The KdV/KP-I limit of the Nonlinear Schrodinger equation

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.