Non-Gaussian statistics of amplifier spontaneous emission

In this paper, we study soliton statistics rigorously deriving the Fokker-Planck equation governing PDF for the four soliton parameters. As the principle example we examine NLS equation with the additive white Gaussian noise (AWGN). First we show that the Gaussian statistics can be justified for short enough propagation distances. For the large distances the Gaussian approximation breaks down and we illustrate it calculating explicitly the PDF for the soliton amplitude.