  To the spectral curves of smooth periodic solutions of the $n$-wave equation
the points with infinite energy are added. The resulting spaces are considered
as generalized Riemann surfcae. In general the genus is equal to infinity,
nethertheless these Riemann surfaces are similar to compact Riemann surfaces.
After proving a Riemann Roch Theorem we can carry over most of the
constructions of the finite gap potentials to all smooth periodic potentials.
The symplectic form turns out to be closely related to Serre duality. Finally
we prove that all non-linear PDE's, which belong to the focussing case of the
non-linear Schr\"odinger equation, have global solutions for arbitrary smooth
periodic inital potantials.
