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Symmetry operators of a Hartree-type equation with quadratic potential

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Abstract

We study the symmetry properties of a nonstationary one-dimensional Hartree-type equation with quadratic periodic potential and nonlocal nonlinearity. We find an explicit form of a nonlinear evolution operator for this equation and obtain a solution to a Cauchy problem in the class of semiclassically concentrated functions. We find parametric families of nonlinear symmetry operators of a Hartree-type equation (keeping invariant the set of solutions to this equation). Using the symmetry operators, we construct families of exact solutions to the equation. This approach constructively extends the ideas and methods of group analysis to the case of nonlinear integro-differential equations.

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References

  1. L. D. Landau E. M. Lifshits (1989) Quantum Mechanics Nauka Moscow

    Google Scholar 

  2. L. P. Pitaevskii (1988) ArticleTitleBose-Einstein condensation in magnetic traps. Introduction to the theory Uspekhi Fiz. Nauk 168 641–653

    Google Scholar 

  3. Fuk Vo Khagn V. M. Chetverikov (1978) ArticleTitleGeneralized solitons of the Schröodinger equation with unitary nonlinearity Teor. Mat. Fizika 36 IssueID3 345–351

    Google Scholar 

  4. L. V. Ovsyannikov (1978) Group Analysis of Differential Equations Nauka Moscow

    Google Scholar 

  5. V. K. Andreev O. V. Kaptsov V. V. Pukhnachov A. A. Rodionov (1994) Application of Group-Theoretic Methods to Hydrodynamics Nauka Novosibirsk

    Google Scholar 

  6. N. Kh. Ibragimov (1983) Transformation Groups in Mathematical Physics Nauka Moscow

    Google Scholar 

  7. P. J. Olver (1986) Application of Lie Groups to Differential Equations Springer-Verlag New York

    Google Scholar 

  8. W. I. Fushchich A. G. Nikitin (1987) Symmetries of Maxwell Equations Reidel Dordrecht

    Google Scholar 

  9. G. Gaeta (1994) Nonlinear Symmetry and Nonlinear Equations Kluwer Academic Press Dordrecht; Boston; London

    Google Scholar 

  10. V. V. Belov A. Yu. Trifonov A. V. Shapovalov (2002) ArticleTitleThe trajectory-coherent approximation and the system of moments for the Hartree type equation Internat. J. Math. Math. Sci. 32 IssueID6 325–370

    Google Scholar 

  11. V. V. Belov A. Yu. Trifonov A. V. Shapovalov (2002) ArticleTitleThe semiclassical trajectory-coherent approximation for the Hartree type equation Teor. Mat. Fizika 130 IssueID3 460–492

    Google Scholar 

  12. V. P. Maslov (1977) Complex WKB Method in Nonlinear Equations Nauka Moscow

    Google Scholar 

  13. V. V. Belov S. Yu. Dobrokhotov (1988) ArticleTitleSemiclassical Maslov asymptotics with complex phases. I. General approach Teor. Mat. Fizika 92 IssueID2 215–254

    Google Scholar 

  14. V. G. Bagrov V. V. Belov A. Yu. Trifonov (1996) ArticleTitleSemiclassical trajectory-coherent approximation in quantum mechanics: I. High order corrections to multidimensional time-dependent equations of Schrödinger type Ann. of Phys. (NY) 246 IssueID2 231–280

    Google Scholar 

  15. V. V. Pukhnachov (1987) ArticleTitleEquivalence transformations and hidden symmetry of evolution equations Dokl. Akad. Nauk SSSR 294 535–538

    Google Scholar 

  16. P. Ehrenfest (1927) ArticleTitleBemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik Z. Phys. 45 455–457

    Google Scholar 

  17. V. P. Maslov M. V. Fedoryuk (1976) Semiclassical Approximation for Equations of Quantum Mechanics Nauka Moscow

    Google Scholar 

  18. H. P. Robertson (1934) ArticleTitleAn indeterminacy relation for several observables and its classical interpretation Phys. Rev. 46 IssueID9 794–801

    Google Scholar 

  19. V. V. Dodonov V. I. Man’ko (1983) ArticleTitleGeneralization of indeterminacy relation in quantum mechanics Tr. FIAN 152 145–193

    Google Scholar 

  20. V. V. Belov M. F. Kondrat’eva (1994) ArticleTitleHamilton systems of equations for quantum means Mat. Zametki 56 IssueID6 27–39

    Google Scholar 

  21. Malkin M. A. and Man’ko V. I., Dynamical Symmetries and Coherent States of Quantum Systems [in Russian], Nauka, Moscow (1979).

  22. A. M. Perelomov (1987) Generalized Coherent States and Some of Their Applications Nauka Moscow

    Google Scholar 

  23. H. Bateman A. Erdélyi (1966) Higher Transcendental Functions Nauka Moscowx

    Google Scholar 

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Authors and Affiliations

  1. Tomsk Polytechnical University, Russia

    A. L. Lisok, A. Yu. Trifonov & A. V. Shapovalov

Authors
  1. A. L. Lisok
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  2. A. Yu. Trifonov
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  3. A. V. Shapovalov
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Additional information

Original Russian Text Copyright © 2005 Lisok A. L., Trifonov A. Yu., and Shapovalov A. V.

The authors were partially supported by the President of the Russian Federation (Grants NSh-1743.2003.2 and MD-246.2003.02). A. L. Lisok was supported by a fellowship of the Noncommercial Foundation “Dynasty” at the International Center for Fundamental Physics in Moscow and by the Ministry for Education of the Russian Federation (Grant A03-2.8-794).

Translated from Sibirski\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath} \) Matematicheski\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath} \) Zhurnal, Vol. 46, No. 1, pp. 149–165, January–February, 2005.

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Lisok, A.L., Trifonov, A.Y. & Shapovalov, A.V. Symmetry operators of a Hartree-type equation with quadratic potential. Sib Math J 46, 119–132 (2005). https://doi.org/10.1007/s11202-005-0013-2

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  • DOI: https://doi.org/10.1007/s11202-005-0013-2

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