**Memoirs of the American Mathematical Society**

2015;
95 pp;
Softcover

MSC: Primary 35; 37;

Print ISBN: 978-1-4704-1424-5

Product Code: MEMO/238/1125

List Price: $76.00

Individual Member Price: $45.60

**Electronic ISBN: 978-1-4704-2613-2
Product Code: MEMO/238/1125.E**

List Price: $76.00

Individual Member Price: $45.60

# Stability of Line Solitons for the KP-II Equation in $\mathbb{R}^{2}$

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*Tetsu Mizumachi*

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

# Table of Contents

## Stability of Line Solitons for the KP-II Equation in $\mathbb{R}^{2}$

- Cover Cover11 free
- Title page i2 free
- Acknowledgments vii8 free
- Chapter 1. Introduction 110 free
- Chapter 2. The Miura transformation and resonant modes of the linearized operator 716 free
- Chapter 3. Semigroup estimates for the linearized KP-II equation 1928
- Chapter 4. Preliminaries 2736
- Chapter 5. Decomposition of the perturbed line soliton 3140
- Chapter 6. Modulation equations 3746
- Chapter 7. À priori estimates for the local speed and the local phase shift 4554
- Chapter 8. The 𝐿²(\R²) estimate 5362
- Chapter 9. Decay estimates in the exponentially weighted space 5766
- Chapter 10. Proof of Theorem 1.1 6372
- Chapter 11. Proof of Theorem 1.4 6574
- Chapter 12. Proof of Theorem 1.5 6978
- Appendix A. Proof of Lemma 6.1 7786
- Appendix B. Operator norms of 𝑆^{𝑗}_{𝑘} and ̃𝐶_{𝑘} 7988
- Appendix C. Proofs of Claims 6.2, 6.3 and 7.1 8392
- Appendix D. Estimates of 𝑅^{𝑘} 8594
- Appendix E. Local well-posedness in exponentially weighted space 91100
- Bibliography 93102
- Back Cover Back Cover1110