  We consider the Schrodinger equation with an external potential and a cubic
nonlinearity, in the semiclassical limit. The initial data are sums of WKB
states, with smooth phases and smooth, compactly supported initial amplitudes,
with disjoint supports. We show that like in the linear case, a superposition
principle holds on some time interval independent of the semiclassical
parameter, in several regimes in term of the size of initial data with respect
to the semiclassical parameter. For large data, we invoke properties of
compressible Euler equations. For smaller data, we show that there may be no
nonlinear interferences on some time interval independent of the semiclassical
parameter, and interferences for later time, thanks to explicit computations
available for particular phases.
