The fractional nonlinear magnetoinductive impurity

We study a one-dimensional split-ring resonator array containing a single linear/nonlinear magnetic impurity where the usual discrete Laplacian is replaced by a fractional one. In the absence of the impurity, the dispersion relation for magnetoinductive waves is obtained in closed form, with a bandwidth that decreases with a decrease in the fractional exponent. Next, by using lattice Green functions, we obtain the bound state energy and its spatial profile, as a function of the impurity strength. We demonstrate that, at large impurity strengths, the bound state energy becomes linear with impurity strength for both linear and nonlinear impurity cases. The transmission of plane waves is computed semi-analytical, showing a qualitative similarity between the linear and nonlinear impurity cases. Finally, we compute the amount of magnetic energy remaining at the impurity site after evolving the system from a completely initially localized condition at the impurity site. For both cases, linear and nonlinear impurities, it is found that for a fixed fractional exponent, there is trapping of magnetic energy, which increases with an increase in impurity strength. The trapping increases with a decreased fractional exponent for a fixed magnetic strength.