  In the wind-driven wave regime, the Miles mechanism gives an estimate of the
growth rate of the waves under the effect of wind. We consider the case where
this growth rate, normalised with respect to the frequency of the carrier wave,
is of the order of the wave steepness. Using the method of multiple scales, we
calculate the terms which appear in the nonlinear Schr\"odinger (NLS) equation
in this regime of fast-growing waves. We define a coordinate transformation
which maps the forced NLS equation into the standard NLS with constant
coefficients, that has a number of known analytical soliton solutions. Among
these solutions, the Peregrine and the Akhmediev solitons show an enhancement
of both their lifetime and maximum amplitude which is in qualitative agreement
with the results of tank experiments and numerical simulations of dispersive
focusing under the action of wind.
