2009/09/07 by Marius Beceanu, Beceanu, Marius · 1 citation
Mathematics · #35Q51 #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.0909.1180
openalex publication_date 2009/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the focusing cubic semilinear Schroedinger equation in R3 i ∂t ψ+ Δψ+ | ψ|2 ψ= 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 R3 --- by the work of Marzuola-Simpson and Costin-Huang-Schlag.