  We consider the small time semi-classical limit for nonlinear Schrodinger
equations with defocusing, smooth, nonlinearity. For a super-cubic
nonlinearity, the limiting system is not directly hyperbolic, due to the
presence of vacuum. To overcome this issue, we introduce new unknown functions,
which are defined nonlinearly in terms of the wave function itself. This
approach provides a local version of the modulated energy functional introduced
by Y.Brenier. The system we obtain is hyperbolic symmetric, and the
justification of WKB analysis follows.
