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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)
Published electronically: July 11, 2013

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Table of Contents


  • 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 $H$, $\tilde H$; the $X,L X$ spaces
  • Chapter 5. The linear $\tilde H$ Schrödinger equation
  • Chapter 6. The time dependent linear evolution
  • Chapter 7. Analysis of the gauge elements in $X,LX$
  • Chapter 8. The nonlinear equation for $\psi $
  • Chapter 9. The bootstrap estimate for the $\lambda $ parameter.
  • Chapter 10. The bootstrap argument
  • Chapter 11. The $\dot H^1$ instability result


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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