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Open AccessArticle

The Gross–Pitaevskii Equation with a Nonlocal Interaction in a Semiclassical Approximation on a Curve

1
Department of Theoretical Physics, Tomsk State University, 1 Novosobornaya Sq., Tomsk 634050, Russia
2
Department of Physics and Mathematics, Tomsk State Pedagogical University, 60 Kievskaya St., Tomsk 634041, Russia
3
Department of Mathematics and Informatics, Tomsk Polytechnic University, 30 Lenin A., Tomsk 634050, Russia
4
V.E. Zuev Institute of Atmospheric Optics, Siberian Branch of the Russian Academy of Sciences, 1 Academician Zuev Sq., Tomsk 634055, Russia
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2020, 12(2), 201; https://doi.org/10.3390/sym12020201
Received: 28 December 2019 / Revised: 10 January 2020 / Accepted: 25 January 2020 / Published: 1 February 2020
(This article belongs to the Special Issue Cosmology and Extragalactic Astronomy)
We propose an approach to constructing semiclassical solutions for the generalized multidimensional Gross–Pitaevskii equation with a nonlocal interaction term. The key property of the solutions is that they are concentrated on a one-dimensional manifold (curve) that evolves over time. The approach reduces the Cauchy problem for the nonlocal Gross–Pitaevskii equation to a similar problem for the associated linear equation. The geometric properties of the resulting solutions are related to Maslov’s complex germ, and the symmetry operators of the associated linear equation lead to the approximation of the symmetry operators for the nonlocal Gross–Pitaevskii equation.
Keywords: Gross–Pitaevskii equation; nonlocal interaction; Bose–Einstein condensate; semiclassical approximation; complex germ; symmetry operators Gross–Pitaevskii equation; nonlocal interaction; Bose–Einstein condensate; semiclassical approximation; complex germ; symmetry operators
MDPI and ACS Style

Shapovalov, A.V.; Kulagin, A.E.; Trifonov, A.Y. The Gross–Pitaevskii Equation with a Nonlocal Interaction in a Semiclassical Approximation on a Curve. Symmetry 2020, 12, 201.

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