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

SIGMA 9 (2013), 066, 21 pages      arXiv:1302.3326      https://doi.org/10.3842/SIGMA.2013.066

### Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation

Aleksandr L. Lisok a, Aleksandr V. Shapovalov a, b and Andrey Yu. Trifonov a, b
a) Mathematical Physics Department, Tomsk Polytechnic University, 30 Lenin Ave., Tomsk, 634034 Russia
b) Theoretical Physics Department, Tomsk State University, 36 Lenin Ave., Tomsk, 634050 Russia

Received February 15, 2013, in final form October 26, 2013; Published online November 06, 2013

Abstract
We consider the symmetry properties of an integro-differential multidimensional Gross-Pitaevskii equation with a nonlocal nonlinear (cubic) term in the context of symmetry analysis using the formalism of semiclassical asymptotics. This yields a semiclassically reduced nonlocal Gross-Pitaevskii equation, which can be treated as a nearly linear equation, to determine the principal term of the semiclassical 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.

Key words: symmetry operators; intertwining operators; nonlocal Gross-Pitaevskii equation; semiclassical asymptotics; exact solutions.

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