  We justify rigorously the convergence of the amplitude of solutions of
Nonlinear-Schr\"odinger type Equations with non zero limit at infinity to an
asymptotic regime governed by the Korteweg-de Vries equation in dimension 1 and
the Kadomtsev-Petviashvili I equation in dimensions 2 and more. We get two
types of results. In the one-dimensional case, we prove directly by energy
bounds that there is no vortex formation for the global solution of the NLS
equation in the energy space and deduce from this the convergence towards the
unique solution in the energy space of the KdV equation. In arbitrary
dimensions, we use an hydrodynamic reformulation of NLS and recast the problem
as a singular limit for an hyperbolic system. We thus prove that smooth $H^s$
solutions exist on a time interval independent of the small parameter. We then
pass to the limit by a compactness argument and obtain the KdV/KP-I equation.
