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Inventiones mathematicae

, Volume 156, Issue 3, pp 565–672 | Cite as

On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation

  • Frank MerleEmail author
  • Pierre Raphael
Article

Abstract

We consider finite time blow-up solutions to the critical nonlinear Schrödinger equation iu t =-Δu-|u|4/N u with initial condition u 0H 1. Existence of such solutions is known, but the complete blow-up dynamic is not understood so far. For a specific set of initial data, finite time blow-up with a universal sharp upper bound on the blow-up rate has been proved in [22], [23].

We establish in this paper the existence of a universal blow-up profile which attracts blow-up solutions in the vicinity of blow-up time. Such a property relies on classification results of a new type for solutions to critical NLS. In particular, a new characterization of soliton solutions is given, and a refined study of dispersive effects of (NLS) in L 2 will remove the possibility of self similar blow-up in energy space H 1.

Keywords

Soliton Initial Data Classification Result Finite Time Soliton Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de Cergy-PontoiseFrance
  2. 2.Institut Universitaire de FranceFrance

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