2009/06/19 by E. Kirr, Kirr, E., Özgür MIZRAK +2
Mathematics · Physics and Astronomy · #35B40 #35Q55 #37K40 #46N20 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Spectral Theory in Mathematical Physics #math-ph #math.AP #math.MP #msc:35B40 #msc:35Q55 #msc:37K40 #msc:46N20
paper · pdf · doi:10.48550/arxiv.0906.3732
40 pages
arxiv created 2009/06/19 · openalex publication_date 2009/06/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a class of nonlinear Schrodinger equation in four and five space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L2) nonlinearities. We show that the center manifold formed by localized in space periodic in time solutions (bound states) is an attractor for all solutions with a small initial data. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a one parameter family of bound states that "shadows" the nonlinear evolution of the system. The methods we employ are an extension to higher dimensions, hence different linear dispersive behavior, and to rougher nonlinearities of our previous results [10, 11, 7].