Quasiparticles Described by the Gross–Pitaevskii Equation in the Semiclassical Approximation

Abstract

Semiclassical asymptotics of the two-dimensional nonlocal Gross-Pitaevskii equation are constructed. The dynamics of the initial state, being a superposition of two wave packets, is investigated. The discrepancy of the obtained solution is investigated. The constructed asymptotic solutions are interpreted as a description of the interaction of two quasiparticles in the semiclassical approximation. A system of equations for the quasiparticle dynamics is obtained. The nonlocal Gross-Pitaevskii equation (GPE) is a nonlinear integrodifferential equation of the Hartree type that is used as a method for solving multiparticle problems in quantum mechanics. Both the local and the nonlocal Gross-Pitaevskii equations have found application in physics in models of the Bose-Einstein condensate (1). The nonlocal GPE is free of the restriction to interactions of contact type between the particles of the condensate at the price of replacing the local cubic-nonlinear term in the GPE by a nonlinear integral term, which creates definite difficulties for finding solutions of this equation by analytical methods. The semiclassical approximation is an effective tool for constructing analytical solutions of the nonlinear Gross-Pitaevskii equation (2-8). In this case the asymptotic solutions are constructed with respect to a formal small parameter that is proportional to the Planck constant  . Among the asymptotic solutions of the nonlocal GPE, localized semiclassical wave packets are of special interest. These wave packets are called semiclassical soliton-like solutions (9, 10). Belov et al. (6) noted that the condition under which the semiclassical wave packets of the nonlinear GPE do not spread out coincides with the conditions for the existence of solitons of the corresponding local GPE. Examples were given in (4, 11) of self-similar soliton-like solutions of the nonlocal GPE. It is characteristic of solitons that their dynamics is similar to the dynamics of classical particles. In the present paper, we investigate the two-dimensional dynamics of localized semiclassical wave packets of the nonlocal GPE that can be considered as wave functions of two interacting quasiparticles. We have performed an analysis of the discrepancy of the obtained asymptotic solution. A system of ordinary differential equations (a Hamilton-Ehrenfest system) has been obtained that describes the dynamics of the quasiparticles as centers of mass of these wave packets.