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The Sine-Gordon Equation in the Semiclassical Limit: Dynamics of Fluxon Condensates
 
Robert J. Buckingham University of Cincinnati, Cincinnati, OH
Peter D. Miller University of Michigan, Ann Arbor, MI
The Sine-Gordon Equation in the Semiclassical Limit: Dynamics of Fluxon Condensates
eBook ISBN:  978-1-4704-1059-9
Product Code:  MEMO/225/1059.E
List Price: $74.00
MAA Member Price: $66.60
AMS Member Price: $44.40
The Sine-Gordon Equation in the Semiclassical Limit: Dynamics of Fluxon Condensates
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The Sine-Gordon Equation in the Semiclassical Limit: Dynamics of Fluxon Condensates
Robert J. Buckingham University of Cincinnati, Cincinnati, OH
Peter D. Miller University of Michigan, Ann Arbor, MI
eBook ISBN:  978-1-4704-1059-9
Product Code:  MEMO/225/1059.E
List Price: $74.00
MAA Member Price: $66.60
AMS Member Price: $44.40
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2252013; 136 pp
    MSC: Primary 35;

    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
     
     
    • Chapters
    • 1. Introduction
    • 2. Formulation of the Inverse Problem for Fluxon Condensates
    • 3. Elementary Transformations of J$(w)$
    • 4. Construction of $g(w)$
    • 5. Use of $g(w)$
    • A. Proofs of Propositions Concerning Initial Data
    • B. Details of the Outer Parametrix in Cases $\mathsf {L}$ and $\mathsf {R}$
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2252013; 136 pp
MSC: Primary 35;

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.

  • Chapters
  • 1. Introduction
  • 2. Formulation of the Inverse Problem for Fluxon Condensates
  • 3. Elementary Transformations of J$(w)$
  • 4. Construction of $g(w)$
  • 5. Use of $g(w)$
  • A. Proofs of Propositions Concerning Initial Data
  • B. Details of the Outer Parametrix in Cases $\mathsf {L}$ and $\mathsf {R}$
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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