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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
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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

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 $$X \subset \dot H^1$$.

• 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