A class of solutions to the 3d cubic nonlinear Schroedinger equation that blow-up on a circle
Analysis of PDEs
2010-03-23 v2
Abstract
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.
Keywords
Cite
@article{arxiv.1002.2407,
title = {A class of solutions to the 3d cubic nonlinear Schroedinger equation that blow-up on a circle},
author = {Justin Holmer and Svetlana Roudenko},
journal= {arXiv preprint arXiv:1002.2407},
year = {2010}
}
Comments
updated introduction, expanded Section 21, added references