### 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 language | English |
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Article number | 007 |

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

Volume | 1 |

DOIs | |

Publication status | Published - 1 Jan 2005 |

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### 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

**Exact solutions and symmetry operators for the nonlocal Gross-Pitaevskii equation with quadratic potential.** / Shapovalov, Aleksandr Vasilievich; Trifonov, Andrey; Lisok, Alexander Leonidovich.

Research output: Contribution to journal › Article

}

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 -