Nonlinear Schrödinger equations on compact manifolds. (English) Zbl 1076.35115

Gérard, P.

Laptev, Ari (ed.), Proceedings of the 4th European congress of mathematics (ECM), Stockholm, Sweden, June 27–July 2, 2004. Zürich: European Mathematical Society (EMS) (ISBN 3-03719-009-4/hbk). 121-139 (2005).

Summary: Nonlinear Schrödinger equations have been studied by mathematicians for about thirty years. However, until recently, most of the contributions concerned the equation on the whole Euclidean space, with the notable exception of J. Bourgain’s contributions on tori. In the case of general Riemannian manifolds, the interaction of geometry with nonlinear operations leads to new phenomena, particularly if the manifold is compact. Here, we review the state of the art concerning the Cauchy problem on such manifolds, and we describe optimal results on spheres, where new estimates on spherical harmonics play a crucial role. The matter of this paper is based on a series of results in collaboration with N. Burq and N. Tzvetkov.
For the entire collection see [Zbl 1064.00004].

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
58J50	Spectral problems; spectral geometry; scattering theory on manifolds
37K05	Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37L50	Noncompact semigroups, dispersive equations, perturbations of infinite-dimensional dissipative dynamical systems

Keywords:

eigenfunction estimates; dispersive equation; spherical harmonics; Schrödinger group; high frequency instability