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arXiv · 1002.2407

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

Justin Holmer·Svetlana Roudenko
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Abstract

We consider the 3d cubic focusing nonlinear Schroedinger equation (NLS) i\partial_t u + Δu + |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 Raphaël's construction \cite{R} of a family of solutions to the 2d quintic focusing NLS, i\partial_t u + Δu + |u|^4 u=0, that blow-up on a circle.

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Justin Holmer, Svetlana Roudenko. 2010-03-22. A class of solutions to the 3d cubic nonlinear Schroedinger equation that blow-up on a circle. https://arxiv.org/abs/1002.2407

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