A Critical Centre-Stable Manifold for Schroedinger's Equation in R^3

TL;DR

This paper constructs a codimension-one manifold of asymptotically stable solutions near the ground state solitons for the focusing cubic NLS in R^3, demonstrating their stability and asymptotic behavior.

Contribution

It introduces a new center-stable manifold for the focusing cubic NLS in R^3, with novel linearization and endpoint Strichartz estimates, advancing understanding of solution stability.

Findings

Existence of a codimension-one invariant manifold of stable solutions.

Solutions in this manifold decompose into a soliton and radiation asymptotically.

Linearized Hamiltonian has no nonzero real eigenvalues or resonances.

Abstract

Consider the focusing cubic semilinear Schroedinger equation in R^3 i \partial_t \psi + \Delta \psi + | \psi |^2 \psi = 0. It admits an eight-dimensional manifold of special solutions called ground state solitons. We exhibit a codimension-one critical real-analytic manifold N of asymptotically stable solutions in a neighborhood of the soliton manifold. We then show that N is centre-stable, in the dynamical systems sense of Bates-Jones, and globally-in-time invariant. Solutions in 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 close to the soliton manifold for all time is in 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 R^3 --- by the work of Marzuola-Simpson and Costin-Huang-Schlag.