A dynamical model equation for interfacial gravity-capillary (GC) waves between two semi-infinite fluid layers, with a lighter fluid lying above a heavier one, is derived. The model proposed is based on the fourth-order truncation of the kinetic energy in the Hamiltonian of the full problem, and on weak transverse variations, in the spirit of the Kadomtsev-Petviashvilli equation. It is well known that for the interfacial GC waves in deep water, there is a critical density ratio where the associated cubic nonlinear Schrödinger equations changes type. Our numerical results reveal that, when the density ratio is below the critical value, the bifurcation diagram of plane solitary waves behaves in a way similar to that of the free-surface GC waves on deep water. However, the bifurcation mechanism in the vicinity of the minimum of the phase speed is essentially similar to that of free-surface gravity-flexural waves on deep water, when the density ratio is in the supercritical regime. Different types of lump solitary waves, which are fully localized in both transverse and longitudinal directions, are also computed using our model equation. Some dynamical experiments are carried out via a marching-in-time algorithm.
Wang, Z., Vanden-Broeck, J-M., & Meng, H. (2014). A quasi-planar model for gravity-capillary interfacial waves in deep water. Studies in Applied Mathematics, 133(2), 232-256. https://doi.org/10.1111/sapm.12043