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Stability of line solitons for the KP-II equation in $\mathbb {R}^2$
About this Title
Tetsu Mizumachi, Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739-8521 Japan
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: https://doi.org/10.1090/memo/1125
Published electronically: May 12, 2015
Keywords: KP-II,
line soliton,
stability
MSC: Primary 35B35, 37K40; Secondary 35Q35
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
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.- J. C. Alexander, R. L. Pego, and R. L. Sachs, On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation, Phys. Lett. A 226 (1997), no. 3-4, 187–192. MR 1435907, DOI 10.1016/S0375-9601(96)00921-8
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