  Stability of inviscid shear shallow water flows with free surface is studied
in the framework of the Benney equations. This is done by investigating the
generalized hyperbolicity of the integrodifferential Benney system of
equations. It is shown that all shear flows having monotonic convex velocity
profiles are stable. The hydrodynamic approximations of the model corresponding
to the classes of flows with piecewise linear continuous and discontinuous
velocity profiles are derived and studied. It is shown that these
approximations possess Hamiltonian structure and a complete system of Riemann
invariants, which are found in an explicit form. Sufficient conditions for
hyperbolicity of the governing equations for such multilayer flows are
formulated. The generalization of the above results to the case of stratified
fluid is less obvious, however, it is established that vorticity has a
stabilizing effect.
