The unified Kadomtsev–Petviashvili equation for interfacial waves

Abstract

In this paper, the propagation of interfacial waves in a two-layered fluid system is investigated. The interfacial waves are weakly nonlinear and dispersive and propagate in a slowly rotating channel with varying topography and sidewalls, and a weak steady background current field. An evolution equation for the interfacial displacement is derived for waves propagating predominantly in the longitudinal direction of the channel. This new evolution equation is called the unified Kadomtsev–Petviashvili (uKP) equation because most of the KP-type equations existing in the literature for both surface water waves and interfacial waves are special cases of the new evolution equation. The Painlevé PDE test is used to find the conditions under which the uKP equation can be solved by the inverse scattering transform. When these conditions are satisfied, elementary transformations are found to reduce the uKP equation to one of the completely integrable equations: the KP, the Korteweg–de Vries (KdV) or the cylindrical KdV equations. The integral invariants associated with the uKP equation for waves propagating in a varying channel are obtained and their relations with the conservation of mass and energy are discussed.