Regularity in time of solutions to nonlinear schrödinger equations

In this paper we consider the regularity of solutions to nonlinear Schrödinger equations (NLS), i ∂_tu + 1 2 Δu = F(u, u), (t, x) ∈ R × Rⁿ, u(0) = φ, x ∈ Rⁿ, where F is a polynomial of degree p with complex coefficients. We prove that if the initial function φ is in some Gevrey class, then there exists a positive constant T such that the solution u of NLS is in the Gevrey class of the same order as in the initial data in time variable t ∈[-T, T]0. In particilar, we show that if the initial function φ has an analytic continuation on the complex domain Γ _A1, _A2 = (z ∈ Cⁿ; z_j=x_j+iy_j, -∞ < x_j < + ∞, -A₂-(tan α) |x_j| <y_j < A₂ + (tan α) |x_j| j = 1, 2., n, A₂ > 0) (see Fig. 1), where 0 < α = sin^-1A₁ < π/2 and 0 < A₁ < 1, then there exists positive constants T and β such that the solution u of NLS is analytic in time variable t ∈ [-T, T]0 and has an analytic continuation on (z₀ = t + iτ; |arg z₀| < β <π/2, |t|<T) where sin β < Min (√2A₁/(1 + √2A₁), 2A₂/(3A₂ + [formula](1 + R))) when |x| < R.