### 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 |

### 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