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.
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.
\Bibitem{LisShaTri13}
\by Aleksandr~L.~Lisok, Aleksandr~V.~Shapovalov, Andrey~Yu.~Trifonov
\paper Symmetry and Intertwining Operators for the Nonlocal Gross--Pitaevskii Equation
\jour SIGMA
\yr 2013
\vol 9
\papernumber 066
\totalpages 21
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\crossref{https://doi.org/10.3842/SIGMA.2013.066}
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This publication is cited in the following 11 articles:
F. Bagarello, F. Gargano, L. Saluto, “Density matrices and entropy operator for non-Hermitian quantum mechanics”, Journal of Mathematical Physics, 66:2 (2025)
Fabio Bagarello, Mathematical Physics Studies, Pseudo-Bosons and Their Coherent States, 2022, 183
Shapovalov A.V., Kulagin A.E., Trifonov A.Yu., “The Gross-Pitaevskii Equation With a Nonlocal Interaction in a Semiclassical Approximation on a Curve”, Symmetry-Basel, 12:2 (2020), 201
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
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
F. Bagarello, F. Gargano, S. Spagnolo, “Bi-squeezed states arising from pseudo-bosons”, J. Phys. A-Math. Theor., 51:45 (2018), 455204
Bagarello F., “Intertwining operators for non-self-adjoint Hamiltonians and bicoherent states”, J. Math. Phys., 57:10 (2016), 103501
Bagarello F., Trapani C., Triolo S., “Gibbs states defined by biorthogonal sequences”, J. Phys. A-Math. Theor., 49:40 (2016), 405202
Bagarello F., “Non-self-adjoint Hamiltonians with complex eigenvalues”, J. Phys. A-Math. Theor., 49:21 (2016), 215304
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
Bagarello F., “Some Results on the Dynamics and Transition Probabilities For Non Self-Adjoint Hamiltonians”, Ann. Phys., 356 (2015), 171–184
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