On scattering for NLS: from Euclidean to hyperbolic space

Valeria Banica (1) , Rémi Carles (2) , Thomas Duyckaerts (3)

1 Département de Mathématiques (Evry)
2 I3M - Institut de Mathématiques et de Modélisation de Montpellier
3 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

We prove asymptotic completeness in the energy space for the nonlinear Schrodinger equation posed on hyperbolic space in the radial case, in space dimension at least 4, and for any energy-subcritical, defocusing, power nonlinearity. The proof is based on simple Morawetz estimates and weighted Strichartz estimates. We investigate the same question on spaces which kind of interpolate between Euclidean space and hyperbolic space, showing that the family of short range nonlinearities becomes larger and larger as the space approaches the hyperbolic space. Finally, we describe the large time behavior of radial solutions to the free dynamics.

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Valeria Banica, Rémi Carles, Thomas Duyckaerts. On scattering for NLS: from Euclidean to hyperbolic space. Discrete and Continuous Dynamical Systems - Series A, 2009, 24 (4), pp.1113-1127. ⟨10.3934/dcds.2009.24.1113⟩. ⟨hal-00204575v2⟩