We consider the 3d cubic focusing nonlinear Schroedinger equation (NLS)
i\partial_t u + \Delta u + |u|^2 u=0, which appears as a model in condensed
matter theory and plasma physics. We construct a family of axially symmetric
solutions, corresponding to an open set in H^1_{axial}(R^3) of initial data,
that blow-up in finite time with singular set a circle in xy plane. Our
construction is modeled on Rapha\"el's construction \cite{R} of a family of
solutions to the 2d quintic focusing NLS, i\partial_t u + \Delta u + |u|^4 u=0,
that blow-up on a circle.Comment: updated introduction, expanded Section 21, added reference