A generalized modified Kadomtsev-Petviashvili equation for interfacial wave propagation near the critical depth level

Propagation of interfacial waves near the critical depth level in a two-layer fluid system is investigated. We first present a generalized modified Kadomtsev-Petviashvili (gmKP) equation for weakly nonlinear and dispersive interfacial waves propagating predominantly in the longitudinal direction of a slowly rotating channel with gradually varying topography and sidewalls. For certain type of non-rotating channels, we find two families of periodic-wave solutions, which include solitary-wave solutions and a shock-like solution as limiting cases, to the variable-coefficient gmKP equation. We also show that in this situation the gmKP equation has only unidirectional N-soliton solutions and does not allow soliton resonance to occur. In a rotating uniform channel, our small-time asymptotic analysis and numerical study of the gmKP equation show that, depending on the relative importance of the cubic nonlinearity to quadratic nonlinearity, the wavefront of a Kelvin solitary wave may curve either forward or backward, trailed by a small train of Poincaré waves. When these two nonlinearities almost balance each other, the wavefront becomes almost straight-crested across the channel, and the trailing Poincaré waves diminish.