Regularity in Time of Solutions to Nonlinear Schrِdinger Equations

In this paper we consider the regularity of solutions to nonlinear Schrödinger equations (NLS), i ∂tu + 12 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; zj=xj+iyj, -∞ < xj < + ∞, -A2-(tan α) |xj| <yj < A2 + (tan α) |xj| 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.