Semi-classical analysis for Hartree equations in some supercritical cases

1 An introduction to semi-classical analysis for a class of nonlinear Schrödinger equations We consider the asymptotic behavior of the solution to the following semi-classical nonlinear Schrödinger equation:    iε∂ t u ε + 1 2 ε 2 ∆u ε = ε α F (u ε), (t, x) ∈ R 1+N u ε |t=0 = u ε 0 (NLS ε) as positive parameter ε → 0. The parameter ε corresponds to the Planck constant and the limit ε → 0 is known as the semi-classical limit. It is relevant when coupling quantum models to classical models. We assume the initial datum of (NLS ε) is of the following form: u ε 0 (x) = e −i |x| 2 2ε f (x). (1) Then, the caustic occurs at origin at t = 1. We shall observe this. Let us consider the approximate solution of the following form u ε (t, x) ∼ a 0 (t, x)e i φ(t,x) ε (ε → 0). (2) If α > 0 then, substituting this to (NLS ε), we obtain the eikonal equation (characteristic equation) ∂ t φ + 1 2 |∇φ| 2 = 0. (3) Then, its bicharacteristics are the system            ˙ x = ξ, ˙ t = 1, ˙ ξ = 0, ˙ τ = 0,