## Abstract

In this paper we consider the regularity of solutions to nonlinear Schrödinger equations (NLS), i ∂_{t}u + 1 2 Δu = F(u, u), (t, x) ∈ R × R^{n}, u(0) = φ, x ∈ R^{n}, 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 ∈ C^{n}; z_{j}=x_{j}+iy_{j}, -∞ < x_{j} < + ∞, -A_{2}-(tan α) |x_{j}| <y_{j} < A_{2} + (tan α) |x_{j}| j = 1, 2., n, A_{2} > 0) (see Fig. 1), where 0 < α = sin^{-1}A_{1} < π/2 and 0 < A_{1} < 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 (z_{0} = t + iτ; |arg z_{0}| < β <π/2, |t|<T) where sin β < Min (√2A_{1}/(1 + √2A_{1}), 2A_{2}/(3A_{2} + [formula](1 + R))) when |x| < R.

Original language | English |
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Pages (from-to) | 253-277 |

Number of pages | 25 |

Journal | Journal of Functional Analysis |

Volume | 128 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1995 Mar |