Symmetry and intertwining operators for the nonlocal gross-pitaevskii equation

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8 Citations (Scopus)

Abstract

We consider the symmetry properties of an integro-differential multidimensional Gross-Pitaevskii equation with a non local nonlinear (cubic) term in the context of symmetry analysis using the formalism of semi classical asymptotics. This yields a semi classically reduced non local Gross-Pitaevskii equation, which can be treated as a nearly linear equation, to determine the principal term of the semi classical asymptotic solution. Our main result is an approach which allows one to construct a class of symmetry operators for the reduced Gross-Pitaevskii equation. These symmetry operators are determined by linear relations including intertwining operators and additional algebraic conditions. The basic ideas are illustrated with a 1D reduced Gross-Pitaevskii equation. The symmetry operators are found explicitly, and the corresponding families of exact solutions are obtained.

Original languageEnglish
Article number066
JournalSymmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume9
DOIs
Publication statusPublished - 6 Nov 2013

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Gross-Pitaevskii Equation
Intertwining Operators
Nonlocal Equations
Symmetry
Operator
Linear Relation
Asymptotic Solution
Term
Linear equation
Exact Solution

Keywords

  • Exact solutions
  • Intertwining operators
  • Nonlocal gross-pitaevskii equation
  • Semiclassical asymptotics
  • Symmetry operators

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology
  • Mathematical Physics

Cite this

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title = "Symmetry and intertwining operators for the nonlocal gross-pitaevskii equation",
abstract = "We consider the symmetry properties of an integro-differential multidimensional Gross-Pitaevskii equation with a non local nonlinear (cubic) term in the context of symmetry analysis using the formalism of semi classical asymptotics. This yields a semi classically reduced non local Gross-Pitaevskii equation, which can be treated as a nearly linear equation, to determine the principal term of the semi classical asymptotic solution. Our main result is an approach which allows one to construct a class of symmetry operators for the reduced Gross-Pitaevskii equation. These symmetry operators are determined by linear relations including intertwining operators and additional algebraic conditions. The basic ideas are illustrated with a 1D reduced Gross-Pitaevskii equation. The symmetry operators are found explicitly, and the corresponding families of exact solutions are obtained.",
keywords = "Exact solutions, Intertwining operators, Nonlocal gross-pitaevskii equation, Semiclassical asymptotics, Symmetry operators",
author = "Lisok, {Alexander Leonidovich} and Shapovalov, {Aleksandr Vasilievich} and Yu Andrey",
year = "2013",
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language = "English",
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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",

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AU - Lisok, Alexander Leonidovich

AU - Shapovalov, Aleksandr Vasilievich

AU - Andrey, Yu

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Y1 - 2013/11/6

N2 - We consider the symmetry properties of an integro-differential multidimensional Gross-Pitaevskii equation with a non local nonlinear (cubic) term in the context of symmetry analysis using the formalism of semi classical asymptotics. This yields a semi classically reduced non local Gross-Pitaevskii equation, which can be treated as a nearly linear equation, to determine the principal term of the semi classical asymptotic solution. Our main result is an approach which allows one to construct a class of symmetry operators for the reduced Gross-Pitaevskii equation. These symmetry operators are determined by linear relations including intertwining operators and additional algebraic conditions. The basic ideas are illustrated with a 1D reduced Gross-Pitaevskii equation. The symmetry operators are found explicitly, and the corresponding families of exact solutions are obtained.

AB - We consider the symmetry properties of an integro-differential multidimensional Gross-Pitaevskii equation with a non local nonlinear (cubic) term in the context of symmetry analysis using the formalism of semi classical asymptotics. This yields a semi classically reduced non local Gross-Pitaevskii equation, which can be treated as a nearly linear equation, to determine the principal term of the semi classical asymptotic solution. Our main result is an approach which allows one to construct a class of symmetry operators for the reduced Gross-Pitaevskii equation. These symmetry operators are determined by linear relations including intertwining operators and additional algebraic conditions. The basic ideas are illustrated with a 1D reduced Gross-Pitaevskii equation. The symmetry operators are found explicitly, and the corresponding families of exact solutions are obtained.

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