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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
Daniel Tataru University of California, Berkeley, Berkeley, CA
Front Cover for Near Soliton Evolution for Equivariant Schrodinger Maps in Two Spatial Dimensions
Available Formats:
Electronic ISBN: 978-1-4704-1481-8
Product Code: MEMO/228/1069.E
List Price: $76.00
MAA Member Price: $68.40
AMS Member Price: $45.60
Front Cover for Near Soliton Evolution for Equivariant Schrodinger Maps in Two Spatial Dimensions
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  • Front Cover for Near Soliton Evolution for Equivariant Schrodinger Maps in Two Spatial Dimensions
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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
Daniel Tataru University of California, Berkeley, Berkeley, CA
Available Formats:
Electronic ISBN:  978-1-4704-1481-8
Product Code:  MEMO/228/1069.E
List Price: $76.00
MAA Member Price: $68.40
AMS Member Price: $45.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2282014; 108 pp
    MSC: Primary 58; Secondary 35;

    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
     
     
    • Chapters
    • 1. Introduction
    • 2. An outline of the paper
    • 3. The Coulomb gauge representation of the equation
    • 4. Spectral analysis for the operators $H$, $\tilde H$; the $X,L X$ spaces
    • 5. The linear $\tilde H$ Schrödinger equation
    • 6. The time dependent linear evolution
    • 7. Analysis of the gauge elements in $X,LX$
    • 8. The nonlinear equation for $\psi $
    • 9. The bootstrap estimate for the $\lambda $ parameter.
    • 10. The bootstrap argument
    • 11. The $\dot H^1$ instability result
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Volume: 2282014; 108 pp
MSC: Primary 58; Secondary 35;

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 \(X \subset \dot H^1\).

  • Chapters
  • 1. Introduction
  • 2. An outline of the paper
  • 3. The Coulomb gauge representation of the equation
  • 4. Spectral analysis for the operators $H$, $\tilde H$; the $X,L X$ spaces
  • 5. The linear $\tilde H$ Schrödinger equation
  • 6. The time dependent linear evolution
  • 7. Analysis of the gauge elements in $X,LX$
  • 8. The nonlinear equation for $\psi $
  • 9. The bootstrap estimate for the $\lambda $ parameter.
  • 10. The bootstrap argument
  • 11. The $\dot H^1$ instability result
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