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

Anne De Bouard, Nakao Hayashi, Keiichi Kato

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45 Citations (Scopus)

Abstract

This paper is concerned with regularizing effects of solutions to the (generalized) Korteweg-de Vries equation {∂tu+∂x 3u=λup−1xu,(t,x)∈ℝ×ℝ,u(0)=ϕ,x∈ℝ, and nonlinear Schrödinger equations in one space dimension {i∂tu+12∂x 2u=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}.

Original languageEnglish
Pages (from-to)673-725
Number of pages53
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume12
Issue number6
DOIs
Publication statusPublished - 1995 Nov 1
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Applied Mathematics

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