Semiclassical Spectral Series Localized on a Curve for the Gross–Pitaevskii Equation with a Nonlocal Interaction

Anton E Kulagin,Andrey Y Trifonov,Alexander V Shapovalov

doi:10.3390/sym13071289

Abstract

We propose the approach to constructing semiclassical spectral series for the generalized multidimensional stationary Gross–Pitaevskii equation with a nonlocal interaction term. The eigenvalues and eigenfunctions semiclassically concentrated on a curve are obtained. The curve is described by the dynamic system of moments of solutions to the nonlocal Gross–Pitaevskii equation. We solve the eigenvalue problem for the nonlocal stationary Gross–Pitaevskii equation basing on the semiclassical asymptotics found for the Cauchy problem of the parametric family of linear equations associated with the time-dependent Gross–Pitaevskii equation in the space of extended dimension. The approach proposed uses symmetries of equations in the space of extended dimension.

Highlights

The collective modes of coherent quantum atom ensembles in the Bose–Einstein condensate (BEC) confined by a trap field in the mean-field approximation are described by the nonlinear Schrödinger equation, the Gross–Pitaevskii equation (GPE): Academic Editor: Michel Planat
∂t parameter, Vtr (~x ) is the external field potential, the function Ψ(~x, t) is the macroscopic wavefunction of the condensate such that |Ψ(~x, t)|2 is proportional to the BEC density and its phase gradient is proportional to the BEC velocity
The approach to the eigenvalue problem (4) and (5) is based on [25], where the leading term of semiclassical asymptotics for the Cauchy problem has been constructed explicitly in a class of functions localized in a neighborhood of a curve in the phase space of a dynamic system of moments of the nonlocal GPE solution termed the Hamilton–Ehrenfest system

Summary

Introduction

The collective modes of coherent quantum atom ensembles in the Bose–Einstein condensate (BEC) confined by a trap field in the mean-field approximation are described by the nonlinear Schrödinger equation, the Gross–Pitaevskii equation (GPE) (see, e.g., [1,2,3,4]): Academic Editor: Michel Planat. In [21], the semiclassical approximation was used to study the non-stationary local GPE with the special type of external field and different orders of the nonlinear term. The approach to the eigenvalue problem (4) and (5) is based on [25], where the leading term of semiclassical asymptotics for the Cauchy problem has been constructed explicitly in a class of functions localized in a neighborhood of a curve in the phase space of a dynamic system of moments of the nonlocal GPE solution termed the Hamilton–Ehrenfest system. As opposed to the linear case (κ = 0 in (4)), the variance matrix of the solutions constructed in [25] may be bounded due to the nonlinearity Such solutions of the Cauchy problems are termed the semiclassical soliton-type solutions [29], and we use exactly such particular solutions for the construction of the spectral series of the nonlocal GPE.

Nonlocal Gross–Pitaevskii Equation

Stationary Solutions of the GPE

Semiclassical Spectral Series

Example

Conclusions