| eBook 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 |
| eBook 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 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 228; 2014; 108 ppMSC: 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 \(X \subset \dot H^1\).
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Table of Contents
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Chapters
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1. Introduction
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2. An outline of the paper
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3. The Coulomb gauge representation of the equation
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4. Spectral analysis for the operators $H$, $\tilde H$; the $X,L X$ spaces
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5. The linear $\tilde H$ Schrödinger equation
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6. The time dependent linear evolution
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7. Analysis of the gauge elements in $X,LX$
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8. The nonlinear equation for $\psi $
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9. The bootstrap estimate for the $\lambda $ parameter.
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10. The bootstrap argument
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11. The $\dot H^1$ instability result
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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 \(X \subset \dot H^1\).
-
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
