Fractional dynamics in nonlinear magnetic metamaterials

We examine the existence of nonlinear modes and their temporal dynamics, in arrays split-ring resonators, using a fractional extension Laplacian evolution equation. find closed-form expression for dispersion relation as function exponent well an exact critical coupling between rings, beyond which no magnetoinductive wave can exist. also low-lying families bulk surface bifurcation diagrams. Here phenomenology is similar all exponents resembles what has been observed other discrete equations, such DNLS. The propagation initially localized magnetic excitation always ballistic, with `speed' that computed form exponent. For given exponent, it increases increase up to value, ballistic speed could diverge inside interval $[0,1]$. Examination modulational instability shows tends where decay proceeds via formation filamentary structures merge eventually pure radiation. dynamical selftrapping around but degree trapping linear limit. This decrease be explained by near-degeneracy considerations.