Fractionality and PT-symmetry in an electrical transmission line

Abstract

We examine the stability of a 1D electrical transmission line in the simultaneous presence of PT-symmetry and fractionality. The array contains a binary gain/loss distribution ${\gamma }_n$ and a fractional Laplacian characterized by a fractional exponent $\alpha$. For an infinite periodic chain, the spectrum is computed in closed form, and its imaginary sector is examined to determine the stable/unstable regions as a function of the gain/loss strength and fractional exponent. In contrast to the non-fractional case where all eigenvalues are complex for any gain/loss, here we observe that a stable region can exist when gain/loss is small, and the fractional exponent is below a critical value, $0<\alpha <{\alpha }_{c1}$ . As the fractional exponent is decreased further, the spectrum acquires a gap with two nearly-flat bands. We also examined numerically the case of a finite chain of size N. Contrary to what happens in the infinite chain, here the stable region always lies above a critical value ${\alpha }_{c2}<\alpha <1$. An increase in gain/loss or $N$ always reduces the width of this stable region until it disappears completely.

6 pages, 6 figures

References in corpus (6)

• Fractional Quantum Mechanics
• Experimental Study of Active LRC Circuits with PT-Symmetries
• Gain-Driven Discrete Breathers in PT-Symmetric Nonlinear Metamaterials
• Fractional derivatives in Dengue epidemics
• Bounded dynamics of finite PT-symmetric magnetoinductive arrays
• Interplay of fractionality and $\mathcal{P}\mathcal{T}$- symmetry on a 1D lattice