Symmetry and Intertwining Operators for the Nonlocal Gross–Pitaevskii Equation

Aleksandr L. Lisok ; Aleksandr V. Shapovalov ; Andrey Yu. Trifonov

Symmetry, integrability and geometry: methods and applications, Tome 9 (2013)

Résumé

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.

Détail

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

Bibliographie

[1] Agrawal G. P., Nonlinear fiber optics, 5th ed., Elsevier Inc., London, 2013

[2] Anderson R. L., Ibragimov N. H., Lie–{B}äcklund transformations in applications, SIAM Studies in Applied Mathematics, 1, SIAM, Philadelphia, Pa., 1979 | MR | Zbl

[3] Bagrov V. G., Belov V. V., Trifonov A. Yu., “Semiclassical trajectory-coherent approximation in quantum mechanics. I: {H}igh-order corrections to multidimensional time-dependent equations of {S}chrödinger type”, Ann. Physics, 246 (1996), 231–290 | DOI | MR | Zbl

[4] Belov V. V., Dobrokhotov S. Y., “Semiclassical {M}aslov asymptotics with complex phases. I: {G}eneral approach”, Theoret. and Math. Phys., 92 (1992), 843–868 | DOI | MR

[5] Belov V. V., Litvinets F. N., Trifonov A. Yu., “Semiclassical spectral series of a {H}artree-type operator corresponding to a rest point of the classical {H}amilton–{E}hrenfest system”, Theoret. and Math. Phys., 150 (2007), 21–33 | DOI | MR | Zbl

[6] Belov V. V., Trifonov A. Yu., Shapovalov A. V., “The trajectory-coherent approximation and the system of moments for the {H}artree type equation”, Int. J. Math. Math. Sci., 32 (2002), 325–370, arXiv: math-ph/0012046 | DOI | MR | Zbl

[7] Bluman G. W., Anco S. C., Symmetry and integration methods for differential equations, Applied Mathematical Sciences, 154, Springer-Verlag, New York, 2002 | MR | Zbl

[8] Bluman G. W., Cheviakov A. F., Anco S. C., Applications of symmetry methods to partial differential equations, Applied Mathematical Sciences, 168, Springer, New York, 2010 | DOI | MR | Zbl

[9] Bryuning J., Dobrokhotov S. Y., Nekrasov R. V., Shafarevich A. I., “Propagation of {G}aussian wave packets in thin periodic quantum waveguides with nonlocal nonlinearity”, Theoret. and Math. Phys., 155 (2008), 689–707 | DOI | MR

[10] Dalfovo F., Giorgini S., Pitaevskii L. P., Stringari S., “Theory of Bose–Einstein condensation in trapped gases”, Rev. Modern Phys., 71 (1999), 463–512, arXiv: cond-mat/9806038 | DOI

[11] Dodonov V. V., Kurmyshev E. V., Man'ko V. I., “Correlated coherent states”, Classical and Quantum Effects in Electrodynamics, Sov. Phys. –Lebedev Inst. Rep., 176, 1986, 128–150 | MR

[12] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F. G., Higher transcendental functions, v. I, II, McGraw-Hill Book Company, Inc., New York – Toronto – London, 1953

[13] Frantzeskakis D. J., “Dark solitons in atomic {B}ose–{E}instein condensates: from theory to experiments”, J. Phys. A: Math. Theor., 43 (2010), 213001, 68 pp., arXiv: 1004.4071 | DOI | MR | Zbl

[14] Fushchich W. I., Nikitin A. G., Symmetries of equations of quantum mechanics, Allerton Press Inc., New York, 1994 | MR | Zbl

[15] Karasev M. V., Maslov V. P., “Algebras with general commutation relations and their applications. II: Unitary-nonlinear operator equations”, J. Sov. Math., 15 (1981), 273–368 | DOI

