### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 1 (2005), 007, 14 pages      math-ph/0511010      https://doi.org/10.3842/SIGMA.2005.007

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

Alexander Shapovalov a, b, c, Andrey Trifonov b, c and Alexander Lisok c
a) Tomsk State University, 36 Lenin Ave., 634050 Tomsk, Russia
b) Tomsk Polytechnic University, 30 Lenin Ave., 634050 Tomsk, Russia
c) Math. Phys. Laboratory, Tomsk Polytechnic University, 30 Lenin Ave., 634050 Tomsk, Russia

Received July 27, 2005, in final form October 06, 2005; Published online October 17, 2005

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.

Key words: WKB-Maslov complex germ method; semiclassical asymptotics; Gross-Pitaevskii equation; the Cauchy problem; nonlinear evolution operator; trajectory concentrated functions; symmetry operators.

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