The trajectory-coherent approximation and the system of moments for the hartree type equation

Research output: Contribution to journalArticle

28 Citations (Scopus)

Abstract

The general construction of semiclassically concentrated solutions to the Hartree type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter □ (□→0), are constructed with a power accuracy of O (□ N/2), where N is any natural number. In constructing the semiclassically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of semiclassically concentrated solutions of Hartree type equations. The results obtained are exemplified by a one-dimensional Hartree type equation with a Gaussian potential.

Original languageEnglish
Pages (from-to)325-370
Number of pages46
JournalInternational Journal of Mathematics and Mathematical Sciences
Volume32
Issue number6
DOIs
Publication statusPublished - 1 Jan 2002

Fingerprint

Trajectory
Moment
Approximation
Formal Solutions
Natural number
Small Parameter
Superposition
Cauchy Problem

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

Cite this

@article{f9dbbad1c32b4acc8ff0dd1ab06fbdb4,
title = "The trajectory-coherent approximation and the system of moments for the hartree type equation",
abstract = "The general construction of semiclassically concentrated solutions to the Hartree type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter □ (□→0), are constructed with a power accuracy of O (□ N/2), where N is any natural number. In constructing the semiclassically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of semiclassically concentrated solutions of Hartree type equations. The results obtained are exemplified by a one-dimensional Hartree type equation with a Gaussian potential.",
author = "Belov, {V. V.} and Trifonov, {A. Y U} and Shapovalov, {Aleksandr Vasilievich}",
year = "2002",
month = "1",
day = "1",
doi = "10.1155/S0161171202112142",
language = "English",
volume = "32",
pages = "325--370",
journal = "International Journal of Mathematics and Mathematical Sciences",
issn = "0161-1712",
publisher = "Hindawi Publishing Corporation",
number = "6",

}

TY - JOUR

T1 - The trajectory-coherent approximation and the system of moments for the hartree type equation

AU - Belov, V. V.

AU - Trifonov, A. Y U

AU - Shapovalov, Aleksandr Vasilievich

PY - 2002/1/1

Y1 - 2002/1/1

N2 - The general construction of semiclassically concentrated solutions to the Hartree type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter □ (□→0), are constructed with a power accuracy of O (□ N/2), where N is any natural number. In constructing the semiclassically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of semiclassically concentrated solutions of Hartree type equations. The results obtained are exemplified by a one-dimensional Hartree type equation with a Gaussian potential.

AB - The general construction of semiclassically concentrated solutions to the Hartree type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter □ (□→0), are constructed with a power accuracy of O (□ N/2), where N is any natural number. In constructing the semiclassically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of semiclassically concentrated solutions of Hartree type equations. The results obtained are exemplified by a one-dimensional Hartree type equation with a Gaussian potential.

UR - http://www.scopus.com/inward/record.url?scp=2442451497&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2442451497&partnerID=8YFLogxK

U2 - 10.1155/S0161171202112142

DO - 10.1155/S0161171202112142

M3 - Article

VL - 32

SP - 325

EP - 370

JO - International Journal of Mathematics and Mathematical Sciences

JF - International Journal of Mathematics and Mathematical Sciences

SN - 0161-1712

IS - 6

ER -