Forward-modulated damping estimates and nonlocalized stability of periodic Lugiato–Lefever waves

IF 1.8 1区 数学 Q1 MATHEMATICS, APPLIED
K. Zumbrun
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引用次数: 1

Abstract

In an interesting recent analysis, Haragus-Johnson-Perkins-de Rijk have shown modulational stability under localized perturbations of steady periodic solutions of the Lugiato-Lefever equation (LLE), in the process pointing out a difficulty in obtaining standard"nonlinear damping estimates"on modulated perturbation variables to control regularity of solutions. Here, we point out that in place of standard"inverse-modulated"damping estimates, one can alternatively carry out a damping estimate on the"forward-modulated"perturbation, noting that norms of forward- and inverse-modulated variables are equivalent modulo absorbable errors, thus recovering the classical argument structure of Johnson-Noble-Rodrigues-Zumbrun for parabolic systems. This observation seems of general use in situations of delicate regularity. Applied in the context of (LLE) it gives the stronger result of stability and asymptotic behavior with respect to nonlocalized perturbations.
周期Lugiato-Lefever波的前调阻尼估计和非定域稳定性
在最近一项有趣的分析中,Haragus-Johnson-Perkins-de Rijk证明了lugito - lefever方程(LLE)的稳定周期解在局域扰动下的调制稳定性,并指出在获得调制扰动变量的标准“非线性阻尼估计”以控制解的正则性方面存在困难。在这里,我们指出,代替标准的“逆调制”阻尼估计,人们可以交替地对“前调”扰动进行阻尼估计,注意到正调制和逆调制变量的范数是等效模可吸收误差,从而恢复抛物系统的johnson - noble - rodriguez - zumbrun的经典参数结构。这一观察似乎普遍适用于微妙规律的情况。应用于(LLE)的背景下,给出了关于非定域扰动的稳定性和渐近性的较强结果。
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来源期刊
CiteScore
4.10
自引率
5.30%
发文量
62
审稿时长
>12 weeks
期刊介绍: The Nonlinear Analysis section of the Annales de l''Institut Henri Poincaré is an international journal created in 1983 which publishes original and high quality research articles. It concentrates on all domains concerned with nonlinear analysis, specially applicable to PDE, mechanics, physics, economy, without overlooking the numerical aspects.
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