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arXiv · 1302.3326

Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation

Aleksandr L. Lisok·Aleksandr V. Shapovalov·Andrey Yu. Trifonov
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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.

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Aleksandr L. Lisok, Aleksandr V. Shapovalov, Andrey Yu. Trifonov. 2013-11-06. Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation. https://doi.org/10.3842/sigma.2013.066

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