The Trajectory-Coherent Approximation and the System of Moments for the Hartree-Type Equation

TL;DR

This paper develops a method for constructing quasi-classically concentrated solutions to the Hartree-type equation using the complex WKB-Maslov approach, providing asymptotic solutions with controllable accuracy and illustrating the method with a Gaussian potential example.

Contribution

It introduces a systematic construction of quasi-classical solutions for the Hartree-type equation using the complex WKB-Maslov method and Hamilton-Ehrenfest equations, including a nonlinear superposition principle.

Findings

Constructed asymptotic solutions with O(ħ^{N/2}) accuracy.

Formulated a nonlinear superposition principle for these solutions.

Demonstrated the method with a Gaussian potential example.

Abstract

The general construction of quasi-classically 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 \h (\h\to0), are constructed with a power accuracy of O(\h^{N/2}), where N is any natural number. In constructing the quasi-classically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for middle or centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of quasi-classically concentrated solutions of the Hartree-type equations. The results obtained are exemplified by the one-dimensional equation Hartree-type with a Gaussian potential.Comments: 6 pages, 4 figures, LaTeX Report no: Subj-class: Accelerator Physics