Travelling waves in nonlinear magneto-inductive lattices

Abstract

We consider a lattice equation modelling one-dimensional metamaterials formed by a discrete array of nonlinear resonators. We focus on periodic travelling waves due to the presence of a periodic force. The existence and uniqueness results of periodic travelling waves of the system are presented. Our analytical results are found to be in good agreement with direct numerical computations

Figures (5)

• Figure 1: The graph of functions $\frac{37}{1225}$ and $\frac{1}{x^2(1-2\lambda\cos xp)}$ with values at nonzero integers depicted with asterisks.
• Figure 2: (a) The oscillation amplitude of periodic solutions of \ref{['eq2']} as a function of the driving amplitude $h_0$ for $\omega=1.23$ with $\gamma = 0$ (black thick lines) and $\gamma=0.001$ (blue thin lines). (b) Solution profiles at the driving amplitudes indicated in the legend for vanishing $\gamma$.
• Figure 3: The same as Fig. \ref{['figw2']}(a) for $\omega=\sqrt{37}/5$ with (a) $\gamma=0$ and (b) $\gamma=0.1$. In both panels, the solid black and the dashed blue lines are the numerics and the analytical results \ref{['exp2']} (a) and \ref{['exp1']}, \ref{['exp3']} (b), respectively.
• Figure 4: The same as Fig. \ref{['figw2']}(a), but for the periodic drive \ref{['h2']}. The dashed lines are the analytical amplitudes from \ref{['exp4']} and \ref{['eqUsmallest']}. Here, $\gamma=0$.
• Figure 5: Floquet multipliers of periodic solutions for some parameter values (see the text). The dashed curve is the unit circle, showing as guide to the eye.

Theorems & Definitions (18)

• Lemma 1
• Lemma 2
• proof
• Theorem 1
• proof
• Remark 1
• Theorem 2
• proof
• Theorem 3
• proof