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A centre-stable manifold for the focussing cubic NLS in \({\mathbb{R}}^{1+3\star}\). (English) Zbl 1148.35082

Summary: Consider the focussing cubic nonlinear Schrödinger equation in \({\mathbb{R}}^3\): \[ i\psi_t+\Delta\psi = -|\psi|^2 \psi. \tag{0.1} \] It admits special solutions of the form \(e^{it \alpha} \varphi\), where \(\varphi \in {\mathcal{S}}({\mathbb{R}}^3)\) is a positive \((\varphi > 0)\) solution of \[ -\Delta \varphi + \alpha\varphi = \varphi^3. \tag{0.2} \] 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 \(e^{i(v \cdot + \Gamma)} \varphi(\cdot - y, \alpha)\). 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 P. W. Bates and C. K. R. T. Jones [in Dynamics reported, Vol. 2, 1–38, Wiley, Chichester (1989; Zbl 0674.58024)]. 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 \[ \mathcal H = \left (\begin{matrix}\Delta + 2\varphi(\cdot, \alpha)^2 - \alpha & \varphi(\cdot, \alpha)^2 \\ -\varphi(\cdot, \alpha)^2 & -\Delta - 2 \varphi(\cdot, \alpha)^2 + \alpha \end{matrix}\right) \tag{0.3} \] has no embedded eigenvalues in the interior of its essential spectrum \((-\infty, -\alpha) \cup (\alpha, \infty)\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems

Citations:

Zbl 0674.58024
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References:

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