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Near Soliton Evolution for Equivariant Schrödinger Maps in Two Spatial Dimensions
Ioan Bejenaru, University of California, San Diego, La Jolla, CA, and Daniel Tataru, University of California, Berkeley, Berkeley, CA
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Memoirs of the American Mathematical Society
2014; 108 pp; softcover
Volume: 228
ISBN-10: 0-8218-9215-0
ISBN-13: 978-0-8218-9215-2
List Price: US$76
Individual Members: US$45.60
Institutional Members: US$60.80
Order Code: MEMO/228/1069
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The authors consider the Schrödinger Map equation in \(2+1\) dimensions, with values into \(\mathbb{S}^2\). This admits a lowest energy steady state \(Q\), namely the stereographic projection, which extends to a two dimensional family of steady states by scaling and rotation. The authors prove that \(Q\) is unstable in the energy space \(\dot H^1\). However, in the process of proving this they also show that within the equivariant class \(Q\) is stable in a stronger topology \(X \subset \dot H^1\).

Table of Contents

  • Introduction
  • An outline of the paper
  • The Coulomb gauge representation of the equation
  • Spectral analysis for the operators \(H\), \(\tilde H\); the \(X,L X\) spaces
  • The linear \(\tilde H\) Schrödinger equation
  • The time dependent linear evolution
  • Analysis of the gauge elements in \(X,LX\)
  • The nonlinear equation for \(\psi\)
  • The bootstrap estimate for the \(\lambda\) parameter
  • The bootstrap argument
  • The \(\dot H^1\) instability result
  • Bibliography
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