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

TL;DR

The paper develops a semiclassical, quasiparticle-based framework for the nonlocal NLSE with an anti-Hermitian term, formulating a Maslov complex germ approach in which the wavefunction concentrates near $K$ moving trajectories with time-dependent masses $\mu_s(t)$. By reducing the Cauchy problem to a Hamilton–Ehrenfest system for $(Z_s(t),\mu_s(t))$ and coupling it to an associated linear NLSE, the authors derive an explicit semiclassical evolution operator $\hat{U}(t)$ and Green-function representation for leading-order dynamics. A physically motivated one-dimensional example with dipole-dipole interactions, optical lattice potential, and damping illustrates how long-range nonlocality and non-Hermiticity yield qualitatively new, soliton-like, two-quasiparticle patterns that differ markedly from the closed-Hermitian case. The results provide a tractable analytic framework for open, nonlocal NLSEs and point to rich dynamics arising from the interplay of long-range interactions and dissipation, with potential applications to open BECs and nonlinear optics.

Abstract

We deal with the $n$-dimensional nonlinear Schrödinger 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.

Figures (6)

• Figure 1: Phase trajectory of the first quasiparticle for $\varkappa=2$ and $\Lambda=0$
• Figure 2: Phase trajectory of the first quasiparticle for $\varkappa=2$ and various $\Lambda$
• Figure 3: Evolution of the mass of a quasiparticle for $\varkappa=2$ and $\Lambda=1$
• Figure 4: Dispersion of the wave function of a quasiparticle for $\varkappa=2$ and $\Lambda=1$
• Figure 5: Squared absolute value of the asymptotic solution to \ref{['exameq1']} for two quasiparticles, $\varkappa=2$, and $\Lambda=1$

Theorems & Definitions (2)

• Definition 2.1
• Definition A.1