A Class of Solutions to the 3d Cubic Nonlinear Schrödinger Equation That Blow-up on a Circle
نویسندگان
چکیده
We consider the 3d cubic focusing nonlinear Schrödinger equation (NLS) i∂tu+∆u+ |u|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 axial(R) of initial data, that blow-up in finite time with singular set a circle in xy plane. Our construction is modeled on Raphaël’s construction [33] of a family of solutions to the 2d quintic focusing NLS, i∂tu+∆u+|u|u = 0, that blow-up on a circle.
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