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A critical center-stable manifold for Schrödinger’s equation in three dimensions. (English) Zbl 1234.35240

Summary: Consider the focusing \(\dot{H}^{1/2}\)-critical semilinear Schrödinger equation in \(\mathbb R^3\) \[ i\partial_t\psi + \Delta\psi + |\psi|^2\psi=0.\tag{1} \] It admits an eight-dimensional manifold of special solutions called ground state solitons.
We exhibit a codimension-1 critical real analytic manifold \(\mathcal N\) of asymptotically stable solutions of (1) in a neighborhood of the soliton manifold. We then show that \(\mathcal N\) is center-stable, in the dynamical systems sense of Bates and Jones, and globally-in-time invariant.
Solutions in \(\mathcal N\) 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 \(\mathcal N\).
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 \(\mathbb R^3\) – by the work of J. L. Marzuola and G. Simpson [Nonlinearity 24, No. 2, 389–429 (2011; Zbl 1213.35371)] and O. Costin, M. Huang and W. Schlag [“On the spectral properties of \(l_\pm\) in three dimensions” (in preparation)].

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35C08 Soliton solutions
35J62 Quasilinear elliptic equations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Citations:

Zbl 1213.35371
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