Numerical study of surface solitary wave and Kelvin solitary wave propagation in a wide channel

The Petrov-Galerkin finite-element method is used to solve the unified Kadomtsev-Petviashvili equation [Chen and Liu, J. Fluid Mech. 288 (1995) 383]. Numerical experiments have been focused on studying the effect of slowly varying topography on the propagation of surface solitary waves in a stationary channel and Kelvin solitary waves in a rotating channel. We find that in the absence of rotation, an oblique incident solitary wave propagating over a three-dimensional shelf in a straight wide channel will eventually develop into a series of uniform straight-crested solitary waves, together with a train of small oscillatory waves moving upstream. With proper phase shifts, the shapes of these final two-dimensional solitary waves coincide with those of solitary waves emerging from a corresponding normal incident solitary wave propagating over the corresponding two-dimensional shelf. In a two-layered rotating channel, the variation of topography does not have much effect on the propagation of a Kelvin solitary wave of depression, whereas it can have a significant influence on the propagation of a Kelvin solitary wave of elevation. Explanations for these numerical findings are given.