  We show that transonic one dimensional flows which are analogous to black
holes obey no-hair theorems both at the level of linear perturbations and in
non-linear regimes. Considering solutions of the Gross-Pitaevskii (or
Korteweg-de Vries) equation, we show that stationary flows which are
asymptotically uniform on both sides of the horizon are stable and act as
attractors. Using Whitham's modulation theory, we analytically characterize the
emitted waves when starting from uniform perturbations. Numerical simulations
confirm the validity of this approximation and extend the results to more
general perturbations and to the (non-integrable) cubic-quintic
Gross-Pitaevskii equation. When considering time reversed flows that correspond
to white holes, the asymptotically uniform flows are unstable to sufficiently
large perturbations and emit either a macroscopic undulation in the supersonic
side, or a non-linear superposition of soliton trains.
