Conditional probability calculations for the nonlinear Schr\"odinger equation with additive noise

TL;DR

This paper develops a method to compute the conditional probability density function for the nonlinear Schrödinger equation with additive noise, providing analytical results in the weakly nonlinear limit and illustrating applications in fiber-optic communications.

Contribution

It introduces a constructive approach to calculate conditional probability densities for the nonlinear Schrödinger equation with noise, including analytical derivations for weak nonlinearity.

Findings

Derived the conditional probability density function in the small noise limit.

Analytically obtained results for weakly nonlinear cases.

Applied the theory to fiber-optic communication scenarios.

Abstract

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