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Near soliton evolution for equivariant Schrödinger Maps in two spatial dimensions


About this Title

Ioan Bejenaru and Daniel Tataru

Publication: Memoirs of the American Mathematical Society
Publication Year 2014: Volume 228, Number 1069
ISBNs: 978-0-8218-9215-2 (print); 978-1-4704-1481-8 (online)
DOI: http://dx.doi.org/10.1090/memo/1069
Published electronically: July 11, 2013

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. An outline of the paper
  • Chapter 3. The Coulomb gauge representation of the equation
  • Chapter 4. Spectral analysis for the operators , ; the spaces
  • Chapter 5. The linear Schrödinger equation
  • Chapter 6. The time dependent linear evolution
  • Chapter 7. Analysis of the gauge elements in
  • Chapter 8. The nonlinear equation for
  • Chapter 9. The bootstrap estimate for the parameter.
  • Chapter 10. The bootstrap argument
  • Chapter 11. The instability result

Abstract


We consider the Schrödinger Map equation in dimensions, with values into . This admits a lowest energy steady state , namely the stereographic projection, which extends to a two dimensional family of steady states by scaling and rotation. We prove that is unstable in the energy space . However, in the process of proving this we also show that within the equivariant class is stable in a stronger topology .

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