A Centre-Stable Manifold for the Focussing Cubic NLS in $R^{1+3}$

Abstract

Consider the focussing cubic nonlinear Schrödinger equation in $R^3$: $$ iψ_t+Δψ= -|ψ|^2 ψ. $$ It admits special solutions of the form $e^{itα}ϕ$, where $ϕ$ 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 $e^{i(v \cdot + Γ)} ϕ(\cdot - y, α)$. We prove that any solution starting sufficiently close to a standing wave in the $Σ= W^{1, 2}(R^3) \cap |x|^{-1}L^2(R^3)$ 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 Schrödinger operator obtained by linearizing around a standing wave solution.