Existence of a finite-dimensional global attractor for a damped parametric nonlinear Schrödinger equation
type: Item
instance of: scholarly article
title: Existence of a finite-dimensional global attractor for a damped parametric nonlinear Schrödinger equation
zbMATH Open document ID: 1260.37052
author: Olivier Goubet
author: Ludovic Legry
published in: Advances in Differential Equations
publication date: 2012-11-16T00:00:00Z
review text: In this work, the authors consider a parametric nonlinear Schrödinger equation of the form \(u_t+au-iu_{xx}+i\lambda u+i|u|^2u+i\gamma\overline{u}=0\), where \(a,\lambda>0\) and \(\gamma\) is a real function, defined on the real line. Under the assumption that \(\lim_{|x|\to +\infty}\gamma(x)=0\), they prove the existence of a global attractor in the Sobolev space \(H^1(\mathbb{R})\) for the dynamical system generated by this equation, which is also compact in \(H^3(\mathbb{R})\). Assuming also that \(\int_{\mathbb{R}}(1+x^2)\gamma(x)^2dx<+\infty\), they prove that the global attractor has finite Hausdorff dimension.
reviewed by: Matheus Cheque Bortolan
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35Q41
zbMATH DE Number: 6108228
zbMATH Keywords: parametric Schrödinger equation
zbMATH Keywords: global attractor
zbMATH Keywords: Hausdorff dimension
MaRDI profile type: Publication
Recommended article: Global attractor for weakly damped nonlinear Schrödinger equations in \(L^2(\mathbb R)\)
Recommended article: Finite dimensional global attractor for a parametric nonlinear Schrödinger system with a trapping potential
Recommended article: Existence and finite dimensionality of the global attractor for a class of nonlinear dissipative equations
Recommended article: Existence and congruence of global attractors for damped and forced integrable and nonintegrable discrete nonlinear Schrödinger equations
Recommended article: Q4012822
Recommended article: Finite dimensional behavior of global attractors for weakly damped nonlinear Schrödinger-Boussinesq equations
Recommended article: Q4984495
Recommended article: On global attractors for 2D damped driven nonlinear Schrödinger equations
Recommended article: Global attractor for the weakly damped driven Schrödinger equation in \(H^2 (\mathbb R)\)
Recommended article: Finite dimensional global attractor for dissipative Schrödinger-Boussinesq equations
description: scientific article; zbMATH DE number 6108228