Nonlinear spectral characterization of integrable turbulence

We study the statistical properties of the discrete spectra of the Zakharov–Shabat eigenvalue problem corresponding to the uniform constant background perturbed by weak correlated noise. The properties of these spectra are important for the studies of the later stages of modulation instability and soliton turbulence in a wide class of systems, from optical fibers to Bose–Einstein condensates. We show that for correlated disorder, the distribution of discrete eigenvalues is anisotropic, favoring the creation of bound multi-soliton states (breathers), while in the limit of delta-correlated disorder, most solutions are asymptotically free.