Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On scattering for NLS: From Euclidean to hyperbolic space

Pages: 1113 - 1127, Volume 24, Issue 4, August 2009      doi:10.3934/dcds.2009.24.1113

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Valeria Banica - Département de Mathématiques, Univ. Evry, Bd. F. Mitterrand, 91025 Evry, France (email)
Rémi Carles - Université Montpellier 2, Mathématiques, CC051, 34095 Montpellier, CNRS, UMR 5149, 34095 Montpellier, France (email)
Thomas Duyckaerts - Département de Mathématiques, Univ. Cergy-Pontoise, CNRS UMR 8088, 2 avenue Adolphe Chauvin, BP 222, Pontoise, 95302 Cergy-Pontoise cedex, France (email)

Abstract: We prove asymptotic completeness in the energy space for the nonlinear Schrödinger equation posed on hyperbolic space $\mathbb H^n$ in the radial case, for $n\ge 4$, and 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 sort 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.

Keywords:  Nonlinear Schrödinger equation, scattering theory, hyperbolic space.
Mathematics Subject Classification:  Primary: 35P25, 35Q55; Secondary: 35B40, 58J50.

Received: March 2008;      Revised: January 2009;      Published: May 2009.