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# memo_has_moved_text();Near soliton evolution for equivariant Schrödinger Maps in two spatial dimensions

Ioan Bejenaru and Daniel Tataru

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 228, Number 1069
ISBNs: 978-0-8218-9215-2 (print); 978-1-4704-1481-8 (online)
DOI: http://dx.doi.org/10.1090/memo/1069
Published electronically: July 11, 2013

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Chapters

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

### Abstract

We 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. We prove that $Q$ is unstable in the energy space $\dot H^1$. However, in the process of proving this we also show that within the equivariant class $Q$ is stable in a stronger topology $X \subset \dot H^1$.