  We consider the problem of the error in soliton transmission in long-haul
optical fibers caused by the spontaneous emission of noise inherent to
amplification. We study two types of noises driving the stochastic focusing
cubic one dimensional nonlinear Schr\"{o}dinger equation which appears in
physics in that context. We focus on the fluctuations of the mass and arrival
time or timing jitter. We give the small noise asymptotic of the tails of these
two quantities for the two types of noises. We are then able to prove several
results from physics among which the Gordon--Haus effect which states that the
fluctuation of the arrival time is a much more limiting factor than the
fluctuation of the mass. The physical results had been obtained with arguments
difficult to fully justify mathematically.
