alternating dispersive and nonlinear segments, a period being of the same order of magnitude as the soliton’s dispersion length. The model including fiber loss and gain can always be reduced to its lossless version. First, we develop a variational approximation that makes it possible to explain the existence of SSM solitons that were originally found solely in numerical form. Overall dynamic behavior of a SSM is described by a phase diagram that identifies an established state (stationary soliton, breather with long-period oscillations, splitting into several pulses, or decay into radiation) depending on the amplitude and the width of the initial pulse. In particular, strong saturation in the dependence of the amplitude of the established soliton on the amplitude of the initial pulse is found. The results clearly show some similarities and drastic differences between the SSM and the ordinary soliton model based on the nonlinear Schrödinger equation. A random version of the SSM is introduced, with the length of the system’s cell uniformly distributed in some interval, which is a relevant case for applications to fiber-optic telecommunication networks. It is found that the dynamics of the SSM solitons as well as interactions between them in random systems (both single-channel and multichannel systems) are virtually the same as in their regular counterparts.
Noise-enhanced stability in fluctuating metastable states
Physical Review E, 2004
We derive general equations for the nonlinear relaxation time of Brownian diffusion in randomly switching potential with a sink. For piece-wise linear dichotomously fluctuating potential with metastable state, we obtain the exact average lifetime as a function of the potential parameters and the noise intensity. Our result is valid for arbitrary white noise intensity and for arbitrary fluctuation rate of the potential. We find noise enhanced stability phenomenon in the system investigated: the average lifetime of the metastable state is greater than the time obtained in the absence of additive white noise. We obtain the parameter region of the fluctuating potential where the effect can be observed. The system investigated also exhibits a maximum of the lifetime as a function of the fluctuation rate of the potential.
Physica A: Statistical Mechanics and its Applications, 2013
The stochastic resonance (SR) in bistable systems has been extensively discussed with the use of phenomenological Langevin models. By using the microscopic, generalized Caldeira-Leggett (CL) model, we study in this paper, SR of an open bistable system coupled to a bath with a nonlinear system-bath interaction. The adopted CL model yields the non-Markovian Langevin equation with nonlinear dissipation and state-dependent diffusion which preserve the fluctuationdissipation relation (FDR). From numerical calculations, we find the following: (1) the spectral power amplification (SPA) exhibits SR not only for a and b but also for τ while the stationary probability distribution function is independent of them where a (b) denotes the magnitude of multiplicative (additive) noise and τ expresses the relaxation time of colored noise; (2) the SPA for coexisting additive and multiplicative noises has a single-peak but two-peak structure as functions of a, b and/or τ. These results (1) and (2) are qualitatively different from previous ones obtained by phenomenological Langevin models where the FDR is indefinite or not held.