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Quasiclassical soliton solutions of the Hartree equation

Abstract

One considers the three-dimensional Hartree equations

, ΔU=¦ψ¦2. One constructs the asymptotic (h→0) solution ψ of soliton type, localized mod 0 (h) in a compact domain. One finds the corresponding asymptotics of the self-consistent potential U. One obtains quantification conditions on the energy of the soliton.

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Literature cited

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    E. H. Lieb and B. Simon, “The Hartree-Fock theory for Coulomb systems,” Commun. Math. Phys.,53, No. 3, 185–194 (1977).

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    R. T. Glassey, “Asymptotic behavior of solutions of certain nonlinear Schrodinger-Hartree equations,” Commun. Math. Phys.,53, No. 1, 9–18 (1977).

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    V. P. Maslov, Complex Markov Chains and the Feynman Continual Integral [in Russian], Nauka, Moscow (1976).

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    V. P. Maslov, “Equations of the self-consistent field,” in: Current Problems in Mathematics [in Russian], Vol. 11, Moscow (1978), pp. 153–234.

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    V. P. Maslov, Perturbation Theory and Asymptotic Methods [in Russian], Moscow State Univ. (1965).

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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk. SSSR, Vol. 84, pp. 108–113, 1979.

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Karasev, M.V., Maslov, V.P. Quasiclassical soliton solutions of the Hartree equation. J Math Sci 21, 328–332 (1983). https://doi.org/10.1007/BF01660588

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Keywords

  • Soliton
  • Quantification Condition
  • Soliton Solution
  • Compact Domain
  • Hartree Equation