### 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 language | English |
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Article number | 066 |

Journal | Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) |

Volume | 9 |

DOIs | |

Publication status | Published - 6 Nov 2013 |

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### 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

**Symmetry and intertwining operators for the nonlocal gross-pitaevskii equation.** / Lisok, Alexander Leonidovich; Shapovalov, Aleksandr Vasilievich; Andrey, Yu.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Symmetry and intertwining operators for the nonlocal gross-pitaevskii equation

AU - Lisok, Alexander Leonidovich

AU - Shapovalov, Aleksandr Vasilievich

AU - Andrey, Yu

PY - 2013/11/6

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.

KW - Exact solutions

KW - Intertwining operators

KW - Nonlocal gross-pitaevskii equation

KW - Semiclassical asymptotics

KW - Symmetry operators

UR - http://www.scopus.com/inward/record.url?scp=84887179969&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887179969&partnerID=8YFLogxK

U2 - 10.3842/SIGMA.2013.066

DO - 10.3842/SIGMA.2013.066

M3 - Article

VL - 9

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

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

SN - 1815-0659

M1 - 066

ER -