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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)
Published electronically: May 12, 2015
Keywords: KP-II, line soliton, stability
MSC: Primary 35B35, 37K40; Secondary 35Q35

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Table of Contents


  • 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


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.

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