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We study the spectrum and transmission coefficient of plane waves propagating along square ribbons of varying widths, containing a square-shaped, PT -symmetric impurity region. We start with a zero-width ribbon (1D chain) and place a PT symmetric dimer. The spectrum is computed numerically and the instability gain is computed as a function of the gain/loss dimer strength. The transmission coefficient is obtained in closed form and examined as a function of wavevector and the gain/loss parameter. Next, we study a ribbon in a narrow ladder configuration containing a square PT impurity. As before, we compute the instability gain numerically and the transmission coefficient in closed form for the two possible input modes. Finally, we repeat the calculations for a wider ladder ribbon containing a Lieb-like impurity in a PT configuration. For all cases and transmission channels, we obtain transmission divergences in wavevector-gain/loss parameter space, whose number increases with the width of the ribbon. • PT -symmetry. • Ribbon lattice. • Square-shaped PT impurity. • Instability rate. • Transmission coefficient. [ABSTRACT FROM AUTHOR]

• Electrical lattice. • Electro-inductive waves. • Excitation transport. • Transmission coefficient. • Fano resonances. We study the scattering properties of an electrical lattice consisting of a one-dimensional array of inductively-coupled LC units. For an initially localized electrical excitation, and in the absence of any impurity, we compute in closed form the mean square displacement of an initially localized electrical excitation for the cases of an infinite and semi-infinite lattice, obtaining a ballistic propagation under very general conditions. For the transport of extended excitations, we compute in closed form the transmission coefficient of electro-inductive plane waves across an impurity region, containing a number of side-coupled units, or a single internal impurity in the presence of coupling to first-and second nearest neighbors, looking for the presence of Fano resonances (FRs). For all cases examined, we obtain a closed-form expression for the position of the FR in terms of the relative strengths of the inductive couplings involved. For the case of two, identical side-coupled impurities, the position of the single FR turns out to be independent of the relative distance between the two impurities. [ABSTRACT FROM AUTHOR]

• A fractional version of discrete Laplacian is inserted into the DNLS equation. • Spectrum of linear waves and root mean square displacement computed in closed form. • Insertion of single general nonlinear impurity. • Use of Green functions to compute the mode energy and profile and the transmission of plane waves. • Effect of fractional exponent on all of the above phenomenology. We use a lattice Green function approach to study the stationary modes of a linear/nonlinear (Kerr) impurity embedded in a periodic one-dimensional lattice where we replace the standard discrete Laplacian by a fractional one. The energies and the mode profiles are computed in closed form, for different fractional exponents and different impurity strengths. The energies of the impurity mode lie outside the linear band whose bandwidth decreases steadily as the fractional exponent decreases. For any fractional exponent values, there is always a single bound state for the linear impurity while for the nonlinear (Kerr) case, up to two bound states are possible, for impurity strengths above certain threshold. The energy of the linear mode (or that of the upper energy nonlinear one), becomes directly proportional to the impurity strength at large impurity strengths. The transmission of plane waves is also computed in closed form for several fractional exponents, and various impurity strengths. We observe that fractionality tends to increase the overall transmission. The selftrapping transition for the nonlinear impurity shifts to lower nonlinearity values as the fractional exponent is decreased. In both cases, linear and nonlinear, we observe a form of trapping at zero impurity strength, which can be explained by the near-degeneracy of the spectrum in the limit of a small fractional exponent. [ABSTRACT FROM AUTHOR]

• 2D split-ring resonator array with resonant frequency disorder. • Localized modes and magnetic transport for uncorrelated and correlated disorder in 2D. • Correlated disorder can create sub-diffusive propagation. We study the effect of a resonant frequency disorder on the eigenstates and the transport of magnetic energy in a two-dimensional (square) array of split-ring resonators (SRRs). In the absence of disorder, we find the dispersion relation of magneto-inductive waves and the mean square displacement (MSD) in closed form, showing that at long times the MSD is ballistic. When disorder is present, we consider two types: the usual Anderson distribution (uncorrelated monomers) and 2 × 2 units assigned at random to lattice sites (correlated tetramers). This is a direct extension to two dimensions of the one-dimensional random dimer model (RDM). For the uncorrelated case, we see saturation of the MSD for all disorder widths, while for the correlated case we find a disorder window, inside which the MSD does not saturate at long times, with an asymptotic sub-diffusive behavior M S D ∼ t 0.26. Outside this disorder window, the MSD shows the same kind of saturation as in the monomer case. We conjecture that the sub-diffusive behavior is a remanent of a weak resonant transmission of a 2D plane wave across a tetramer unit. [ABSTRACT FROM AUTHOR]

