Blow up on a curve for a nonlinear Schrödinger equation on Riemannian surfaces

Details Blow up on a curve for a nonlinear Schrödinger equation on Riemannian surfaces Blow up on a curve for a nonlinear Schrödinger equation on Riemannian surfaces

2012-04-15

arxiv.org/abs/1204.3301v1

Mathematics

Analysis of PDEs

Scientific paper

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 \cite{Rap2006} and our result shows that the log log rate persists in other geometries with the assumption of a radial symmetry of the manifold.

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