The KdV/KP-I Limit of the Nonlinear Schrödinger Equation

International audienceWe justify rigorously the convergence of the amplitude of solutions of nonlinear Schrödinger-type equations with nonzero limit at infinity to an asymptotic regime governed by the Korteweg-de Vries (KdV) equation in dimension 1 and the Kadomtsev-Petviashvili I (KP-I) equation in dimensions 2 and greater. 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 nonlinear Schrödinger equation in the energy space and deduce from this the convergence toward the unique solution in the energy space of the KdV equation. In arbitrary dimensions, we use a hydrodynamic reformulation of the nonlinear Schrödinger equation and recast the problem as a singular limit for a 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