  We introduce a dispersive regularization of the compressible Euler equations
in Lagrangian coordinates, in the one-dimensional torus. We assume a Van der
Waals pressure law, which presents both hyperbolic and elliptic zones. The
dispersive regularization is of Schroedinger type. In particular, the
regularized system is complex-valued. It has a conservation law, which, for
real unknowns, is identical to the energy of the unregularized physical system.
The regularized system supports high-frequency solutions, with an existence
time or an amplitude which depend strongly on the pressure law.
