Conditional probability calculations for the nonlinear Schrödinger equation with additive noise

The method for the computation of the conditional probability density function for the nonlinear Schrödinger equation with additive noise is developed. We present in a constructive form the conditional probability density function in the limit of small noise and analytically derive it in a weakly nonlinear case. The general theory results are illustrated using fiber-optic communications as a particular, albeit practically very important, example

Wave kinetics of random fibre lasers

Traditional wave kinetics describes the slow evolution of systems with many degrees of freedom to equilibrium via numerous weak non-linear interactions and fails for very important class of dissipative (active) optical systems with cyclic gain and losses, such as lasers with non-linear intracavity dynamics. Here we introduce a conceptually new class of cyclic wave systems, characterized by non-uniform double-scale dynamics with strong periodic changes of the energy spectrum and slow evolution from cycle to cycle to a statistically steady state. Taking a practically important example—random fibre laser—we show that a model describing such a system is close to integrable non-linear Schrödinger equation and needs a new formalism of wave kinetics, developed here. We derive a non-linear kinetic theory of the laser spectrum, generalizing the seminal linear model of Schawlow and Townes. Experimental results agree with our theory. The work has implications for describing kinetics of cyclical systems beyond photonics