59 research outputs found
A class of solutions to the 3d cubic nonlinear Schroedinger equation that blow-up on a circle
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
Divergence of infinite-variance nonradial solutions to the 3d NLS equation
We consider solutions to the 3d NLS equation such that and is nonradial.
Denoting by and , the mass and energy, respectively, of a solution
, and by the ground state solution to , we
prove the following: if and , then either blows-up in
finite positive time or exists globally for all positive time and there
exists a sequence of times such that . Similar statements hold for negative time
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