Gevrey regularizing effect for the (generalized) Korteweg-de Vries equation and nonlinear Schrödinger equations

Abstract

This paper is concerned with regularizing effects of solutions to the (generalized) Korteweg-de Vries equation(gKdV){∂tu+∂x3u=λup−1∂xu,(t,x)∈ℝ×ℝ,u(0)=ϕ,x∈ℝ, and nonlinear Schrödinger equations in one space dimension(NLS){i∂tu+12∂x2u=G(u,u¯),(t,x)∈ℝ×ℝ,u(0)=ψ,x∈ℝ, where p is an integer satisfying p ≥ 2, λ ∊ ℂ and G is a polynomial of (u,u¯). We prove that if the initial function ϕ is in a Gevrey class of order 3 defined in Section 1, then there exists a positive time T such that the solution of (gKdV) is analytic in space variable for t ∊ [−T, T]\\{0}, and if the initial function ψ in a Gevrey class of order 2, then there exists a positive time T such that the solution of (NLS) is analytic in space variable for t ∊ [−T, T]\\{0}.