Semiclassical states localized on a one-dimensional manifold and governed by the nonlocal NLSE with an anti-Hermitian term
Anton E. Kulagin, Alexander V. Shapovalov
TL;DR
The paper addresses semiclassical states of a nonlocal NLSE with an anti-Hermitian term that are localized on a time-evolving curve, modeling open quantum-system dynamics with nontrivial geometry. It develops a Maslov complex germ-based framework that lifts the problem to an expanded space, yielding a Hamilton–Ehrenfest-type description of the localization curve and its moments, and constructs asymptotic solutions via trajectory-centered functions. A concrete 2D vortex-model demonstrates quasi-steady vortex states along a rotating circle, derives linearized deformations, and proposes curve convexity as a criterion for transitions to new vortex lattices. The work provides a semi-analytical toolkit for analyzing nonlocal, non-Hermitian NLSEs in open systems, with potential extensions to fractional nonlocal kinetics and more complex geometries.
Abstract
We develop the method for constructing solutions to the nonlocal nonlinear Schrödinger 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.