Exact solutions and symmetry operators for the nonlocal Gross-Pitaevskii equation with quadratic potential

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

The complex WKB-Maslov method is used to consider an approach to the semiclassical integrability of the multidimensional Gross-Pitaevskii equation with an external field and nonlocal nonlinearity previously developed by the authors. Although the WKB-Maslov method is approximate in essence, it leads to exact solution of the Gross- Pitaevskii equation with an external and a nonlocal quadratic potential. For this equation, an exact solution of the Cauchy problem is constructed in the class of trajectory concentrated functions. A nonlinear evolution operator is found in explicit form and symmetry operators (mapping a solution of the equation into another solution) are obtained for the equation under consideration. General constructions are illustrated by examples.

Original languageEnglish
Article number007
JournalSymmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume1
DOIs
Publication statusPublished - 1 Jan 2005

Fingerprint

Gross-Pitaevskii Equation
Nonlocal Equations
Exact Solution
Symmetry
Operator
Evolution Operator
Nonlinear Operator
Integrability
External Field
Cauchy Problem
Nonlinearity
Trajectory

Keywords

  • Gross-pitaevskii equation
  • Nonlinear evolution operator
  • Semiclassical asymptotics
  • Symmetry operators
  • The cauchy problem
  • Trajectory concentrated functions
  • Wkb-maslov complex germ method

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Geometry and Topology

Cite this

@article{734adfdb28284ae1b9cc25d59e7f2bac,
title = "Exact solutions and symmetry operators for the nonlocal Gross-Pitaevskii equation with quadratic potential",
abstract = "The complex WKB-Maslov method is used to consider an approach to the semiclassical integrability of the multidimensional Gross-Pitaevskii equation with an external field and nonlocal nonlinearity previously developed by the authors. Although the WKB-Maslov method is approximate in essence, it leads to exact solution of the Gross- Pitaevskii equation with an external and a nonlocal quadratic potential. For this equation, an exact solution of the Cauchy problem is constructed in the class of trajectory concentrated functions. A nonlinear evolution operator is found in explicit form and symmetry operators (mapping a solution of the equation into another solution) are obtained for the equation under consideration. General constructions are illustrated by examples.",
keywords = "Gross-pitaevskii equation, Nonlinear evolution operator, Semiclassical asymptotics, Symmetry operators, The cauchy problem, Trajectory concentrated functions, Wkb-maslov complex germ method",
author = "Shapovalov, {Aleksandr Vasilievich} and Andrey Trifonov and Lisok, {Alexander Leonidovich}",
year = "2005",
month = "1",
day = "1",
doi = "10.3842/SIGMA.2005.007",
language = "English",
volume = "1",
journal = "Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)",
issn = "1815-0659",
publisher = "Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine",

}

TY - JOUR

T1 - Exact solutions and symmetry operators for the nonlocal Gross-Pitaevskii equation with quadratic potential

AU - Shapovalov, Aleksandr Vasilievich

AU - Trifonov, Andrey

AU - Lisok, Alexander Leonidovich

PY - 2005/1/1

Y1 - 2005/1/1

N2 - The complex WKB-Maslov method is used to consider an approach to the semiclassical integrability of the multidimensional Gross-Pitaevskii equation with an external field and nonlocal nonlinearity previously developed by the authors. Although the WKB-Maslov method is approximate in essence, it leads to exact solution of the Gross- Pitaevskii equation with an external and a nonlocal quadratic potential. For this equation, an exact solution of the Cauchy problem is constructed in the class of trajectory concentrated functions. A nonlinear evolution operator is found in explicit form and symmetry operators (mapping a solution of the equation into another solution) are obtained for the equation under consideration. General constructions are illustrated by examples.

AB - The complex WKB-Maslov method is used to consider an approach to the semiclassical integrability of the multidimensional Gross-Pitaevskii equation with an external field and nonlocal nonlinearity previously developed by the authors. Although the WKB-Maslov method is approximate in essence, it leads to exact solution of the Gross- Pitaevskii equation with an external and a nonlocal quadratic potential. For this equation, an exact solution of the Cauchy problem is constructed in the class of trajectory concentrated functions. A nonlinear evolution operator is found in explicit form and symmetry operators (mapping a solution of the equation into another solution) are obtained for the equation under consideration. General constructions are illustrated by examples.

KW - Gross-pitaevskii equation

KW - Nonlinear evolution operator

KW - Semiclassical asymptotics

KW - Symmetry operators

KW - The cauchy problem

KW - Trajectory concentrated functions

KW - Wkb-maslov complex germ method

UR - http://www.scopus.com/inward/record.url?scp=84889234828&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84889234828&partnerID=8YFLogxK

U2 - 10.3842/SIGMA.2005.007

DO - 10.3842/SIGMA.2005.007

M3 - Article

VL - 1

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SN - 1815-0659

M1 - 007

ER -