  The dynamics of singularity formation on the interface between two ideal
fluids is studied for the Kelvin-Helmholtz instability development within the
Hamiltonian formalism. It is shown that the equations of motion derived in the
small interface angle approximation (gravity and capillary forces are
neglected) admit exact solutions in the implicit form. The analysis of these
solutions shows that, in the general case, weak root singularities are formed
on the interface in a finite time for which the curvature becomes infinite,
while the slope angles remain small. For Atwood numbers close to unity in
absolute values, the surface curvature has a definite sign correlated with the
boundary deformation directed towards the light fluid. For fluids with
comparable densities, the curvature changes its sign in a singular point. In
the particular case of fluids with equal densities, the obtained results are
consistent with those obtained by Moore based on the Birkhoff-Rott equation
analysis.
