The fractional nonlinear PT dimer

TL;DR

This paper investigates a fractional PT-symmetric nonlinear dimer, revealing how fractional derivatives influence exchange dynamics, damping, and self-trapping phenomena through analytical and numerical methods.

Contribution

It introduces the fractional discrete nonlinear Schrödinger dimer with PT symmetry, analyzing its dynamics using Laplace transforms and numerical simulations, highlighting effects of fractional derivatives.

Findings

Fractional derivatives cause a monotonically decreasing amplitude envelope.

Damped oscillations occur at low gain/loss, unbounded growth at high gain/loss.

Nonlinearity reduces self-trapping effectiveness as it exceeds a threshold.

Abstract

We examine a fractional Discrete Nonlinear Schrodinger dimer, where the usual first-order derivative of the time evolution is replaced by a non integer-order derivative. The dimer is nonlinear (Kerr) and PT -symmetric, and we examine the exchange dynamics between both sites. By means of the Laplace transformation technique, the linear PT dimer is solved in closed form in terms of Mittag-Leffler functions, while for the nonlinear regime, we resort to numerical computations using the direct explicit Grunwald algorithm. In general, the main effect of the fractional derivative is the onset of a monotonically decreasing time envelope for the amplitude of the oscillatory exchange. In the presence of PT symmetry, the dynamics shows damped oscillations for small gain/loss in both sites, while at higher gain/loss parameter values, the amplitudes of both sites grows unbounded. In the presence of nonlinearity, selftrapping is still possible although the trapped fraction decreases as the nonlinearity is increased past threshold, in marked contrast with the standard case.