Stability of nonlinear waves on the ideal liquid surface in a tangential electric field

The propagation of waves with arbitrary amplitudes on the free surface of the ideal dielectric liquid in a strong tangential electric field is considered. A dispersion relation for the evolution of small perturbations of a stationary wave profile of the surface of a liquid with a large permittivity is obtained using conformal variables. It is established that, irrespective of the wave geometry, the amplitudes of small-scale perturbations do not increase with the time, which is evidence for the stability of wave solutions.