The nonlinear magnetoinductive dimer

We examine a nonlinear magnetoinductive dimer and compute its linear and nonlinear symmetric, antisymmetric and asymmetric modes in closed-form, in the rotating-wave approximation. A linear stability analysis of these modes reveals that the asymmetric mode is always stable, for any allowed value of the coupling parameter and for both, hard and soft nonlinearity. A numerical computation of the dimer dynamics reveals a magnetic energy selftrapping whose threshold increases for increasing dimer coupling.Comment: 4 double-column pages, 6 figures, submitte

Flat bands and PT-symmetry in quasi-one-dimensional lattices

We examine the effect of adding PT-symmetric gain and loss terms to quasi 1D lattices (ribbons) that possess flat bands. We focus on three representative cases: (a) The Lieb ribbon, (b) The kagome ribbon, and (c) The stub Ribbon. In general we find that the effect on the flat band depends strongly on the geometrical details of the lattice being examined. One interesting and novel result that emerge from an analytical calculation of the band structure of the Lieb ribbon including gain and loss, is that its flat band survives the addition of PT-symmetry for any amount of gain and loss, while for the other two lattices, any presence of gain and loss destroys the flat bands. For all three ribbons, there are finite stability windows whose size decreases with the strength of the gain and loss parameter. For the Lieb and kagome cases, the size of this window converges to a finite value. The existence of finite stability windows, plus the constancy of the Lieb flat band are in marked contrast to the behavior of a pure one-dimensional lattice.Comment: 5 pages, 5 figure

LOcalized modes on an Ablowitz-Ladik nonlinear impurity

We study localized modes on a single Ablowitz-Ladik impurity embedded in the bulk or at the surface of a one-dimensional linear lattice. Exact expressions are obtained for the bound state profile and energy. Dynamical excitation of the localized mode reveals exponentially-high amplitude oscillations of the spatial profile at the impurity location. The presence of a surface increases the minimum nonlinearity to effect a dynamical selftrapping.Comment: 11 pages, 6 figures, accepted in PL

Compact modes in quasi one dimensional coupled magnetic oscillators

In this work we study analytically and numerically the spectrum and localization properties of three quasi-one-dimensional (ribbons) split-ring resonator arrays which possess magnetic flatbands, namely, the stub, Lieb and kagome lattices, and how their spectra is affected by the presence of perturbations that break the delicate geometrical interference needed for a magnetic flatband to exist. We find that the Stub and Lieb ribbons are stable against the three types of perturbations considered here, while the kagome ribbon is, in general, unstable. When losses are incorporated, all flatbands remain dispersionless but become complex, with the kagome ribbon exhibiting the highest loss rate.Comment: 11 pages, 15 figure

Discrete embedded modes in the continuum in 2D lattices

We study the problem of constructing bulk and surface embedded modes (EMs) inside the quasi-continuum band of a square lattice, using a potential engineering approach \`a la Wigner and von Neumann. Building on previous results for the one-dimensional (1D) lattice, and making use of separability, we produce examples of two-dimensional envelope functions and the two-dimensional (2D) potentials that produce them. The 2D embedded mode decays like a stretched exponential, with a supporting potential that decays as a power law. The separability process can cause that a 1D impurity state (outside the 1D band) can give rise to a 2D embedded mode (inside the band). The embedded mode survives the addition of random perturbations of the potential; however, this process introduces other localized modes inside the band, and causes a general tendency towards localization of the perturbed modes.Comment: 7 pages, 6 figure

Fractional electrical impurity

We examine the localized mode and the transmission of plane waves across a capacitive impurity of strength $\Delta$, in a 1D bi-inductive electrical transmission line where the usual discrete Laplacian is replaced by a fractional one characterized by a fractional exponent $s$. In the absence of the impurity, the plane wave dispersion is computed in closed form in terms of hypergeometric functions. It is observed that the bandwidth decreases steadily, as $s$ decreases towards zero, reaching a minimum width at $s=0$. The localized mode energy and spatial profiles are computed in close form v\`{i}a lattice Green functions. The profiles show a remnant of the staggered-unstaggered symmetry that is common in non-fractional chains. The width of the localized mode decreases with decreasing $s$, becoming completely localized at the impurity site at $s=0$. The transmission coefficient of plane waves across the impurity is qualitatively similar to its non-fractional counterpart ($s=1$), except at low $s$ values ($s\ll 1$). For a fixed exponent $s$, the transmission decreases with increasing $\Delta$Comment: 6 pages, 6 figure

Fractionality and PT-symmetry in an electrical transmission line

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.Comment: 6 pages, 6 figure