We use so-called energy dependent Schrödinger operators to establish a link
between special classes of solutions of
N-component systems of evolution
equations and finite dimensional Hamiltonian systems on the moduli spaces of
Riemann surfaces. We also investigate the phase space geometry of these
Hamiltonian systems and introduce deformations of the level sets associated to
conserved quantities, which results in a new class of solutions with monodromy
for
N-component systems of pde's.
After constructing a variety of mechanical systems related to the spatial flows of
nonlinear evolution equations, we investigate their semiclassical limits. In
particular, we obtain semiclassical asymptotics for the Bloch eigenfunctions of
the energy dependent Schrödinger operators, which is of importance in
investigating zero-dispersion limits of N-component systems of pde's.
