Fokker-Planck equation approach to the description of soliton statistics in optical fiber transmission systems

We derive rigorously the Fokker-Planck equation that governs the statistics of soliton parameters in optical transmission lines in the presence of additive amplifier spontaneous emission. We demonstrate that these statistics are generally non-Gaussian. We present exact marginal probability-density functions for soliton parameters for some cases. A WKB approach is applied to describe the tails of the probability-density functions. © 2005 Optical Society of America.

Income and poverty in a developing economy

We present a stochastic agent-based model for the distribution of personal incomes in a developing economy. We start with the assumption that incomes are determined both by individual labour and by stochastic effects of trading and investment. The income from personal effort alone is distributed about a mean, while the income from trade, which may be positive or negative, is proportional to the trader's income. These assumptions lead to a Langevin model with multiplicative noise, from which we derive a Fokker-Planck (FP) equation for the income probability density function (IPDF) and its variation in time. We find that high earners have a power law income distribution while the low-income groups have a Levy IPDF. Comparing our analysis with the Indian survey data (obtained from the world bank website: http://go.worldbank.org/SWGZB45DN0) taken over many years we obtain a near-perfect data collapse onto our model's equilibrium IPDF. Using survey data to relate the IPDF to actual food consumption we define a poverty index (Sen A. K., Econometrica., 44 (1976) 219; Kakwani N. C., Econometrica, 48 (1980) 437), which is consistent with traditional indices, but independent of an arbitrarily chosen "poverty line" and therefore less susceptible to manipulation. Copyright © EPLA, 2010.

Statistics of three interacting optical solitons

We examine the statistics of three interacting optical solitons under the effects of amplifier noise and filtering. We derive rigorously the Fokker-Planck equation that governs the probability distribution of soliton parameters.

Statistics of three interacting optical solitons

We examine the statistics of three interacting optical solitons under the effects of amplifier noise and filtering. We derive rigorously the Fokker-Planck equation that governs the probability distribution of soliton parameters.

Statistics of a noise-driven Manakov soliton

We investigate the statistics of a vector Manakov soliton in the presence of additive Gaussian white noise. The adiabatic perturbation theory for a Manakov soliton yields a stochastic Langevin system which we analyse via the corresponding Fokker-Planck equation for the probability density function (PDF) for the soliton parameters. We obtain marginal PDFs for the soliton frequency and amplitude as well as soliton amplitude and polarization angle. We also derive formulae for the variances of all soliton parameters and analyse their dependence on the initial values of polarization angle and phase. © 2006 IOP Publishing Ltd.

Non-Gaussian statistics of an optical soliton in the presence of amplified spontaneous emission

A study was performed on non-Gaussian statistics of an optical soliton in the presence of amplified spontaneous emission. An approach based on the Fokker-Planck equation was applied to study the optical soliton parameters in the presence of additive noise. The rigorous method not only allowed to reproduce and justify the classical Gordon-Haus formula but also led to new exact results.

Stochastically forced dislocation density distribution in plastic deformation

The dynamical evolution of dislocations in plastically deformed metals is controlled by both deterministic factors arising out of applied loads and stochastic effects appearing due to fluctuations of internal stress. Such type of stochastic dislocation processes and the associated spatially inhomogeneous modes lead to randomness in the observed deformation structure. Previous studies have analyzed the role of randomness in such textural evolution but none of these models have considered the impact of a finite decay time (all previous models assumed instantaneous relaxation which is "unphysical") of the stochastic perturbations in the overall dynamics of the system. The present article bridges this knowledge gap by introducing a colored noise in the form of an Ornstein-Uhlenbeck noise in the analysis of a class of linear and nonlinear Wiener and Ornstein-Uhlenbeck processes that these structural dislocation dynamics could be mapped on to. Based on an analysis of the relevant Fokker-Planck model, our results show that linear Wiener processes remain unaffected by the second time scale in the problem but all nonlinear processes, both Wiener type and Ornstein-Uhlenbeck type, scale as a function of the noise decay time τ. The results are expected to ramify existing experimental observations and inspire new numerical and laboratory tests to gain further insight into the competition between deterministic and random effects in modeling plastically deformed samples.