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

In this paper, we analyze the dynamical evolution of physical systems governed by the one-dimensional nonlinear Schrödinger equation, with initial conditions modelled by white noise. We show that first- and second-order Akhmediev breathers emerge in numerical simulations from such random initial conditions, demonstrating that these analytical solutions — typically constructed using the Darboux transformation — are intrinsic properties of the equation and can be regarded as its fingerprint. We further investigate this problem by analyzing the statistics of local intensity maxima and show that second-order breather solutions are also clearly visible in probability density histograms over intensity intervals. To determine their properties, we employ the peak-height formula, which enables precise identification of the frequencies of the two commensurate components that nonlinearly form the breather. The method is validated using two distinct intensity peaks obtained in numerical simulations with different algorithms and evolution steps. Finally, we provide a brief analysis explaining why third- and higher-order breather solutions are difficult to generate from white noise, based on weak perturbations of carefully constructed initial conditions derived from analytical solutions.