A. L. Lisok, A. V. Shapovalov, A. Yu. Trifonov
We consider symmetry properties of an integro-differential multidimensional Gross-Pitaevskii equation with nonlocal cubic nonlinearity in the context of symmetry analysis using the formalism of semiclassical asymptotics. This yields a semiclassically reduced nonlocal Gross--Pitaevskii equation which determines the principal term of a semiclassical asymptotic solution and can be referred to as a nearly linear equation. 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 by the examples of a 1D reduced Gross-Pitaevskii equation. The symmetry operators are found explicitly, and the corresponding families of exact solutions generated by the symmetry operators are obtained.
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http://arxiv.org/abs/1302.3326
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