  We analyze nonlinear dynamics of the Kelvin-Helmholtz quantum instability of
the He-II free surface, which evolves during counterpropagation of the normal
and superfluid components of liquid helium. It is shown that in the vicinity of
the linear stability threshold, the evolution of the boundary is described by
the $|\phi|^4$ Klein-Gordon equation for the complex amplitude of the excited
wave with cubic nonlinearity. It is important that for any ratio of the
densities of the helium component, the nonlinearity plays a destabilizing role,
accelerating the linear instability evolution of the boundary. The conditions
for explosive growth of perturbations of the free surface are formulated using
the integral inequality approach. Analogy between the Kelvin-Helmholtz quantum
instability and electrohydrodynamic instability of the free surface of liquid
helium charged by electrons is considered.
