  Consider the focusing cubic semilinear Schroedinger equation in R^3 i
\partial_t \psi + \Delta \psi + | \psi |^2 \psi = 0. It admits an
eight-dimensional manifold of special solutions called ground state solitons.
  We exhibit a codimension-one critical real-analytic manifold N of
asymptotically stable solutions in a neighborhood of the soliton manifold. We
then show that N is centre-stable, in the dynamical systems sense of
Bates-Jones, and globally-in-time invariant.
  Solutions in N are asymptotically stable and separate into two asymptotically
free parts that decouple in the limit --- a soliton and radiation. Conversely,
in a general setting, any solution that stays close to the soliton manifold for
all time is in N.
  The proof uses the method of modulation. New elements include a different
linearization and an endpoint Strichartz estimate for the time-dependent
linearized equation.
  The proof also uses the fact that the linearized Hamiltonian has no nonzero
real eigenvalues or resonances. This has recently been established in the case
treated here --- of the focusing cubic NLS in R^3 --- by the work of
Marzuola-Simpson and Costin-Huang-Schlag.
