Blow up on a curve for a nonlinear Schr\"odinger equation on Riemannian surfaces

Abstract

We consider the focusing quintic nonlinear Schr\"odinger equation posed on a rotationally symmetric surface, typically the sphere $S^2$ or the two dimensional hyperbolic space $H^2$. We prove the existence and the stability of solutions blowing up on a suitable curve with the log log speed. The Euclidean case is handled in Rapha\"el (2006) and our result shows that the log log rate persists in other geometries with the assumption of a radial symmetry of the manifold.