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Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2013, том 9
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Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation
Інший ID:
2010 Mathematics Subject Classification: 35Q55; 45K05; 76M60; 81Q20
DOI: http://dx.doi.org/10.3842/SIGMA.2013.066
DOI: http://dx.doi.org/10.3842/SIGMA.2013.066
Посилання:
Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation / A.L. Lisok, A.V. Shapovalov, A.Y. Trifonov // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 34 назв. — англ.
Підтримка:
We would like to thank the anonymous referees who gave a relevant contribution to improve the
paper. The work was supported in part by the Russian Federation programs “Kadry” (contract
No. 16.740.11.0469) and “Nauka” (contract No. 1.604.2011) and by Tomsk State University
project No. 2.3684.2011.
Дата:
2013
Переглядів:
446
Завантажень:
220
Анотація:
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.
