Semiclassical Trajectory-Coherent Approximations of Hartree-Type Equations

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.