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Stability of Line Solitons for the KP-II Equation in $\mathbb{R}^2$
 
Tetsu Mizumachi Kyushu University, Fukuoka, Japan
Stability of Line Solitons for the KP-II Equation in R^2
eBook ISBN:  978-1-4704-2613-2
Product Code:  MEMO/238/1125.E
List Price: $76.00
MAA Member Price: $68.40
AMS Member Price: $45.60
Stability of Line Solitons for the KP-II Equation in R^2
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Stability of Line Solitons for the KP-II Equation in $\mathbb{R}^2$
Tetsu Mizumachi Kyushu University, Fukuoka, Japan
eBook ISBN:  978-1-4704-2613-2
Product Code:  MEMO/238/1125.E
List Price: $76.00
MAA Member Price: $68.40
AMS Member Price: $45.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2382015; 95 pp
    MSC: Primary 35; 37;

    The author proves nonlinear stability of line soliton solutions of the KP-II equation with respect to transverse perturbations that are exponentially localized as \(x\to\infty\). He finds that the amplitude of the line soliton converges to that of the line soliton at initial time whereas jumps of the local phase shift of the crest propagate in a finite speed toward \(y=\pm\infty\). The local amplitude and the phase shift of the crest of the line solitons are described by a system of 1D wave equations with diffraction terms.

  • Table of Contents
     
     
    • Chapters
    • Acknowledgments
    • 1. Introduction
    • 2. The Miura transformation and resonant modes of the linearized operator
    • 3. Semigroup estimates for the linearized KP-II equation
    • 4. Preliminaries
    • 5. Decomposition of the perturbed line soliton
    • 6. Modulation equations
    • 7. À priori estimates for the local speed and the local phase shift
    • 8. The $L^2(\mathbb {R}^2)$ estimate
    • 9. Decay estimates in the exponentially weighted space
    • 10. Proof of Theorem
    • 11. Proof of Theorem
    • 12. Proof of Theorem
    • A. Proof of Lemma
    • B. Operator norms of $S^j_k$ and $\widetilde {C_k}$
    • C. Proofs of Claims , and
    • D. Estimates of $R^k$
    • E. Local well-posedness in exponentially weighted space
  • 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: 2382015; 95 pp
MSC: Primary 35; 37;

The author proves nonlinear stability of line soliton solutions of the KP-II equation with respect to transverse perturbations that are exponentially localized as \(x\to\infty\). He finds that the amplitude of the line soliton converges to that of the line soliton at initial time whereas jumps of the local phase shift of the crest propagate in a finite speed toward \(y=\pm\infty\). The local amplitude and the phase shift of the crest of the line solitons are described by a system of 1D wave equations with diffraction terms.

  • Chapters
  • Acknowledgments
  • 1. Introduction
  • 2. The Miura transformation and resonant modes of the linearized operator
  • 3. Semigroup estimates for the linearized KP-II equation
  • 4. Preliminaries
  • 5. Decomposition of the perturbed line soliton
  • 6. Modulation equations
  • 7. À priori estimates for the local speed and the local phase shift
  • 8. The $L^2(\mathbb {R}^2)$ estimate
  • 9. Decay estimates in the exponentially weighted space
  • 10. Proof of Theorem
  • 11. Proof of Theorem
  • 12. Proof of Theorem
  • A. Proof of Lemma
  • B. Operator norms of $S^j_k$ and $\widetilde {C_k}$
  • C. Proofs of Claims , and
  • D. Estimates of $R^k$
  • E. Local well-posedness in exponentially weighted space
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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