A critical center‐stable manifold for Schrödinger's equation in three dimensions

Abstract Consider the focusing $\dot H^{1/2}$ ‐critical semilinear Schrödinger equation in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^3$ It admits an eight‐dimensional manifold of special solutions called ground state solitons. We exhibit a codimension‐1 critical real analytic manifold ${\cal N}$ of asymptotically stable solutions of (0.1) in a neighborhood of the soliton manifold. We then show that ${\cal N}$ is center‐stable, in the dynamical systems sense of Bates and Jones, and globally‐in‐time invariant. Solutions in ${\cal N}$ are asymptotically stable and separate into two asymptotically free parts that decouple in the limit—a soliton and radiation. Conversely, in a general setting, any solution that stays $\dot H^{1/2}$ ‐close to the soliton manifold for all time is in ${\cal N}$ . The proof uses the method of modulation. New elements include a different linearization and an endpoint Strichartz estimate for the time‐dependent linearized equation. The proof also uses the fact that the linearized Hamiltonian has no nonzero real eigenvalues or resonances. This has recently been established in the case treated here—of the focusing cubic NLS in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^3$ —by the work of Marzuola and Simpson and Costin, Huang, and Schlag. © 2012 Wiley Periodicals, Inc.