Akhmediev breathers emerging from the white noise: evidence and identification using the peak-height formula

In this paper, we analyze the dynamical evolution of the physical systems, governed by the one-dimensional nonlinear Schrödinger equation (NLSE), when initial conditions are modelled by white noise. We report that the Akhmediev breathers (AB) of the first- and the second-order emerge from such random sequence of numbers showing that these analytical solutions, usually derived using the Darboux transformation (DT) scheme, are intrinsic properties of the NLSE and thus can be regarded as the fingerprint of this nonlinear model. We tackle the white noise evolution problem from the other side by computing the statistics of the local intensity maxima. We report that the second-order NLSE structures are also visible from histograms of probability density over intensity regions. To determine the exact properties of the second-order AB we use the peak-height formula (PHF). This simple expression enables the precise computation of the frequencies of two commensurate DT components that nonlinearly build the breather in numerical integration. We demonstrate this method on the two real intensity peaks that were unambiguously obtained in the numerical integration using different algorithms and evolution steps. Finally, the brief analysis was conducted to show why third- and higher-order breather solutions are hard to produce from the white noise by introducing weak perturbation of the carefully tailored initial conditions from analytical DT solutions.