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 ℏ (ℏ → 0), are constructed with a power accuracy of O(ℏ 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.
Highlights
The nonlinear Schrödinger equation− i∂t + Ᏼt, |Ψ |2 Ψ = 0, (1.1)where Ᏼ (t, |Ψ |2) is a nonlinear operator, arises in describing a broad spectrum of physical phenomena
In statistical physics and quantum field theory, the generalized model of the evolution of bosons is described in terms of the second quantization formalism by the Schrödinger equation [24] which, in Hartree’s approximation, leads to the classical multidimensional Schrödinger equation with a nonlocal nonlinearity for one-particle functions, that is, a Hartree type equation
The quantum effects associated with the propagation of an optical pulse in a nonlinear medium are described in the second quantization formalism by the onedimensional Schrödinger equation with a delta-shaped interaction potential
Summary
Where Ᏼ (t, |Ψ |2) is a nonlinear operator, arises in describing a broad spectrum of physical phenomena. The constructed solutions are a generalization of the well-known quantum mechanical coherent and compressed states for linear equations [9, 34] for the case of nonlinear Hartree type equations with variable coefficients. In terms of the approach under consideration, the formal asymptotic solutions of the Cauchy problem for this equation and the evolution operator have been constructed in the class of trajectory-concentrated functions, allowing any accuracy in small parameter , → 0. We introduce a class of functions singularly depending on a small parameter , which is a generalization of the notion of a solitary wave It appears that asymptotic solutions of (2.1) can be constructed based on functions of this class, which depend on the phase trajectory z = Z(t, ), the real function S(t, ) (analogous to the classical action at = 0 in the linear case), and the parameter. In the cases where this does not give rise to ambiguity, we use a shorthand symbol of ᏼt for ᏼt (Z(t, ), S(t, ))
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