### 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 ℏ, ℏ → 0) are constructed with the power-law accuracy O(ℏ^{N/2}), where N ≥ 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.

Original language | English |
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Pages (from-to) | 391-418 |

Number of pages | 28 |

Journal | Theoretical and Mathematical Physics |

Volume | 130 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Dec 2002 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Semiclassical trajectory-coherent approximations of Hartree-type equations.** / Belov, V. V.; Trifonov, A. Yu; Shapovalov, Aleksandr Vasilievich.

Research output: Contribution to journal › Article

*Theoretical and Mathematical Physics*, vol. 130, no. 3, pp. 391-418. https://doi.org/10.1023/A:1014719007121

}

TY - JOUR

T1 - Semiclassical trajectory-coherent approximations of Hartree-type equations

AU - Belov, V. V.

AU - Trifonov, A. Yu

AU - Shapovalov, Aleksandr Vasilievich

PY - 2002/12/1

Y1 - 2002/12/1

N2 - 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 ℏ, ℏ → 0) are constructed with the power-law accuracy O(ℏN/2), where N ≥ 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.

AB - 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 ℏ, ℏ → 0) are constructed with the power-law accuracy O(ℏN/2), where N ≥ 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.

UR - http://www.scopus.com/inward/record.url?scp=0036243766&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036243766&partnerID=8YFLogxK

U2 - 10.1023/A:1014719007121

DO - 10.1023/A:1014719007121

M3 - Article

AN - SCOPUS:0036243766

VL - 130

SP - 391

EP - 418

JO - Theoretical and Mathematical Physics(Russian Federation)

JF - Theoretical and Mathematical Physics(Russian Federation)

SN - 0040-5779

IS - 3

ER -