Annales Henri Poincaré

, Volume 2, Issue 4, pp 605–673

On the Formation of Singularities in Solutions of the Critical Nonlinear Schrödinger Equation

  • G. Perelman
Open Access

DOI: 10.1007/PL00001048

Cite this article as:
Perelman, G. Ann. Henri Poincaré (2001) 2: 605. doi:10.1007/PL00001048


For the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity the Cauchy problem with initial data close to a soliton is considered. It is shown that for a certain class of initial perturbations the solution develops a self-similar singularity infinite time T*, the profile being given by the ground state solitary wave and the limiting self-focusing law being of the form¶¶\( \lambda(t) \sim (ln \mid ln(T^* -t)\mid)^{1/2} (T^* - t)^{-1/2} \)

Copyright information

© Birkhäuser Verlag Basel, 2001

Authors and Affiliations

  • G. Perelman
    • 1
  1. 1.Centre de Mathématiques, Ecole Polytechnique, F-91128 Palaiseau Cedex, France, e-mail: perelman@cmath.polytechnique.frFR

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