Statistics of the nonlinear discrete spectrum of a noisy pulse

In the presence of additive white Gaussian noise, the computation of the statistics of the nonlinear discrete spectrum of a pulse is an open problem. We extend the existing Fourier Collocation method to compute the whole discrete spectrum (eigenvalues and spectral amplitudes) and apply perturbation theory of linear operators to derive analytic expressions for the second-order statistics of the discrete spectrum of a noisy pulse. Our analytic expressions closely match numerical statistics obtained through simulations.