  We investigate theoretically the fundamental phenomenon of the spontaneous,
noise-induced modulational instability (MI) of a plane wave. The long-term
statistical properties of the noise-induced MI have been previously observed in
experiments and in simulations but have not been explained so far. In the
framework of inverse scattering transform (IST), we propose a model of the
asymptotic stage of the noise-induced MI based on $N$-soliton solutions
($N$-SS) of the integrable focusing one-dimensional nonlinear Schr\"odinger
equation (1D-NLSE). These $N$-SS are bound states of strongly interacting
solitons having a specific distribution of the IST eigenvalues together with
random phases. We use a special approach to construct ensembles of
multi-soliton solutions with statistically large number of solitons $N\sim100$.
Our investigation demonstrates complete agreement in spectral (Fourier) and
statistical properties between the long-term evolution of the condensate
perturbed by noise and the constructed multi-soliton bound states. Our results
can be generalised to a broad class of integrable turbulence problems in the
cases when the wave field dynamics is strongly nonlinear and driven by
solitons.

Exactly Solvable and Integrable Systems;Pattern Formation and Solitons