Semiclassical approach to the nonlocal nonlinear Schr\"{o}dinger equation with a non-Hermitian term

TL;DR

This paper develops a semiclassical method using the Maslov approach to construct asymptotic solutions for a nonlocal, multidimensional nonlinear Schrödinger equation with a non-Hermitian term, relevant to open quantum systems.

Contribution

It introduces a novel semiclassical framework for solving nonlocal, non-Hermitian NLSEs, including explicit asymptotic solutions for a model of an atom laser.

Findings

Derived semiclassical nonlinear evolution and symmetry operators.

Constructed explicit asymptotic solutions for the atom laser model.

Provided a linearization approach via auxiliary dynamical systems.

Abstract

The nonlinear Sch\"{o}dinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic solutions to this equation within the framework of semiclassically concentrated states. The semiclassical nonlinear evolution operator and symmetry operators for the leading term of asymptotics are derived. Our approach is based on the solutions of the auxiliary dynamical system that effectively linearize the problem under certain algebraic conditions. The formalism proposed is illustrated with the specific example of the NLSE with a non-Hermitian term that is the model of an atom laser. The analytical asymptotic solution to the Cauchy problem is obtained explicitly for this example.