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

TL;DR

This paper constructs a family of axially symmetric solutions to the 3D cubic focusing nonlinear Schrödinger equation that blow up in finite time along a circle, extending previous 2D results.

Contribution

It introduces a novel class of blow-up solutions for the 3D cubic NLS with singularity on a circle, based on and extending Raphaël's 2D construction.

Findings

Solutions blow up in finite time on a circle

Constructed solutions are axially symmetric

Extends 2D blow-up techniques to 3D setting

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.