Whitham modulation theory for the Kadomtsev-Petviashvili equation
Mark J. Ablowitz, Gino Biondini, Qiao Wang
TL;DR
This paper derives and analyzes the genus-1 Whitham modulation system for the KP equation variants, exploring its properties, integrability, and stability of solutions, revealing instability in KPI and stability in KPII.
Contribution
It introduces the genus-1 KP-Whitham system, discusses its properties and potential integrability, and applies it to analyze the linear stability of solutions.
Findings
Genus-1 KP-Whitham system derived for KPI and KPII.
All genus-1 KPI solutions are linearly unstable.
All genus-1 KPII solutions are linearly stable.
Abstract
The genus-1 KP-Whitham system is derived for both variants of the Kadomtsev-Petviashvili (KP) equation (namely, the KPI and KPII equations). The basic properties of the KP-Whitham system, including symmetries, exact reductions, and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-deVries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable while all genus-1 solutions of KPII {are linearly stable within the context of Whitham theory.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.