Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation
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.
Keywords
Cite
@article{arxiv.1302.3326,
title = {Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation},
author = {Aleksandr L. Lisok and Aleksandr V. Shapovalov and Andrey Yu. Trifonov},
journal= {arXiv preprint arXiv:1302.3326},
year = {2013}
}