  We consider the \emph{focusing} nonlinear Schr\"odinger equation posed on the
one dimensional line, with nonzero background condition at spatial infinity,
given by a homogeneous plane wave. For this problem of physical interest, we
study the initial value problem for perturbations of the background wave in
Sobolev spaces. It is well-known that the associated linear dynamics for this
problem describes a phenomenon known in the literature as \emph{modulational
instability}, also recently related to the emergence of \emph{rogue waves} in
ocean dynamics. In qualitative terms, small perturbations of the background
state increase its size exponentially in time. In this paper we show that, even
if there is no time decay for the linear dynamics due to the modulationally
unstable regime, the equation is still locally well-posed in $H^s$,
$s>\frac12$. We apply this result to give a rigorous proof of the unstable
character of two well-known NLS solutions: the Peregrine and Kuznetsov-Ma
breathers.
