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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

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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