  We consider the nonlinear Schrodinger equation with defocusing, smooth,
nonlinearity. Below the critical Sobolev regularity, it is known that the
Cauchy problem is ill-posed. We show that this is even worse: there is a loss
of regularity, in the same spirit as the result due to G.Lebeau in the case of
the wave equation. We use an isotropic change of variable, which reduces the
problem to a super-critical WKB analysis. For super-cubic, smooth nonlinearity,
this analysis is new, and relies on the introduction of a modulated energy
functional a la Brenier.

Analysis of PDEs