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Stability of line solitons for the KP-II equation in $\mathbb {R}^2$


About this Title

Tetsu Mizumachi

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 238, Number 1125
ISBNs: 978-1-4704-1424-5 (print); 978-1-4704-2613-2 (online)
DOI: http://dx.doi.org/10.1090/memo/1125
Published electronically: May 12, 2015
Keywords:KP-II, line soliton

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Table of Contents


Chapters

  • Acknowledgments
  • Chapter 1. Introduction
  • Chapter 2. The Miura transformation and resonant modes of the linearized operator
  • Chapter 3. Semigroup estimates for the linearized KP-II equation
  • Chapter 4. Preliminaries
  • Chapter 5. Decomposition of the perturbed line soliton
  • Chapter 6. Modulation equations
  • Chapter 7. A priori estimates for the local speed and the local phase shift
  • Chapter 8. The $L^2(\mathbb {R}^2)$ estimate
  • Chapter 9. Decay estimates in the exponentially weighted space
  • Chapter 10. Proof of Theorem 1.1
  • Chapter 11. Proof of Theorem 1.4
  • Chapter 12. Proof of Theorem 1.5
  • Appendix A. Proof of Lemma 6.1
  • Appendix B. Operator norms of $S^j_k$ and $\protect \widetilde {C_k}$
  • Appendix C. Proofs of Claims 6.2, 6.3 and 7.1
  • Appendix D. Estimates of $R^k$
  • Appendix E. Local well-posedness in exponentially weighted space

Abstract


We prove nonlinear stability of line soliton solutions of the KP-II equation with respect to transverse perturbations that are exponentially localized as . We find 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 . 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.

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