**Memoirs of the American Mathematical Society**

2013;
136 pp;
Softcover

MSC: Primary 35;

Print ISBN: 978-0-8218-8545-1

Product Code: MEMO/225/1059

List Price: $74.00

AMS Member Price: $44.40

MAA Member Price: $66.60

**Electronic ISBN: 978-1-4704-1059-9
Product Code: MEMO/225/1059.E**

List Price: $74.00

AMS Member Price: $44.40

MAA Member Price: $66.60

# The Sine-Gordon Equation in the Semiclassical Limit: Dynamics of Fluxon Condensates

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*Robert J. Buckingham; Peter D. Miller*

The authors study the Cauchy problem for the sine-Gordon equation in the semiclassical limit with pure-impulse initial data of sufficient strength to generate both high-frequency rotational motion near the peak of the impulse profile and also high-frequency librational motion in the tails. They show that for small times independent of the semiclassical scaling parameter, both types of motion are accurately described by explicit formulae involving elliptic functions. These formulae demonstrate consistency with predictions of Whitham's formal modulation theory in both the hyperbolic (modulationally stable) and elliptic (modulationally unstable) cases.

#### Table of Contents

# Table of Contents

## The Sine-Gordon Equation in the Semiclassical Limit: Dynamics of Fluxon Condensates

- Chapter 1. Introduction 18 free
- 1.1. Pure impulse initial data for the sine-Gordon equation. Connection to the Zakharov-Shabat scattering problem 714 free
- 1.2. Exact solutions. Impulse threshold for generation of rotational waves 1017
- 1.3. Statement of results 1320
- 1.4. Outline of the rest of the Memoir 1926
- 1.5. Notation and terminology 2027

- Chapter 2. Formulation of the Inverse Problem for Fluxon Condensates 2330
- Chapter 3. Elementary Transformations of J(𝑤) 3138
- Chapter 4. Construction of 𝑔(𝑤) 4754
- Chapter 5. Use of 𝑔(𝑤) 8188
- Appendix A. Proofs of Propositions Concerning Initial Data 101108
- Appendix B. Details of the Outer Parametrix in Cases 𝖫 and 𝖱 105112
- Bibliography 135142