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

Abstract

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.2000 Mathematics Subject Classification: 81Q20, 35Q55, 47G20.

Exact Solutions and Symmetry Operators for the Nonlocal Gross-Pitaevskii Equation with Quadratic Potential

“…Following [13,14], we equate the functional vector-parameter Z(t, ) of the P t class of TCFs (7) with z Ψ (t, ), i.e. p Ψ (t, ) = P (t, ), x Ψ (t, ) = X(t, ).…”

“…In [13,14,15,16] a semiclassical integrability approach was developed for a generalized nonlocal GPE named there a Hartree type equation:…”

“…The Cauchy problem was considered in the P t class of trajectory concentrated functions (TCFs) introduced in [13,14]. Being approximate one in essence, the semiclassical approach results in some cases in exact solutions.…”

Transverse Evolution Operator for the Gross-Pitaevskii Equation in Semiclassical Approximation

“…Following [11,12], consider equation (2) for the (1 + 1)-dimensional case with no external field (U = 0)…”

“…These functions model the external fields imposed on the system described by equation (2). A formalism of semiclassical asymptotics was developed for the GPE with a nonlocal nonlinearity in [11,12]. This equation was named the Hartree type equation.…”