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 SIGMA, 2013, Volume 9, 066, 21 pages (Mi sigma849)

Symmetry and Intertwining Operators for the Nonlocal Gross–Pitaevskii Equation

Aleksandr L. Lisoka, Aleksandr V. Shapovalovab, Andrey Yu. Trifonovab

a Mathematical Physics Department, Tomsk Polytechnic University, 30 Lenin Ave., Tomsk, 634034 Russia
b Theoretical Physics Department, Tomsk State University, 36 Lenin Ave., Tomsk, 634050 Russia

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: symmetry operators; intertwining operators; nonlocal Gross–Pitaevskii equation; semiclassical asymptotics; exact solutions.

DOI: https://doi.org/10.3842/SIGMA.2013.066

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ArXiv: 1302.3326
MSC: 35Q55; 45K05; 76M60; 81Q20
Received: February 15, 2013; in final form October 26, 2013; Published online November 6, 2013
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Citation: Aleksandr L. Lisok, Aleksandr V. Shapovalov, Andrey Yu. Trifonov, “Symmetry and Intertwining Operators for the Nonlocal Gross–Pitaevskii Equation”, SIGMA, 9 (2013), 066, 21 pp.

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Bagarello F., “Some Results on the Dynamics and Transition Probabilities For Non Self-Adjoint Hamiltonians”, Ann. Phys., 356 (2015), 171–184
2. Bagarello F., “Intertwining operators for non-self-adjoint Hamiltonians and bicoherent states”, J. Math. Phys., 57:10 (2016), 103501
3. Bagarello F., Trapani C., Triolo S., “Gibbs states defined by biorthogonal sequences”, J. Phys. A-Math. Theor., 49:40 (2016), 405202
4. Bagarello F., “Non-self-adjoint Hamiltonians with complex eigenvalues”, J. Phys. A-Math. Theor., 49:21 (2016), 215304
5. Shapovalov A.V., Trifonov A.Yu., Lisok A.L., “Symmetry operators of the two-component Gross–Pitaevskii equation with a Manakov-type nonlocal nonlinearity”, XXIII International Conference on Integrable Systems and Quantum Symmetries (ISQS-23), Journal of Physics Conference Series, 670, eds. Burdik C., Navratil O., Posta S., IOP Publishing Ltd, 2016, UNSP 012046
6. F. Bagarello, E. M. F. Curado, J. P. Gazeau, “Generalized Heisenberg algebra and (non linear) pseudo-bosons”, J. Phys. A-Math. Theor., 51:15 (2018), 155201
7. F. Bagarello, F. Gargano, S. Spagnolo, “Bi-squeezed states arising from pseudo-bosons”, J. Phys. A-Math. Theor., 51:45 (2018), 455204
8. Shapovalov A.V., Trifonov A.Yu., “Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher-Kpp Type”, Symmetry-Basel, 11:3 (2019), 366
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