Nonlinear Schrödinger Equations on Compact Manifolds

Nicolas Burq

Université de Paris-Sud, Orsay

ABSTRACT

Nonlinear Schrdinger equations (NLS) have been studied extensively in ${\mathbb R}^n$ and more recently on compact manifolds. The ideas originating in semi-classical quantum mechanics allow solving NLS on all compact surfaces and in some higher dimensional cases. However, when the dimension is too high short time instability occurs.

In this talk I will present the basic ideas behind these developments stressing the importance of global geometry on the existence and stability of solutions.