SPECTRAL FORMULATION OF THE TWO FLUID EULER EQUATIONS WITH A FREE INTERFACE AND LONG WAVE REDUCTIONS

摘要

A nonlocal spectral formulation of classic water waves is derived, and its connection to the classic water wave equations and the Dirichlet-Neumann operator is explored. The nonlocal spectral formulation is also extended to a two-fluid system with a free interface, from which long wave asymptotic reductions are obtained. Of particular interest is an asymptotically distinguished (2+1)-dimensional generalization of the intermediate long wave equation, which includes the Kadomtsev-Petviashvili equation and the Benjamin-Ono equation as limiting cases. Lump-type solutions to this (2+1)-dimensional ILW equation are obtained, and the speed versus amplitude relationship is shown to be linear in the shallow, intermediate, and deep water regimes.