On the point-particle (Newtonian) limit of the nonlinear Hartree equation
type: Item
instance of: scholarly article
title: On the point-particle (Newtonian) limit of the nonlinear Hartree equation
zbMATH Open document ID: 1025.81015
author: Tai-Peng Tsai
author: Horng-Tzer Yau
author: Jürg Fröhlich
published in: Communications in Mathematical Physics
publication date: 2002-05-21T00:00:00Z
review text: The authors explore a classical regime for a class of systems of identical, non-relativistic bosons, with very weak two-body interactions described by the potential \(-k\Phi\) of van der Waals or Newtonian type satisfying certain regularity properties. They consider the nonlinear Hartree equation \[ i\partial_t\psi_t=-\frac{1}{2}\Delta\psi_t+\lambda V\psi_t-\nu(\Phi*|\psi_t|^2)\psi_t, \] here \(\psi_t(x)=\psi(t,x)\), \(t\in\mathbb{R}, x\in\mathbb{R}^n\) is taken from the Sobolev space \(H^1(\mathbb{R}^n)\), \(\lambda\in\mathbb{R}\), \(V\) is smooth, bounded, positive function on \(\mathbb{R}^n\), \(-\Phi\) is a radially symmetric two-body potential, with \(\Phi\in L^p(\mathbb{R}^n,d^x)+L^{\infty}(\mathbb R^n)\), \(p\geq n/2\) and \(*\) denotes the usual convolution. The authors describe the dynamics of the given Hartree equation with given initial condition, under some imposed conditions on the Hamiltonian function. It is shown that a nonlinear Møller wave operator describing the scattering of a soliton and a wave can be defined. The authors also consider the dynamics of a soliton in slowly varying background potential \(W(\epsilon x)\). It is proved that the soliton decomposes into a soliton plus a scattering wave up to times of order \(\epsilon^{-1}\).
Mathematics Subject Classification ID: 81Q15
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 82C10
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 81U20
zbMATH DE Number: 1742843
zbMATH Keywords: nonlinear Hartree equation
zbMATH Keywords: weakly interacting non-relativistic bosons
zbMATH Keywords: Møller wave operator
zbMATH Keywords: soliton
zbMATH Keywords: Newton equation
reviewed by: Farruh Mukhamedov
MaRDI profile type: Publication
full work available at URL: s002200100579
OpenAlex ID: W2056264735
DOI: 10.1007/S002200100579
description: scientific article; zbMATH DE number 1742843