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TMF, 2002, Volume 130, Number 3, Pages 460–492 (Mi tmf313)  

This article is cited in 17 scientific papers (total in 17 papers)

Semiclassical Trajectory-Coherent Approximations of Hartree-Type Equations

V. V. Belova, A. Yu. Trifonovb, A. V. Shapovalovc

a Moscow State Institute of Electronics and Mathematics
b Tomsk Polytechnic University
c Tomsk State University

Abstract: We use the concept of the complex WKB–Maslov method to construct semiclassically concentrated solutions for Hartree-type equations. Formal solutions of the Cauchy problem for this equation that are asymptotic (with respect to a small parameter , $\hbar$, $\hbar \to 0$) are constructed with the power-law accuracy $O(\hbar ^{N/2})$, where $N\ge 3$ is a positive integer. The system of Hamilton–Ehrenfest equations (for averaged and centered moments) derived in this paper plays a significant role in constructing semiclassically concentrated solutions. In the class of semiclassically concentrated solutions of Hartree-type equations, we construct an approximate Green's function and state a nonlinear superposition principle.


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English version:
Theoretical and Mathematical Physics, 2002, 130:3, 391–418

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Received: 19.09.2001

Citation: V. V. Belov, A. Yu. Trifonov, A. V. Shapovalov, “Semiclassical Trajectory-Coherent Approximations of Hartree-Type Equations”, TMF, 130:3 (2002), 460–492; Theoret. and Math. Phys., 130:3 (2002), 391–418

Citation in format AMSBIB
\by V.~V.~Belov, A.~Yu.~Trifonov, A.~V.~Shapovalov
\paper Semiclassical Trajectory-Coherent Approximations of Hartree-Type Equations
\jour TMF
\yr 2002
\vol 130
\issue 3
\pages 460--492
\jour Theoret. and Math. Phys.
\yr 2002
\vol 130
\issue 3
\pages 391--418

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    This publication is cited in the following articles:
    1. A. L. Lisok, A. Yu. Trifonov, A. V. Shapovalov, “Green's Function of a Hartree-Type Equation with a Quadratic Potential”, Theoret. and Math. Phys., 141:2 (2004), 1528–1541  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. Lisok AL, Trifonov AY, Shapovalov AV, “The evolution operator of the Hartree-type equation with a quadratic potential”, Journal of Physics A-Mathematical and General, 37:16 (2004), 4535–4556  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. A. L. Lisok, A. Yu. Trifonov, A. V. Shapovalov, “Symmetry operators of a Hartree-type equation with quadratic potential”, Siberian Math. J., 46:1 (2005), 119–132  mathnet  crossref  mathscinet  zmath  isi
    4. Alexander Shapovalov, Andrey Trifonov, Alexander Lisok, “Exact Solutions and Symmetry Operators for the Nonlocal Gross–Pitaevskii Equation with Quadratic Potential”, SIGMA, 1 (2005), 007, 14 pp.  mathnet  crossref  mathscinet  zmath
    5. Bellucci, S, “Semiclassically concentrated solutions for the one-dimensional Fokker-Planck equation with a nonlocal nonlinearity”, Journal of Physics A-Mathematical and General, 38:7 (2005), L103  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    6. V. V. Belov, E. I. Smirnova, “Localized Asymptotic Solutions of the Self-Consistent Field Equation”, Math. Notes, 80:2 (2006), 296–299  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. Belov, VV, “Semiclassical spectrum for a Hartree-type equation corresponding to a rest point of the Hamilton-Ehrenfest system”, Journal of Physics A-Mathematical and General, 39:34 (2006), 10821  crossref  mathscinet  zmath  adsnasa  isi  scopus
    8. Litvinets, FN, “Berry phases for the nonlocal Gross–Pitaevskii equation with a quadratic potential”, Journal of Physics A-Mathematical and General, 39:5 (2006), 1191  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    9. V. M. Babich, “Space-time ray method and quasiphotons”, J. Math. Sci. (N. Y.), 148:5 (2008), 633–638  mathnet  crossref  mathscinet  zmath
    10. V. V. Belov, F. N. Litvinets, A. Yu. Trifonov, “Semiclassical spectral series of a Hartree-type operator corresponding to a rest point of the classical Hamilton–Ehrenfest system”, Theoret. and Math. Phys., 150:1 (2007), 21–33  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    11. Alexander V. Shapovalov, Roman O. Rezaev, Andrey Yu. Trifonov, “Symmetry Operators for the Fokker–Plank–Kolmogorov Equation with Nonlocal Quadratic Nonlinearity”, SIGMA, 3 (2007), 005, 16 pp.  mathnet  crossref  mathscinet  zmath
    12. Belov, VV, “Semiclassical soliton-type solutions of the Hartree equation”, Doklady Mathematics, 76:2 (2007), 775  crossref  mathscinet  isi  elib  scopus
    13. J. Brüning, S. Yu. Dobrokhotov, R. V. Nekrasov, A. I. Shafarevich, “Propagation of Gaussian wave packets in thin periodic quantum waveguides with a nonlocal nonlinearity”, Theoret. and Math. Phys., 155:2 (2008), 689–707  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    14. M. M. Goncharovskiy, I. V. Shirokov, “An integrable class of differential equations with nonlocal nonlinearity on Lie groups”, Theoret. and Math. Phys., 161:3 (2009), 1604–1615  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    15. V. M. Babich, A. I. Popov, “Quasiphotons of waves on the surface of the heavy liquid”, J. Math. Sci. (N. Y.), 173:3 (2011), 243–253  mathnet  crossref
    16. Belov V.V., Smirnova E.I., Trifonov A.Yu., “Semiclassical Spectral Series for the Two-Component Hartree-Type Equation”, Russian Physics Journal, 54:6 (2011), 639–648  crossref  mathscinet  zmath  adsnasa  isi  elib  elib  scopus
    17. Kulagin A.E., Trifonov A.Yu., Shapovalov A.V., “Quasiparticles Described By the Gross–Pitaevskii Equation in the Semiclassical Approximation”, Russ. Phys. J., 58:5 (2015), 606–615  crossref  zmath  isi  scopus  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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