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

Abstract

Consider the focussing cubic nonlinear Schrödinger equation in $R^3$: $$i{\psi }_t+\Delta \psi =-|\psi {|}^2\psi .$$ It admits special solutions of the form $e^{it\alpha }\varphi$, where $\varphi$ is a Schwartz function and a positive ($\varphi >0$) solution of $$-\Delta \varphi +\alpha \varphi ={\varphi }^3.$$ 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 $e^{i(v\cdot +\Gamma )}\varphi (\cdot -y,\alpha )$. We prove that any solution starting sufficiently close to a standing wave in the $\Sigma =W^{1,2}(R^3)\cap |x{|}^{-1}{L}^2(R^3)$ 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.

56 pages

References in corpus (5)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• Asymptotic stability of N-soliton states of NLS
• Stable manifolds for an orbitally unstable NLS
• Stable manifolds for all monic supercritical NLS in one dimension
• Dispersive estimates for Schroedinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II