The nonlinear dynamics of the free surface of an ideal dielectric fluid placed in an external inclined electric field is studied. A system of nonlinear integro-differential equations describing the evolution of nonlinear waves in the small-angle approximation is obtained within the framework of the Hamiltonian formalism. It is shown that for a liquid with high permittivity, the equations allow the solution in the form of plane waves of arbitrary shape propagating without distortion in the direction of the horizontal component of the external field. The conditions for the implementation of this regime are formulated.
We have studied the nonlinear dynamics of the interface of perfect dielectric liquids in the presence of a tangential break of velocity on the boundary when the stabilizing influence of the horizontal electric field. The possibility of realization of the mode of motion in which liquids move along the field lines is shown. The boundary motion equations corresponding to this regime allow reduction to an arbitrary number of ordinary differential equations describing the propagation and interaction of solitary waves - rational solitons. It is shown that at weak interaction solitary waves after collision restore the form and speed, and at strong interaction can form a wave packet (breather), or mutually to be destroyed.
The exact solutions of the problem of the equilibrium configuration of an uncharged cylindrical jet of a conducting liquid in a transverse electric field are found (in a two-dimensional plane formulation). The cross-section of the jet moving between two flat electrodes is deformed by electrostatic forces (capillary forces play a deterrent role). According to the solutions, initially the circular section of the jet can be significantly (formally unlimited) stretched along the field lines. The domain of existence of the solution depending on the applied potential difference and the length of the interelectrode interval is established. The exact solutions of a close (from a mathematical point of view) problem on possible equilibrium configurations of the free surface of a perfectly conducting liquid in a nonuniform magnetic field of a system of linear conductors with currents are also obtained.
