Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrödinger system

Abstract

In this paper, we establish two new types of invariant sets for the coupled nonlinear Schrödinger system in the Euclidean n -space Rn and derive two sharp thresholds of blow-up and global existence for its solutions. Some analogous results for the nonlinear Schrödinger system posed on the hyperbolic space Hn and on the standard 2-sphere S2 are also presented. Our arguments and constructions are improvements of some previous works on this direction. At the end, we give some heuristic analysis about the strong instability of the solitary waves. The relation between the two types of thresholds is a very interesting problem, and we leave it as an open problem for further study. © 2008 American Institute of Physics.