International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 6, Pages 325-370

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

V. V. Belov,1 A. Yu. Trifonov,2 and A. V. Shapovalov3

1Moscow Institute of Electronics and Mathematics, 3/12 Trekhsvyatitel'sky Lane, Moscow 109028, Russia
2Tomsk Polytechnic University, 30 Lenin Ave., Tomsk 634034, Russia
3Tomsk State University, 36 Lenin Ave., Tomsk 634050, Russia

Received 21 December 2001

Copyright © 2002 V. V. Belov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 (0), are constructed with a power accuracy of O(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.