Global existence of small analytic solutions to schrödinger equations with quadratic nonlinearity

We study the nonlinear Schrödinger equations {i∂_tu + 1/2 Δu = F(u, ∇u, ū, ∇ū), (t, x) ∈ R × Rⁿ u(0, x) = ∈₀φ, x ∈ Rⁿ. (*) When n = 3, 4, F : C²ⁿ⁺² → C is quadratic and ∈₀ is sufficiently small, it is shown that small analytic solutions of (*) exist globally in time under the condition |∂_uF| + |∂ūF| ≤ C|∇u|. Furthermore we give a global existence theorem when n = 2 and F = λ(∂₁u∂₂ū - ∂₁ū∂₂u). We notice that we do not need the condition ∂_∂juF is pure imaginary which ensures the use of classical energy estimates.