Geometric optics and boundary layers for Nonlinear-Schrödinger Equations

where φ0 is real-valued. We are interested in the semiclassical limit ε → 0. The nonlinear Schrödinger equation (1) appears, for instance, in optics, and also as a model for Bose-Einstein condensates, with f(ρ) = ρ − 1, and the equation is termed Gross-Pitaevskii equation, or also with f(ρ) = ρ2 (see [13]). Some more complicated nonlinearities are also used especially in low dimensions, see [12]. At first, let us focus on the case Ω = Rd. To guess the formal limit, when ε goes to zero, it is classical to use the Madelung transform, i.e to seek for a solution of (1) under the form Ψ = √ ρε exp (