Stability of Line Solitons for the KP-II Equation in R(2)

ISBN-13: 9781470414245 / Angielski

Tetsu Mizumachi

Opis

The author proves nonlinear stability of line soliton solutions of the KP-II equation with respect to transverse perturbations that are exponentially localized as $x oinfty$. 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 $y=pminfty$. 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.

Spis treści

• Introduction
• The Miura transformation and resonant modes of the linearized operator
• Semigroup estimates for the linearized KP-II equation
• Preliminaries
• Decomposition of the perturbed line soliton
• Modulation equations
• A priori estimates for the local speed and the local phase shift
• The $L^2(\mathbb^2)$ estimate
• Decay estimates in the exponentially weighted space
• Proof of Theorem 1.1
• Proof of Theorem 1.4
• Proof of Theorem 1.5
• Appendix A. Proof of Lemma 6.1
• Appendix B. Operator norms of $S^j_k$ and $\widetilde$
• 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
• Bibliography