A class of solutions to the 3d cubic nonlinear Schroedinger equation that blow-up on a circle

Details A class of solutions to the 3d cubic nonlinear Schroedinger equation that blow-up on a circle A class of solutions to the 3d cubic nonlinear Schroedinger equation that blow-up on a circle

2010-02-11

arxiv.org/abs/1002.2407v2

Mathematics

Analysis of PDEs

updated introduction, expanded Section 21, added references

Scientific paper

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.

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