Quasiparticle solutions for the nonlocal NLSE with an anti-Hermitian term in semiclassical approximation

TL;DR

This paper develops semiclassical asymptotic solutions for a nonlocal nonlinear Schrödinger equation with an anti-Hermitian term, revealing how long-range interactions influence localized quasiparticle patterns in open quantum systems.

Contribution

It introduces a novel semiclassical method for solving a nonlocal NLSE with anti-Hermitian terms, including a reduction to ODEs and linear equations, and applies it to a complex 1D model.

Findings

Localized solutions form spatial patterns that move along trajectories

Long-range interactions significantly alter the behavior of solutions

The formalism applies to models with periodic potentials and damping

Abstract

We deal with the $n$-dimensional nonlinear Schr\"{o}dinger equation (NLSE) with a cubic nonlocal nonlinearity and an anti-Hermitian term, which is widely used model for the study of open quantum system. We construct asymptotic solutions to the Cauchy problem for such equation within the formalism of semiclassical approximation based on the Maslov complex germ method. Our solutions are localized in a neighbourhood of few points for every given time, i.e. form some spatial pattern. The localization points move over trajectories that are associated with the dynamics of semiclassical quasiparticles. The Cauchy problem for the original NLSE is reduced to the system of ODEs and auxiliary linear equations. The semiclassical nonlinear evolution operator is derived for the NLSE. The general formalism is applied to the specific one-dimensional NLSE with a periodic trap potential, dipole-dipole interaction, and phenomenological damping. It is shown that the long-range interactions in such model, which are considered through the interaction of quasiparticles in our approach, can lead to drastic changes in the behaviour of our asymptotic solutions.