On nonlinear effects in multiphase WKB analysis for the nonlinear Schrödinger equation *

We consider the Schrödinger 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 régimes in term of the size of initial data with respect to the semiclassical parameter. When nonlinear effects are strong in terms of the semiclassical parameter, we invoke properties of compressible Euler equations. For weaker nonlinear effects, 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.