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

Abstract

Consider the focussing cubic nonlinear Schrödinger equation in ${\mathbb{R}}^3$ :

$$i\psi_t+\Delta\psi = -|\psi|^2 \psi. \quad (0.1) $$

It admits special solutions of the form e ^itα ϕ, where $\phi \in {\mathcal{S}}({\mathbb{R}}^3)$ is a positive (ϕ > 0) solution of

$$-\Delta \phi + \alpha\phi = \phi^3. \quad (0.2)$$

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

$$\mathcal H = \left ( \begin{array}{cc}\Delta + 2\phi(\cdot, \alpha)^2 - \alpha &\quad \phi(\cdot, \alpha)^2 \\ -\phi(\cdot, \alpha)^2 &\quad -\Delta - 2 \phi(\cdot, \alpha)^2 + \alpha \end{array}\right ) \quad (0.3)$$

has no embedded eigenvalues in the interior of its essential spectrum $(-\infty, -\alpha) \cup (\alpha, \infty)$ .