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

Abstract.

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} $.

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Submitted 20/10/00, accepted 08/03/01

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Perelman, G. On the Formation of Singularities in Solutions of the Critical Nonlinear Schrödinger Equation. Ann. Henri Poincaré 2, 605–673 (2001). https://doi.org/10.1007/PL00001048

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Keywords

  • Initial Data
  • Cauchy Problem
  • Solitary Wave
  • Initial Perturbation
  • Critical Power