Nonlinear Spectra of Integrable Turbulence in a Periodic Box

Investigated in this paper are the nonlinear spectra of integrable turbulence with periodic boundary conditions. The model is the focusing nonlinear Schrödinger equation with partially coherent waves as the initial conditions. For our parameters, the spectra are found dominated by the pointlike spectral bands representing solitons, while the small-amplitude finite bands across and near the real axis of the complex plane seem to be minors. Statistical distribution of the main spectral eigenvalues of those pointlike bands is studied for different correlation lengths of the initial waves. It is observed that for both the small and large correlation lengths the real parts of the eigenvalues follow a Gaussian distribution with zero mean and width almost inversely proportional to the correlation length, while the imaginary parts are Rayleigh-distributed with some fixed parameter for enough large correlation length, but deviate from this distribution for small correlation length. Our results may facilitate a spectral understanding of the random waves in nonlinear integrable systems with the setting of periodic boundaries.