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The trajectory-coherent approximation and the system of moments for the Hartree type equation

Author

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  • V. V. Belov
  • A. Yu. Trifonov
  • A. V. Shapovalov

Abstract

The general construction of semiclassically concentrated solutions to the Hartree type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter ℏ ( ℏ → 0 ) , are constructed with a power accuracy of O ( ℏ N / 2 ) , where N is any natural number. In constructing the semiclassically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of semiclassically concentrated solutions of Hartree type equations. The results obtained are exemplified by a one-dimensional Hartree type equation with a Gaussian potential.

Suggested Citation

  • V. V. Belov & A. Yu. Trifonov & A. V. Shapovalov, 2002. "The trajectory-coherent approximation and the system of moments for the Hartree type equation," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 32, pages 1-46, January.
    • Handle: RePEc:hin:jijmms:931236
    DOI: 10.1155/S0161171202112142
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    Cited by:

    1. Alexander V. Shapovalov & Anton E. Kulagin, 2021. "Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media," Mathematics, MDPI, vol. 9(23), pages 1-17, November.
    2. Anton E. Kulagin & Alexander V. Shapovalov, 2024. "A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term," Mathematics, MDPI, vol. 12(4), pages 1-22, February.

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