A Critical Centre-Stable Manifold for the Cubic Focusing Schroedinger Equation in Three Dimensions

Abstract

Consider the H^{1/2}-critical Schroedinger equation with a cubic nonlinearity in R^3, i \partial_t \psi + \Delta \psi + |\psi|^2 \psi = 0. It admits an eight-dimensional manifold of periodic solutions called solitons e^{i(\Gamma + vx - t|v|^2 + \alpha^2 t)} \phi(x-2tv-D, \alpha), where \phi(x, \alpha) is a positive ground state solution of the semilinear elliptic equation -\Delta \phi + \alpha^2\phi = \phi^3. We prove that in the neighborhood of the soliton manifold there exists a H^{1/2} real analytic manifold N of asymptotically stable solutions of the Schroedinger equation, meaning they are the sum of a moving soliton and a dispersive term. Furthermore, a solution starting on N remains on N for all positive time and for some finite negative time and N can be identified as the centre-stable manifold for this equation. The proof is based on the method of modulation, introduced by Soffer and Weinstein and adapted by Schlag to the L^2-supercritical case. Novel elements include a different linearization and new Strichartz-type estimates for the linear Schroedinger equation. The main result depends on a spectral assumption concerning the absence of embedded eigenvalues. We also establish several new estimates for solutions of the time-dependent and time-independent linear Schroedinger equation, which hold under sharper or more general conditions than previously known. Several of these estimates are based on a new approach that makes use of Wiener's Theorem in the context of function spaces.