[16] Karasëv M. V., Maslov V. P., Nonlinear {P}oisson brackets. Geometry and quantization, Translations of Mathematical Monographs, 119, American Mathematical Society, Providence, RI, 1993 | MR | Zbl

[17] Karasev M. V., Pereskokov A. V., “The quantization rule for equations of a self-consistent field with a local rapidly decreasing nonlinearity”, Theoret. and Math. Phys., 79 (1989), 479–486 | DOI | MR

[18] Levchenko E. A., Shapovalov A. V., Trifonov A. Yu., “Symmetries of the {F}isher–{K}olmogorov–{P}etrovskii–{P}iskunov equation with a nonlocal nonlinearity in a semiclassical approximation”, J. Math. Anal. Appl., 395 (2012), 716–726 | DOI | MR | Zbl

[19] Lisok A. L., Trifonov A. Yu., Shapovalov A. V., “The evolution operator of the {H}artree-type equation with a quadratic potential”, J. Phys. A: Math. Gen., 37 (2004), 4535–4556, arXiv: math-ph/0312004 | DOI | MR | Zbl

[20] Lushnikov P. M., “Collapse of Bose–Einstein condensates with dipole-dipole interactions”, Phys. Rev. A, 6 (2002), 051601(R), 4 pp., arXiv: cond-mat/0208312 | DOI | MR

[21] Malkin M. A., Manko V. I., Dynamic symmetries and coherent states of quantum systems, Nauka, M., 1979 (in Russian)

[22] Maslov V. P., Complex Markov chains and the continual Feinmann integral, Nauka, M., 1976 (in Russian) | MR

[23] Maslov V. P., “Equations of the self-consistent field”, J. Sov. Math., 11 (1979), 123–195 | DOI | MR | Zbl

[24] Maslov V. P., The complex {WKB} method for nonlinear equations, v. I, Progress in Physics, 16, Linear theory, Birkhäuser Verlag, Basel, 1994 | DOI | MR | Zbl

[25] Maslov V. P., Fedoryuk M. V., The semiclassical approximation for quantum mechanics equations, Reidel, Boston, 1981 | Zbl

[26] Meirmanov A. M., Pukhnachov V. V., Shmarev S. I., Evolution equations and {L}agrangian coordinates, de Gruyter Expositions in Mathematics, 24, Walter de Gruyter Co., 1997 | DOI | MR

[27] Novikov S., Manakov S. V., Pitaevskiĭ L. P., Zakharov V. E., Theory of solitons. The inverse scattering method, Contemporary Soviet Mathematics, Plenum, New York, 1984 | MR | Zbl

[28] Novoa D., Malomed B. A., Humberto Michinel H., Pérez-García V. M., “Supersolitons: solitonic excitations in atomic soliton chains”, Phys. Rev. Lett., 101 (2008), 144101, 4 pp., arXiv: 0804.1927 | DOI

[29] Olver P. J., Applications of {L}ie groups to differential equations, Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1986 | DOI | MR | Zbl

[30] Ovsiannikov L. V., Group analysis of differential equations, Academic Press Inc., New York, 1982 | MR | Zbl

[31] Perelomov A., Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986 | MR | Zbl

[32] Robertson H. P., “An indeterminacy relation for several observables and its classical interpretation”, Phys. Rep., 46 (1934), 794–801 | DOI

[33] Shapovalov A. V., Trifonov A. Yu., Lisok A. L., “Exact solutions and symmetry operators for the nonlocal {G}ross–{P}itaevskii equation with quadratic potential”, SIGMA, 1 (2005), 007, 14 pp., arXiv: math-ph/0511010 | DOI | MR | Zbl

[34] Vakulenko S. A., Maslov V. P., Molotkov I. A., Shafarevich A. I., “Asymptotic solutions of the {H}artree equation that are concentrated, as {$h\to 0$}, in a small neighborhood of a curve”, Dokl. Akad. Nauk, 345 (1995), 743–745 (in Russian) | MR | Zbl