**Memoirs of the American Mathematical Society**

2014;
108 pp;
Softcover

MSC: Primary 58;
Secondary 35

Print ISBN: 978-0-8218-9215-2

Product Code: MEMO/228/1069

List Price: $76.00

AMS Member Price: $45.60

MAA Member Price: $68.40

**Electronic ISBN: 978-1-4704-1481-8
Product Code: MEMO/228/1069.E**

List Price: $76.00

AMS Member Price: $45.60

MAA Member Price: $68.40

# Near Soliton Evolution for Equivariant Schrödinger Maps in Two Spatial Dimensions

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*Ioan Bejenaru; Daniel Tataru*

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

# Table of Contents

## Near Soliton Evolution for Equivariant Schrodinger Maps in Two Spatial Dimensions

- Chapter 1. Introduction 18 free
- Chapter 2. An outline of the paper 714
- 2.1. The frame method and the Coulomb gauge 714
- 2.2. The reduced field 𝜓 714
- 2.3. Linearizations and the operators 𝐻, 𝐻 815
- 2.4. The 𝑋 and 𝐿𝑋 spaces 1017
- 2.5. The elliptic transition between 𝑢 and its reduced field 𝜓 1017
- 2.6. The nonlinear Schrödinger equation for 𝜓: Take 1 [local] 1118
- 2.7. The functions (𝛼(𝑡),𝜆(𝑡)) 1118
- 2.8. The nonlinear Schrödinger equation for 𝜓: Take 2 [global] 1219
- 2.9. The instability result 1320

- Chapter 3. The Coulomb gauge representation of the equation 1522
- Chapter 4. Spectral analysis for the operators 𝐻, 𝐻; the 𝑋,𝐿𝑋 spaces 2532
- 4.1. Spectral theory for the operator 𝐻 2532
- 4.2. Spectral theory for the operator 𝐻 2633
- 4.3. The spaces 𝑋 and 𝐿𝑋 2835
- 4.4. A companion space 3239
- 4.5. Littlewood-Paley projectors in the 𝐻 frame 3441
- 4.6. Time dependent frames and the transference identity 3542
- 4.7. Compositions of Littlewood-Paley projectors 3946
- 4.8. Nonresonant quadrilinear forms 4148

- Chapter 5. The linear 𝐻 Schrödinger equation 4552
- Chapter 6. The time dependent linear evolution 5764
- Chapter 7. Analysis of the gauge elements in 𝑋,𝐿𝑋 7178
- Chapter 8. The nonlinear equation for 𝜓 8592
- Chapter 9. The bootstrap estimate for the ł parameter. 95102
- Chapter 10. The bootstrap argument 99106
- Chapter 11. The 𝐻¹ instability result 103110
- Bibliography 107114