Nonlinear waves on the surface of a charged liquid. Instability, bifurcation, and nonequilibrium shapes of the charged surface (Q1060114)

An analysis is made of the propagation of nonlinear waves on the surface of a charged liquid placed between two flat parallel plates used as electrodes which are held at a fixed voltage difference. Using the method of multiple scales, it is shown that for long wave lengths the wave propagation is governed by an equation of Korteweg-de Vries type which admits solution in the form of a soliton. The presence of surface charges reduces the velocity of the soliton which may be either an elevation or a depression depending on the strength of the voltage, thickness of the liquid layer, distance between the upper plate and the undisturbed free surface and the surface tension. The electric field can either decrease or increase the width \(\Delta_ 0\) of the soliton and may even lead to the break up of the soliton \((\Delta_ 0\to \infty)\). It is further shown that in the case of one-dimensional propagation (plane case), there is a loss of stability leading to hard bifurcation. Finally the problem of the equilibrium shapes of a charged liquid is investigated in the nonlinear formulation of the dynamics of solitary forms (e.g. lunes or trenches) on the surface.