Consider the focussing cubic nonlinear Schrödinger equation in
({mathbb{R}}^3) :
$ipsi_t+Deltapsi = -|psi|^2 psi. quad (0.1) $
It admits special solutions of the form
e itα ?, where
(phi in {mathcal{S}}({mathbb{R}}^3)) is a positive (
? > 0) solution of
$-Delta phi + alphaphi = phi^3. quad (0.2)$
The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the 8-dimensional manifold that consists of functions of the form
(e^{i(v cdot + Gamma)} phi(cdot - y, alpha)) . We prove that any solution starting sufficiently close to a standing wave in the
(Sigma = W^{1, 2}({mathbb{R}}^3) cap |x|^{-1}L^2({mathbb{R}}^3)) norm and situated on a certain codimension-one local Lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves. Furthermore, we show that
({mathcal{N}}) is invariant under the Hamiltonian flow, locally in time, and is a centre-stable manifold in the sense of Bates, Jones [BatJon]. The proof is based on the modulation method introduced by Soffer and Weinstein for the
L 2-subcritical case and adapted by Schlag to the
L 2-supercritical case. An important part of the proof is the Keel-Tao endpoint Strichartz estimate in
({mathbb{R}}^3) for the nonselfadjoint Schrödinger operator obtained by linearizing (0.1) around a standing wave solution. All results in this paper depend on the standard spectral assumption that the Hamiltonian
$mathcal H = left ( begin{array}{cc}Delta + 2phi(cdot, alpha)^2 - alpha &;quad phi(cdot, alpha)^2 -phi(cdot, alpha)^2 &;quad -Delta - 2 phi(cdot, alpha)^2 + alpha end{array}right ) quad (0.3)$
has no embedded eigenvalues in the interior of its essential spectrum
((-infty, -alpha) cup (alpha, infty)) .
