Self-focusing and envelope pulse generation in nonlinear magnetic metamaterials

The self-modulation of waves propagating in nonlinear magnetic metamaterials is investigated. Considering the propagation of a modulated amplitude magnetic field in such a medium, we show that the self-modulation of the carrier wave leads to a spontaneous energy localization via the generation of localized envelope structures (envelope solitons), whose form and properties are discussed. These results are also supported by numerical calculations.

Figure 1

(Color online) (a) Linear dispersion $\omega =\omega \left(k\right)$, and (b) dispersion coefficient $P=P\left(k\right)$, for $\lambda =0.05$ (black-solid curve), $0.2$ (red-dotted curve), $0.35$ (green-dashed curve).

Figure 2

(Color online) Bright-type modulated wave packets, for $\lambda =0.20$, ${ϵ}_{\ell }=2$, ${\psi }_0^{\prime }=0.1$, $N=100$, and (a) $\alpha =-1$, $k=0.86<k_{cr}(\omega =1.16)$, $L=9.02$; (b) $\alpha =+1$, $k=1.12>k_{cr}(\omega =1.10)$, $L=7.48$. The filled circles correspond to the $q_n$’s, with $q_n$ the charge at site number $n$, while the black-dotted curves serve as a guide to the eye. The red-solid curves are the envelope Eq. (8), with parameters as in (a) and (b), respectively.

Figure 3

(Color online) Dark-type modulated wave packets, for $\lambda =0.20$, ${ϵ}_{\ell }=2$, ${\psi }_0^{\prime }=0.1$, $N=100$, and (a) $\alpha =-1$, $k=1.12>k_{cr}(\omega =1.10)$, $L=3.74$; (b) $\alpha =+1$, $k=0.86<k_{cr}(\omega =1.16)$, $L=4.51$. The filled circles correspond to the $q_n$’s, with $q_n$ the charge at site number $n$, while the black-dotted curve serves as a guide to the eye. The red-solid curves are the envelope Eq. (10), and the green-dashed curves are the envelope Eq. (12), with parameters as in (a) and (b), respectively and $d=0.995$.