Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, United Kingdom.
Chaos. 2013 Mar;23(1):013126. doi: 10.1063/1.4792268.
There exist two versions of the Kadomtsev-Petviashvili (KP) equation, related to the Cartesian and cylindrical geometries of the waves. In this paper, we derive and study a new version, related to the elliptic cylindrical geometry. The derivation is given in the context of surface waves, but the derived equation is a universal integrable model applicable to generic weakly nonlinear weakly dispersive waves. We also show that there exist nontrivial transformations between all three versions of the KP equation associated with the physical problem formulation, and use them to obtain new classes of approximate solutions for water waves.
存在两种卡姆登-彼得斯维里（KP）方程版本，分别与波的笛卡尔和圆柱几何有关。在本文中，我们推导出并研究了一个与椭圆圆柱几何有关的新版本。推导是在表面波的背景下进行的，但所得到的方程是一个通用的可积模型，适用于一般的弱非线性弱色散波。我们还表明，与物理问题的公式化相关联的 KP 方程的所有三个版本之间存在非平凡的变换，并利用它们为水波获得新的近似解类。
