Consider the focussing cubic nonlinear Schr\"odinger equation in R 3 :
iψ t +Δψ=−|ψ| 2 ψ.
It admits special solutions of the form e itα ϕ , whereϕ is a Schwartz function and a positive (ϕ>0 ) solution of
−Δϕ+αϕ=ϕ 3 .
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 eight-dimensional manifold that consists of functions of the form e i(v⋅+Γ) ϕ(⋅−y,α) . We prove that any solution starting sufficiently close to a standing wave in the Σ=W 1,2 (R 3 )∩|x| −1 L 2 (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 $\mc N$ is invariant under the Hamiltonian flow, locally in time, and is a centre-stable manifold in the sense of Bates, Jones. 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 R 3 for the nonselfadjoint Schr\"odinger operator obtained by linearizing around a standing wave solution.
Beceanu, Marius and Goldberg, Michael, "Schrödinger Dispersive Dstimates for a Dcaling-Critical Class of Potentials" (2010). Mathematics and Statistics Faculty Scholarship. 13.