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We consider the Schrodinger equation with an external potential and a cubicnonlinearity, in the semiclassical limit. The initial data are sums of WKBstates, with smooth phases and smooth, compactly supported initial amplitudes,with disjoint supports. We show that like in the linear case, a superpositionprinciple holds on some time interval independent of the semiclassicalparameter, in several regimes in term of the size of initial data with respectto the semiclassical parameter. For large data, we invoke properties ofcompressible Euler equations. For smaller data, we show that there may be nononlinear interferences on some time interval independent of the semiclassicalparameter, and interferences for later time, thanks to explicit computationsavailable for particular phases.

On nonlinear effects in multiphase WKB analysis for the nonlinear Schrodinger equation