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Exact Solutions and Symmetry Operators for the Nonlocal Gross–Pitaevskii Equation with Quadratic Potential
Symmetry, integrability and geometry: methods and applications, Tome 1 (2005) Cet article a éte moissonné depuis la source Math-Net.Ru

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The complex WKB–Maslov method is used to consider an approach to the semiclassical integrability of the multidimensional Gross–Pitaevskii equation with an external field and nonlocal nonlinearity previously developed by the authors. Although the WKB–Maslov method is approximate in essence, it leads to exact solution of the Gross–Pitaevskii equation with an external and a nonlocal quadratic potential. For this equation, an exact solution of the Cauchy problem is constructed in the class of trajectory concentrated functions. A nonlinear evolution operator is found in explicit form and symmetry operators (mapping a solution of the equation into another solution) are obtained for the equation under consideration. General constructions are illustrated by examples.
Keywords: WKB–Maslov complex germ method; semiclassical asymptotics; Gross–Pitaevskii equation; the Cauchy problem; nonlinear evolution operator; trajectory concentrated functions; symmetry operators. 
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     title = {Exact {Solutions} and {Symmetry} {Operators} for the {Nonlocal} {Gross{\textendash}Pitaevskii} {Equation} with {Quadratic} {Potential}},
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Alexander Shapovalov; Andrey Trifonov; Alexander Lisok. Exact Solutions and Symmetry Operators for the Nonlocal Gross–Pitaevskii Equation with Quadratic Potential. Symmetry, integrability and geometry: methods and applications, Tome 1 (2005). http://geodesic.mathdoc.fr/item/SIGMA_2005_1_a6/

[1] Cornell E. A., Wieman C. E., “Nobel lecture: Bose–Einstein condensation in a dilute gas, the first 70 years some recent experiments”, Rev. Mod. Phys., 74 (2002), 875–893 | DOI

[2] Pitaevskii L. P., “Vortex lines in an imperfect Bose gas”, Zh. Eksper. Teor. Fiz., 40 (1961), 646–651 (in Russian)

[3] Gross E. P., “Structure of a quantized vortex in boson systems”, Nuovo Cimento, 20:3 (1961), 454–477 | DOI | MR | Zbl

[4] Kivshar Y. S., Pelinovsky D. E., “Self-focusing and transverse instabilities of solitary waves”, Phys. Rep., 331:4 (2000), 117–195 | DOI | MR

[5] Bang O., Krolikowski W., Wyller J., Rasmussen J. J., Collapse arrest and soliton stabilization in nonlocal nonlinear media, arXiv:nlin.PS/0201036

[6] Sov. Phys. JETP, 34 (1971), 62–69 | MR

[7] Plenum, New York, 1984 | MR | Zbl

[8] Academic Press, New York, 1982 | MR

[9] Anderson R. L., Ibragimov N. H., Lie–Bäcklund transformations in applications, SIAM, Philadelphia, 1979 | MR

[10] Olver P. J., Application of Lie groups to differential equations, Springer, New York, 1986 | MR

[11] Fushchich W. I., Shtelen W. M., Serov N. I., Symmetry analysis and exact solutions of equations of nonlinear mathematical physics, Kluwer, Dordrecht, 1993 | MR | Zbl

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

[13] 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. and Math. Sci., 32:6 (2002), 325–370 | DOI | MR | Zbl

[14] Theor. Math. Phys., 130:3 (2002), 391–418 | DOI | MR | Zbl

[15] Shapovalov A. V., Trifonov A. Yu., Lisok A. L., “Semiclassical approach to the geometric phase theory for the Hartree type equation”, Proceedinds of Fifth International Conference “Symmetry in Nonlinear Mathematical Physics”, Part 3 (June 23–29, 2003, Kyiv), Proceedings of Institute of Mathematics, 50, eds. A. G. Nikitin, V. M. Boyko, R. O. Popovych and I. A. Yehorchenko, Kyiv, 2004, 1454–1465 | MR | Zbl

[16] 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 | DOI | MR | Zbl

[17] Nonlinear Poisson brackets: geometry and quantization, Translations of Mathematical Monographs, 119, Amer. Math. Soc., Providence, RI, 1993 | MR | MR | Zbl | Zbl

[18] The complex WKB method for nonlinear equations. I. Linear theory, Birkhauser Verlag, Basel–Boston–Berlin, 1994 | MR | Zbl

[19] Theor. Math. Phys., 92:2 (1992), 843–868 | DOI | MR

[20] Ehrenfest P., “Bemerkung über die angenherte Gültigkeit der klassishen Mechanik innerhalb der Quanten Mechanik”, Zeits. Phys., 45 (1927), 455–457 | DOI | Zbl

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

[22] Perelomov A. M., Generalized coherent states and their application, Springer-Verlag, Berlin, 1986 | MR

[23] Meirmanov A. M., Pukhnachov V. V., Shmarev S. I., Evolution equations and Lagrangian coordinates, Walter de Gruyter, New York–Berlin, 1994 | MR