Nonlinear Schrodinger equation on real hyperbolic spaces

By Jean-Philippe Anker, Vittoria Pierfelice. 2008

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Abstract We consider the Schrodinger equation with no radial assumption on real hyperbolic spaces H n . We obtain in all dimensions n ⩾ 2 sharp dispersive and Strichartz estimates for a large family of admissible pairs. As a first consequence, we obtain strong well-posedness results for NLS. Specifically, for small initial data, we prove L 2 and H 1 global well-posedness for any subcritical power (in contrast with the Euclidean case) and with no gauge invariance assumption on the nonlinearity F. On the other hand, if F is gauge invariant, L 2 charge is conserved and hence, as in the Euclidean case, it is possible to extend local L 2 solutions to global ones. The corresponding argument in H 1 requires conservation of energy, which holds under the stronger condition that F is defocusing. Recall that global well-posedness in the gauge invariant case was already proved by Banica, Carles and Staffilani, for small radial L 2 data or for large radial H 1 data. The second application of our global Strichartz estimates is scattering for NLS both in L 2 and in H 1 , with no radial or gauge invariance assumption. Notice that, on Euclidean spaces R n , this is only possible for the critical power γ = 1 + 4 n and can be false for subcritical powers while, on hyperbolic spaces H n , global existence and scattering of small L 2 solutions hold for all powers 1 γ ⩽ 1 + 4 n . If we restrict to defocusing nonlinearities F, we can extend the H 1 scattering results of Banica, Carles and Staffilani to the nonradial case. Also there is no distinction anymore between short range and long range nonlinearities: the geometry of hyperbolic spaces makes every power-like nonlinearity short range.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a415c044d633bc5920af79403ecd055ce7432505

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Citations in the corpus

On the global well-posedness of energy-critical Schr\"odinger equations in curved spaces (2010, background): “This heuristic can be made precise, see [2, 5, 6, 7, 8, 9, 19, 37, 45] for theorems concerning subcritical nonlinear Schrödinger equations in hyperbolic spaces (or other spaces that interpolate between Euclidean and hyperbolic spaces).”
The wave equation on hyperbolic spaces (2010, methodology): “In this paper we pursue our study of dispersive equations on hyperbolic spaces, initiated with the Schrödinger equation [1], by considering the shifted wave equation (4) on H.”