Exact Solutions and Symmetry Operators for the Nonlocal Gross-Pitaevskii Equation with Quadratic Potential

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2005
• Main Authors: Shapovalov, A., Trifonov, A., Lisok, A.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2005
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/209353
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Exact Solutions and Symmetry Operators for the Nonlocal Gross-Pitaevskii Equation with Quadratic Potential / A. Shapovalov, A. Trifonov, A. Lisok // Symmetry, Integrability and Geometry: Methods and Applications. — 2005. — Т. 1. — Бібліогр.: 23 назв. — англ.

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.