Existence of global solutions and global attractor for the third order Lugiato-Lefever equation on \(\mathbf T\)
type: Item
instance of: scholarly article
title: Existence of global solutions and global attractor for the third order Lugiato-Lefever equation on \(\mathbf T\)
zbMATH Open document ID: 1391.35357
author: Tomoyuki Miyaji
author: Yoshio Tsutsumi
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2017-11-21T00:00:00Z
review text: The main concern of this paper is a Cauchy problem that consists of a third order Lugiato-Lefever equation and a classical initial condition. In fact, the equation is a generalized Lugiato-Lefever equation which has been attracting a great interest especially in the field of nonlinear optics. The authors show that this equation appears in different mathematical models, as such, for Kerr frequency comb generation in a whispering gallery mode resonator, octave-spanning Kerr frequency comb in a micro-ring resonator or cavity solitons in micro-ring resonator near zero group-dispersion. For the formulated Cauchy problem, the authors prove the global well-posedness in \(L^2\) and investigate the nonlinear smoothing effect. Based on the smoothing effect, the authors can obtain the existence of the global attractor in \(L^2\) for flows generated by the third order Lugiato-Lefever equation. In the last part it is proved that, without damping and forcing, the solution of the Cauchy problem satisfies the three conservations laws, namely, the mass, the momentum and the energy conservations laws.
reviewed by: Marin I. Marin
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 6810596
zbMATH Keywords: third order Lugiato-Lefever equation
zbMATH Keywords: Strichartz' estimate on one dimensional torus
zbMATH Keywords: nonlinear smoothing effect
zbMATH Keywords: global attractor
MaRDI profile type: Publication
full work available at URL: j.anihpc.2016.12.004
OpenAlex ID: W2564847386
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DOI: 10.1016/J.ANIHPC.2016.12.004
description: scientific article; zbMATH DE number 6810596