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# memo_has_moved_text();Stability of line solitons for the KP-II equation in $\mathbb {R}^2$

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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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 $x\to \infty$. 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 $y=\pm \infty$. 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.