Standing Ring Blowup Solutions for Cubic NLS

Abstract

We prove there exist solutions to the focusing cubic nonlinear Schrödinger equation in three dimensions that blowup on a circle, in the sense of L^2 concentration on a ring, bounded H^1 norm outside any surrounding toroid, and growth of the global H^1 norm with the log-log rate. Analogous behaviour occurs in any higher dimension. That is, there exists data for which the corresponding evolution by cubic NLS explodes on a set of co-dimension two. To simplify the exposition, the proof is presented in dimension three, with remarks to indicate the adaptations in higher dimension.

54 pages. Introduction Updated.

References in corpus (3)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• On blow-up solutions to the 3D cubic nonlinear Schroedinger equation
• A class of solutions to the 3d cubic nonlinear Schroedinger equation that blow-up on a circle

Cited by in corpus (1)

• Blow-up solutions on a sphere for the 3d quintic NLS in the energy space