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

TL;DR

This paper extends the understanding of blow-up solutions for the focusing quintic nonlinear Schr"odinger equation to rotationally symmetric surfaces, demonstrating existence and stability of blow-up on curves with a specific rate.

Contribution

It proves the existence and stability of blow-up solutions on curves for the nonlinear Schr"odinger equation on curved surfaces, generalizing Euclidean results to spherical and hyperbolic geometries.

Findings

Blow-up solutions exist on curves with log log speed.

The log log blow-up rate persists beyond Euclidean space.

Stability of these solutions is established on symmetric 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.