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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 Click above image for expanded view 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 $X \subset \dot H^1$.

• 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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