TY - JOUR

T1 - Gevrey regularizing effect for the (generalized) Korteweg-de Vries equation and nonlinear Schrödinger equations

AU - De Bouard, Anne

AU - Hayashi, Nakao

AU - Kato, Keiichi

N1 - Publisher Copyright:
© 2016 L'Association Publications de l'Institut Henri Poincaré

PY - 1995/11/1

Y1 - 1995/11/1

N2 - This paper is concerned with regularizing effects of solutions to the (generalized) Korteweg-de Vries equation {∂tu+∂x 3u=λup−1∂xu,(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}.

AB - This paper is concerned with regularizing effects of solutions to the (generalized) Korteweg-de Vries equation {∂tu+∂x 3u=λup−1∂xu,(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}.

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U2 - 10.1016/S0294-1449(16)30148-2

DO - 10.1016/S0294-1449(16)30148-2

M3 - Article

AN - SCOPUS:85011573784

SN - 0294-1449

VL - 12

SP - 673

EP - 725

JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

IS - 6

ER -