Ablowitz Mark J, Biondini Gino, Wang Qiao
Department of Applied Mathematics, University of Colorado, Boulder, CO 80303, USA.
Department of Mathematics, State University of New York, Buffalo, NY 14260, USA.
Proc Math Phys Eng Sci. 2017 Aug;473(2204):20160695. doi: 10.1098/rspa.2016.0695. Epub 2017 Aug 2.
The genus-1 Kadomtsev-Petviashvili (KP)-Whitham system is derived for both variants of the 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-de Vries 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.
针对Kadomtsev-Petviashvili（KP）方程的两种变体，即KPI方程和KPII方程，推导了1类KP-Whitham系统。讨论了KP-Whitham系统的基本性质，包括对称性、精确约化及其可能的完全可积性，以及Korteweg-de Vries方程一维黎曼问题的适当推广。最后，利用KP-Whitham系统研究了KPI方程和KPII方程1类解的线性稳定性性质；结果表明，在Whitham理论的背景下，KPI方程的所有1类解都是线性不稳定的，而KPII方程的所有1类解都是线性稳定的。
