Geodesic
Semiclassical Trajectory-Coherent Approximations of Hartree-Type Equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 130 (2002) no. 3, pp. 460-492

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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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     author = {V. V. Belov and A. Yu. Trifonov and A. V. Shapovalov},
     title = {Semiclassical {Trajectory-Coherent} {Approximations} of {Hartree-Type} {Equations}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     volume = {130},
     number = {3},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2002_130_3_a6/}
}
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V. V. Belov; A. Yu. Trifonov; A. V. Shapovalov. Semiclassical Trajectory-Coherent Approximations of Hartree-Type Equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 130 (2002) no. 3, pp. 460-492. http://geodesic.mathdoc.fr/item/TMF_2002_130_3_a6/