Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity
type: Item
instance of: scholarly article
title: Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity
zbMATH Open document ID: 1155.35089
author: Jaeyoung Byeon
author: Louis Jeanjean
published in: Discrete and Continuous Dynamical Systems
publication date: 2008-01-18T00:00:00Z
review text: The work is dealing with a stationary nonlinear equation in the space of dimension \(N\geq 3\), \[ (\hbar^2/2)\Delta v -(V({\mathbf r}) - E)v + f(v) = 0, \] where a function \(f(v)\) is real, \(\Delta\) is the \(N\)-dimensional Laplacian, and, in a typical case, \(f(v)=-v^3\). This equation appears as the stationary version of the multidimensional Gross-Pitaevskii equation for Bose-Einstein condensates. It is assumed that potential \(V({\mathbf r})\) is chosen in such a form that, in the Thomas-Fermi approximation, which corresponds to \(\hbar^2 \to 0\), a solution reduces to a set of strongly localized noninteracting peaks, pinned to local minima of the potential. The objective of the work is to produce a rigorous proof of the fact that the solution keeps essentially the same form at small finite values of \(\hbar^2\), by ``gluing'' togehter the strongly localized peaks. This is done by means of a purely variational method.
reviewed by: B. A. Malomed
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 5228386
zbMATH Keywords: Gross-Pitaevskii equation
zbMATH Keywords: Thomas-Fermi approximation
zbMATH Keywords: Schrödinger equation
MaRDI profile type: Publication
full work available at URL: dcds.2007.19.255
OpenAlex ID: W2068590741
DOI: 10.3934/DCDS.2007.19.255
description: scientific article