  We study numerically the integrable turbulence in the framework of the
one-dimensional nonlinear Schrodinger equation (1D-NLSE) of the focusing type
using a new approach called the "growing of turbulence". In this approach, we
add a small linear pumping term to the equation and start evolution from
statistically homogeneous Gaussian noise. After reaching a certain level of
average intensity, we switch off the pumping and examine the resulting
integrable turbulence. For sufficiently small initial noise and pumping
coefficient, and also for not very wide simulation box (basin length), we
observe that the turbulence grows in a universal adiabatic regime, moving
successively through the statistically stationary states of the integrable
1D-NLSE, which do not depend on the pumping coefficient, amplitude of the
initial noise or basing length. Waiting longer in the growth stage, we transit
from weakly nonlinear states to strongly nonlinear ones, characterized by a
high frequency of rogue waves. Using the inverse scattering transform (IST)
method to monitor the evolution, we observe that the solitonic part of the
wavefield becomes dominant even when the (linear) dispersion effects are still
leading in the dynamics and with increasing average intensity the wavefield
approaches a dense bound-state soliton gas, whose properties are defined by the
Fourier spectrum of initial noise. Regimes deviating from the universal
adiabatic growth also lead to solitonic states, but solitons in these states
have noticeably different velocities and a significantly wider distribution by
amplitude, while the statistics of wavefield indicates a much more frequent
appearance of very large waves.
