Nonlinear asymptotic calculation of the Kelvin–Helmholtz instability

The problem of periodic capillary-gravitational wave motion on the uniformly charged interface between two ideal immiscible incompressible liquids is solved in the third order of smallness. The lower liquid is assumed to be ideally conducting, while the upper one is a dielectric executing translational motion parallel to the interface with a constant velocity. A nonlinear frequency correction in the resonance form is found. It is shown that the positions of internal nonlinear resonances depend on the sum of the field and Weber parameters, the density ratio of the liquids, and the wave number. When the upper liquid is denser than the lower one, resonances are absent.