Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 066, 21 pages      arXiv:1302.3326      https://doi.org/10.3842/SIGMA.2013.066

Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation

Aleksandr L. Lisok a, Aleksandr V. Shapovalov a, b and Andrey Yu. Trifonov a, b
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

Received February 15, 2013, in final form October 26, 2013; Published online November 06, 2013

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.

Key words: symmetry operators; intertwining operators; nonlocal Gross-Pitaevskii equation; semiclassical asymptotics; exact solutions.

pdf (452 kb)   tex (29 kb)

References

  1. Agrawal G.P., Nonlinear fiber optics, 5th ed., Elsevier Inc., London, 2013.
  2. Anderson R.L., Ibragimov N.H., Lie-Bäcklund transformations in applications, SIAM Studies in Applied Mathematics, Vol. 1, SIAM, Philadelphia, Pa., 1979.
  3. Bagrov V.G., Belov V.V., Trifonov A.Yu., Semiclassical trajectory-coherent approximation in quantum mechanics. I. High-order corrections to multidimensional time-dependent equations of Schrödinger type, Ann. Physics 246 (1996), 231-290.
  4. Belov V.V., Dobrokhotov S.Y., Semiclassical Maslov asymptotics with complex phases. I. General approach, Theoret. and Math. Phys. 92 (1992), 843-868.
  5. Belov V.V., Litvinets F.N., Trifonov A.Yu., Semiclassical spectral series of a Hartree-type operator corresponding to a rest point of the classical Hamilton-Ehrenfest system, Theoret. and Math. Phys. 150 (2007), 21-33.
  6. Belov V.V., Trifonov A.Yu., Shapovalov A.V., The trajectory-coherent approximation and the system of moments for the Hartree type equation, Int. J. Math. Math. Sci. 32 (2002), 325-370, arXiv:math-ph/0012046.
  7. Bluman G.W., Anco S.C., Symmetry and integration methods for differential equations, Applied Mathematical Sciences, Vol. 154, Springer-Verlag, New York, 2002.
  8. Bluman G.W., Cheviakov A.F., Anco S.C., Applications of symmetry methods to partial differential equations, Applied Mathematical Sciences, Vol. 168, Springer, New York, 2010.
  9. Bryuning J., Dobrokhotov S.Y., Nekrasov R.V., Shafarevich A.I., Propagation of Gaussian wave packets in thin periodic quantum waveguides with nonlocal nonlinearity, Theoret. and Math. Phys. 155 (2008), 689-707.
  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, cond-mat/9806038.
  11. Dodonov V.V., Kurmyshev E.V., Man'ko V.I., Correlated coherent states, in Classical and Quantum Effects in Electrodynamics, Sov. Phys. - Lebedev Inst. Rep., Vol. 176, 1986, 128-150.
  12. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York - Toronto - London, 1953.
  13. Frantzeskakis D.J., Dark solitons in atomic Bose-Einstein condensates: from theory to experiments, J. Phys. A: Math. Theor. 43 (2010), 213001, 68 pages, arXiv:1004.4071.
  14. Fushchich W.I., Nikitin A.G., Symmetries of equations of quantum mechanics, Allerton Press Inc., New York, 1994.
  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.
  16. Karasëv M.V., Maslov V.P., Nonlinear Poisson brackets. Geometry and quantization, Translations of Mathematical Monographs, Vol. 119, American Mathematical Society, Providence, RI, 1993.
  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.
  18. Levchenko E.A., Shapovalov A.V., Trifonov A.Yu., Symmetries of the Fisher-Kolmogorov-Petrovskii-Piskunov equation with a nonlocal nonlinearity in a semiclassical approximation, J. Math. Anal. Appl. 395 (2012), 716-726.
  19. Lisok A.L., Trifonov A.Yu., Shapovalov A.V., The evolution operator of the Hartree-type equation with a quadratic potential, J. Phys. A: Math. Gen. 37 (2004), 4535-4556, math-ph/0312004.
  20. Lushnikov P.M., Collapse of Bose-Einstein condensates with dipole-dipole interactions, Phys. Rev. A 66 (2002), 051601(R), 4 pages, cond-mat/0208312.
  21. Malkin M.A., Manko V.I., Dynamic symmetries and coherent states of quantum systems, Nauka, Moscow, 1979 (in Russian).
  22. Maslov V.P., Complex Markov chains and the continual Feinmann integral, Nauka, Moscow, 1976, (in Russian).
  23. Maslov V.P., Equations of the self-consistent field, J. Sov. Math. 11 (1979), 123-195.
  24. Maslov V.P., The complex WKB method for nonlinear equations. I. Linear theory, Progress in Physics, Vol. 16, Birkhäuser Verlag, Basel, 1994.
  25. Maslov V.P., Fedoryuk M.V., The semiclassical approximation for quantum mechanics equations, Reidel, Boston, 1981.
  26. Meirmanov A.M., Pukhnachov V.V., Shmarev S.I., Evolution equations and Lagrangian coordinates, de Gruyter Expositions in Mathematics, Vol. 24, Walter de Gruyter & Co., Berlin, 1997.
  27. Novikov S., Manakov S.V., Pitaevski L.P., Zakharov V.E., Theory of solitons. The inverse scattering method, Contemporary Soviet Mathematics, Plenum, New York, 1984.
  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 pages, arXiv:0804.1927.
  29. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1986.
  30. Ovsiannikov L.V., Group analysis of differential equations, Academic Press Inc., New York, 1982.
  31. Perelomov A., Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986.
  32. Robertson H.P., An indeterminacy relation for several observables and its classical interpretation, Phys. Rep. 46 (1934), 794-801.
  33. Shapovalov A.V., Trifonov A.Yu., Lisok A.L., Exact solutions and symmetry operators for the nonlocal Gross-Pitaevskii equation with quadratic potential, SIGMA 1 (2005), 007, 14 pages, math-ph/0511010.
  34. Vakulenko S.A., Maslov V.P., Molotkov I.A., Shafarevich A.I., Asymptotic solutions of the Hartree equation that are concentrated, as h→0, in a small neighborhood of a curve, Dokl. Akad. Nauk 345 (1995), 743-745 (in Russian).

Previous article  Next article   Contents of Volume 9 (2013)