We consider the focusing one-dimensional nonlinear Schrodinger equation (NLSE), which can be completely integrated using the Inverse Scattering Transform (IST) method. The IST allows decomposing of nonlinear wave fields into solitons and continuous spectrum waves characterized by scattering data. The NLSE has a simple plane wave solution – condensate- among the most well-known examples of strongly nonlinear wave fields with the dominant role of solitons in its behavior. The condensate is unstable to long-wave perturbation due to modulation instability. Modulation instability plays an important role in physics, and the description of its nonlinear stage of development represents a fundamental challenge for the IST theory. In our study, we model the modulation instability development with a specific multi-soliton solution of the NLSE, neglecting the impact of the continuous spectrum radiation. We show that the solitonic model can describe the initial condensate and the final stage of spontaneous (i.e., noise-induced) modulation instability development [1,2]. Finally, we discuss our recent results on statistical saturation of the spontaneous modulation instability and its explanation using the ill-conditioned properties of the multi-soliton solutions. [1] A. Gelash, D. Agafontsev, V. Zakharov, G. El, S. Randoux and P. Suret,. Bound state soliton gas dynamics underlying the noise-induced modulational instability, Phys. Rev. Lett., 2019.[2] A. Gelash, D. Agafontsev, P. Suret, S. Randoux, Solitonic model of the condensate. Physical Review E, 2021.

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