A Class of Solutions to the 3D Cubic Nonlinear Schrodinger Equation that Blows Up on a Circle

摘要

We consider the 3D cubic focusing nonlinear Schrodinger (NLS) equation iθ_tu+ Δu+ |u|~2u=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_(axial)~1(R~3) of initial data, that blow up in finite time with a singular circle in the xy-plane. Our construction is modeled on Raphael’s [33] construction of a family of solutions to the 2D quintic focusing NLS, iθ_tu+ Δu+ |u|~4u=0, that blows up on a circle.