• A simplified model for a magnetic metamaterial. • One-dimensional array of split-ring resonators. • Transport properties in the presence of correlated disorder. • Correlated disorder induces a degree of magnetic transport. • Power-law decrease of transmission of planes waves across disordered magnetic array. We examine the transport of magnetic energy in a simplified model of a magnetic metamaterial, consisting of a one-dimensional array of split-ring resonators (SRR), in the presence of correlated disorder in the resonant frequencies. The computation of the average participation ratio (PR) reveals that on average, the modes for the correlated disorder system are less localized than in the uncorrelated case. The numerical computation of the mean square displacement of an initially localized magnetic excitation shows an asymptotic behavior 〈 n 2 〉 ∼ t α , where α ∼ 0 for uncorrelated disorder, and α > 0 for the correlated case, spanning a whole range of behavior ranging from localization to super-diffusive behavior at finite system sizes. The transmission coefficient of a plane wave across a single magnetic dimer reveals the existence of well-defined regions in disorder strength-magnetic coupling space, where unit transmission for some wavevector(s) is possible. This explains the finite degree of mobility for magnetic excitations in the presence of correlated disorder, in the spirit of the random dimer model (Dunlap et al., 1990 [14]). Finally, the transmission of plane waves across an array with correlated disorder shows a power-law decrease with system size, in contrast to the exponential decrease found in the standard Anderson model. [ABSTRACT FROM AUTHOR]

We examine a fractional version of the discrete nonlinear Schrödinger (dnls) equation, where the usual discrete laplacian is replaced by a fractional discrete laplacian. This leads to the replacement of the usual nearest-neighbor interaction to a long-range intersite coupling that decreases asymptotically as a power-law. For the linear case, we compute both, the spectrum of plane waves and the mean square displacement of an initially localized excitation in closed form, in terms of regularized hypergeometric functions, as a function of the fractional exponent. In the nonlinear case, we compute numerically the low-lying nonlinear modes of the system and their stability, as a function of the fractional exponent of the discrete laplacian. The selftrapping transition threshold of an initially localized excitation shifts to lower values as the exponent is decreased and, for a fixed exponent and zero nonlinearity, the trapped fraction remains greater than zero. • A fractional version of discrete laplacian is inserted into the dnls equation. • Spectrum of linear waves and root mean square displacement computed in closed form. • Comparison of nonlinear modes between pure dnls modes and fractional ones. • Shifting of selftrapping transition with fractional exponent. [ABSTRACT FROM AUTHOR]

We examine the transport of extended and localized excitations in one-dimensional linear chains populated by linear and nonlinear symmetric identical n-mers (with n=3, 4, 5, and 6), randomly distributed. First, we examine the transmission of plane waves across a single linear n-mer, paying attention to its resonances, and looking for parameters that allow resonances to merge. Within this parameter regime we examine the transmission of plane waves through a disordered and nonlinear segment composed by n-mers randomly placed inside a linear chain. It is observed that nonlinearity tends to inhibit the transmission, which decays as a power law at long segment lengths. This behavior still holds when the n-mer parameters do not obey the resonance condition. On the other hand, the mean square displacement exponent of an initially localized excitation does not depend on nonlinearity at long propagation distances z, and shows a superdiffusive behavior ~z1.8 for all n-mers, when parameters obey the resonance merging condition; otherwise the exponent reverts back to the random dimer model value ~z1.5. [ABSTRACT FROM AUTHOR]