We study the Cauchy problem for a system of two coupled nonlinear focusing Schrödinger equations arising in nonlinear optics. We discuss when the solutions are global in time or blow-up in finite time. Some results, in dependence of the data of the problem, are proved; in particular we give a bound, depending on the coupling parameter, for the blow-up threshold. 2000 Mathematics Subject Classif...

The dissipative quasi-geostrophic equation with dispersive forcing is considered. By striking new balances between the effects of and smoothing viscous dissipation, we obtain global well-posedness for Cauchy problem arbitrary initial data, provided that parameter large enough.

Cauchy problems and stochastic integral equations Bachelor thesis, June 21, 2011 Thesis supervisor: dr. O. van Gaans Mathematisch Instituut, Universiteit Leiden