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Long time stability of plane wave solutions to Schrödinger equation on Torus
Applicable Analysis ( IF 1.1 ) Pub Date : 2020-09-23 , DOI: 10.1080/00036811.2020.1823375
Lufang Mi 1 , Yingte Sun 2 , Peizhen Wang 3
Affiliation  

ABSTRACT

We prove the long time orbital stability of the plane wave solutions to the nonlinear Schrödinger equation (NLS) in the defocusing (λ=1) or focusing (λ=1) case, iut=Δu+λ|u|2u,xTd,tR. More precisely, in a Gevrey space Gσ:=u: | uσ2=aZde2σ|a||ua|2< for some positive constant σ, we show that solution with the initial datum in the 4ϵ-neighborhood of the plane wave solution still stays in the Cϵ-neighborhood ( C>4 ) of the plane wave solution for a subexponential long time |t|ϵζ|lnϵ|ϱ, where ζ=min{14,σσ},σ>σ>0 and 0<ϱ<1/6.



中文翻译:

圆环上薛定谔方程平面波解的长期稳定性

摘要

我们证明了在散焦中非线性薛定谔方程(NLS)的平面波解的长期轨道稳定性(λ=1) 或聚焦 (λ=-1) 案子,一世=-Δ+λ||2,Xd,R.更准确地说,在 Gevrey 空间中Gσ:= | σ2=一个Zde2σ|一个||一个|2<对于一些正常数σ,我们用初始数据在4ε-平面波解的邻域仍然停留在Cε-次指数长时间内平面波解的邻域 ( C >4 )||ε-ζ|lnε|ϱ,在哪里ζ=分钟{14,σ-σ'},σ>σ'>00<ϱ<1/6.

更新日期:2020-09-23
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