  We rigorously analyze the bifurcation of stationary so called nonlinear Bloch
waves (NLBs) from the spectrum in the Gross-Pitaevskii (GP) equation with a
periodic potential, in arbitrary space dimensions. These are solutions which
can be expressed as finite sums of quasi-periodic functions, and which in a
formal asymptotic expansion are obtained from solutions of the so called
algebraic coupled mode equations. Here we justify this expansion by proving the
existence of NLBs and estimating the error of the formal asymptotics. The
analysis is illustrated by numerical bifurcation diagrams, mostly in 2D. In
addition, we illustrate some relations of NLBs to other classes of solutions of
the GP equation, in particular to so called out--of--gap solitons and truncated
NLBs, and present some numerical experiments concerning the stability of these
solutions.
