Abstract
We consider the nonlinear electrohydrodynamic stability of wave packets of capillary-gravity waves in fluids of any depth; they travel predominantly in one direction, however the wave amplitudes are modulated slowly in both horizontal directions. The method of multiple time scales is used to obtain a nonlinear Schrodinger equation describing the behaviour of the perturbed system. The envelope solutions of steady form are obtained in terms of the Jacobian elliptic functions. It follows that various types of envelope solutions of the modulated Stokes waves may exist depending on the relative signs of terms representing dispersive and nonlinear effects; the solitary and periodic envelope solutions for the general case of any liquid depth are described. It is also shown that the evolution of the envelope is governed by two coupled partial differential equations with cubic nonlinearity. The stability of solitons with respect to transverse perturbations is investigated It is found that such wave packets are stable to short waves, and unstable to long disturbances, and the envelope solitons and the waveguides are always unstable, and the stability of the system depends on the values of the dielectric constant ratio, the electric field the wavenumber, and the depth of the fluid.