  The boundary conditions at the deformable interface between two contacting
fluids are derived for the general case of the large-amplitude perturbations.
The interface is modeled as perturbed free boundary that evolves in time, and
the non-linear description is performed and analyzed in a wide range of
physical situations. The differential equations of the interfacial motion thus
obtained may be useful in research of the non-linear development of the
classical hydrodynamic instabilities. They should play an important role in the
understanding of the hydrodynamic phenomena associated with the flows involving
complex interfacial evolution including parametric control of the boundaries in
continua (for example, with electromagnetic field or/and vibration).
