eBook ISBN:  9781470426132 
Product Code:  MEMO/238/1125.E 
List Price:  $76.00 
MAA Member Price:  $68.40 
AMS Member Price:  $45.60 
eBook ISBN:  9781470426132 
Product Code:  MEMO/238/1125.E 
List Price:  $76.00 
MAA Member Price:  $68.40 
AMS Member Price:  $45.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 238; 2015; 95 ppMSC: Primary 35; 37;
The author proves nonlinear stability of line soliton solutions of the KPII 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

Chapters

Acknowledgments

1. Introduction

2. The Miura transformation and resonant modes of the linearized operator

3. Semigroup estimates for the linearized KPII 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 wellposedness in exponentially weighted space


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The author proves nonlinear stability of line soliton solutions of the KPII 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.

Chapters

Acknowledgments

1. Introduction

2. The Miura transformation and resonant modes of the linearized operator

3. Semigroup estimates for the linearized KPII 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 wellposedness in exponentially weighted space