Semiclassical states localized on a one-dimensional manifold and governed by the nonlocal NLSE with an anti-Hermitian term

TL;DR

This paper introduces a method for constructing semiclassically localized solutions to a nonlocal nonlinear Schrödinger equation with an anti-Hermitian term, focusing on their evolution along a one-dimensional manifold with geometric influence.

Contribution

It develops a novel approach to analyze localized solutions on curves for a nonlocal NLSE with anti-Hermitian terms, linking quantum system dynamics to geometric properties.

Findings

Vortex states evolve according to integro-differential equations tied to curve geometry

Semiclassical vortex behavior is largely influenced by the localization curve's shape

The approach models open quantum systems with nontrivial geometric constraints

Abstract

We develop the method for constructing solutions to the nonlocal nonlinear Schr\"{o}dinger equation (NLSE) with an anti-Hermitian term that are semiclassically localized on a one-dimensional manifold (a curve). The evolution of the curve is given by the closed system of integro-differential equations that can be treated as the "classical"\, analog of the open quantum system with the nontrivial geometry. Using our approach, we consider the evolution of vortex states in the open quantum system described by the specific model NLSE. The semiclassical stage of the vortex evolution can be treated as a quasi-steady vortex state. We show that the behaviour of this state is largely determined by the geometry of the localization curve.