From solitons to rogue waves in nonlinear left-handed metamaterials

In the present work, we explore soliton and roguelike wave solutions in the transmission line analog of a nonlinear left-handed metamaterial. The nonlinearity is expressed through a voltage-dependent, symmetric capacitance motivated by recently developed ferroelectric barium strontium titanate thin-film capacitor designs. We develop both the corresponding nonlinear dynamical lattice and its reduction via a multiple scales expansion to a nonlinear Schrödinger (NLS) model for the envelope of a given carrier wave. The reduced model can feature either a focusing or a defocusing nonlinearity depending on the frequency (wave number) of the carrier. We then consider the robustness of different types of solitary waves of the reduced model within the original nonlinear left-handed medium. We find that both bright and dark solitons persist in a suitable parametric regime, where the reduction to the NLS model is valid. Additionally, for suitable initial conditions, we observe a rogue wave type of behavior that differs significantly from the classic Peregrine rogue wave evolution, including most notably the breakup of a single Peregrine-like pattern into solutions with multiple wave peaks. Finally, we touch upon the behavior of generalized members of the family of the Peregrine solitons, namely, Akhmediev breathers and Kuznetsov-Ma solitons, and explore how these evolve in the left-handed transmission line.

Figure 1

A sketch of the unit cell of the transmission line emulating the left-handed metamaterial.

Figure 2

Left panel: The linear dispersion relation [cf. Eq. (7)]. Right panel: The dependence of the factor $PQ$ (which determines the focusing or defocusing nature of the model) on the frequency $\omega$; see also the text. When $PQ>0$, the nonlinearity is self-focusing, while for the opposite sign it is self-defocusing.

Figure 3

Contour plots depicting the space (node)-time ($t$) evolution of a bright soliton propagating through the left-handed medium. Note that $v_p>0$ and $v_g<0$, as seen by the slopes of the lower and upper straight lines, respectively. In both panels, insets zooming in the initial stage of the evolution highlight the fact that $v_p{v}_g<0$, as is the case for a left-handed medium. The soliton clearly propagates with the prescribed group velocity. The initial data are obtained from the bright soliton solution of Eq. (10) with $u_0=1$ and $c=0$. For the quasicontinuum, long-wavelength approximation, we use $\epsilon =0.1$, and $(k,\omega )\approx (1.3823,0.7712)$ in the left panel and $(k,\omega )\approx (0.4650,1.9305)$ in the right panel; respective values in physical units are $(k,f)\approx (1.3823{\mathrm{cm}}^{-1},200\mathrm{kHz})$ and $(k,f)\approx (0.4650{\mathrm{cm}}^{-1},501\mathrm{kHz}$). The maximum soliton amplitudes are 0.93 V (left panel) and 6.7 V (right panel); in both cases, $g=0.056$.

Figure 4

Amplitude (i.e., maximum voltage) of the bright soliton as a function of time for Fig. 3. Despite the modulation induced by the LH medium, notice the robustness of the bright soliton wave form.

Figure 5

Dark soliton space-time contour plot evolution for $g=0.056$. The initial data for the dark soliton solution are obtained on the basis of Eq. (11) using $u_0=0.1, A=0, B=1, K=0, \epsilon =0.1$, and $(k,\omega )\approx (0.1649,3.4681)$ [i.e., $(k,f)\approx (0.1649{\mathrm{cm}}^{-1},900\mathrm{kHz})$, in physical units]. The background voltage amplitude is 0.53 V. We observe the nearly undistorted propagation of the dark soliton which is now supported by the defocusing NLS model.

Figure 6

Evolution of a Peregrine soliton in the left-handed metamaterial lattice, for parameter values as in the left panel of Fig. 3. The left panel shows the space-time contour plot of the lattice, while the right panel shows the evolution of the maximum voltage amplitude. Note that, in physical units, the initial and maximum values of the voltage are 1.03 and 2.7 V, respectively.

Figure 7

Evolution through the left-handed metamaterial lattice of a Peregrine soliton for $\omega =1.926$ ($f\approx 503$ kHz in physical units) and $g=0.056$. Here, Eq. (12) is utilized to obtain the initial condition in the original variables. The top left panel shows the evolution of the solution's maximum. The top right, bottom left, and bottom right panels show a very long space-time contour plot for the evolution of the voltage. For the formation of a large-amplitude (extreme) event and the subsequent splitting, see details in the text.

Figure 8

Evolution through the left-handed metamaterial lattice of an Akhmediev breather for $f\approx 200\mathrm{kHz}$ and $g=0.056$. We use Eq. (13) with $a=0.1131$ to obtain the initial condition in the original variables. Here, we start from the time before the formation of the maximum amplitude of the Akhmediev breather. Once again, the space-time contour plots of the voltage (left) and of the maximal evolution of the voltage amplitude over time (right) are shown. In this case, in physical units, the initial and maximum values of the voltage are 0.98 and 3.6 V, respectively.

Figure 9

Same as in Fig. (8) but for initial data using Eq. (13) at $T=0$.

Figure 10

Evolution through the left handed metamaterial lattice of a Kuznetsov-Ma soliton for $f\approx 200\mathrm{kHz}$ and $g=0.056$. The solution of Eq. (13) with $a=0.9$ is utilized to obtain the initial condition in the original variables. The same diagnostics, namely, voltage space-time contour plot (left panel) and maximal voltage vs. time (right panel) are depicted. In this case, in physical units, the initial and maximum values of the voltage are 3.4 and 4.0 V, respectively.