Discrete breathers in a nonlinear electric line: Modeling, computation, and experiment

We study experimentally and numerically the existence and stability properties of discrete breathers in a periodic nonlinear electric line. The electric line is composed of single cell nodes, containing a varactor diode and an inductor, coupled together in a periodic ring configuration through inductors and driven uniformly by a harmonic external voltage source. A simple model for each cell is proposed by using a nonlinear form for the varactor characteristics through the current and capacitance dependence on the voltage. For an electrical line composed of 32 elements, we find the regions, in driver voltage and frequency, where $n$-peaked breather solutions exist and characterize their stability. The results are compared to experimental measurements with good quantitative agreement. We also examine the spontaneous formation of $n$-peaked breathers through modulational instability of the homogeneous steady state. The competition between different discrete breathers seeded by the modulational instability eventually leads to stationary $n$-peaked solutions whose precise locations is seen to sensitively depend on the initial conditions.

Figure 1

Schematic circuit diagram of the electrical transmission line.

Figure 2

Experimental data ($*$) and numerical approximation thereof (continuous line) corresponding to $C\left(V\right)$ and $I_D\left(V\right)$ (linear and semilog plots) for the nonlinear varactor.

Figure 3

Single cell element model.

Figure 4

Nonlinear resonance curves corresponding to a single element. Dots (orange/gray) correspond to experimental data while the continuous and dashed black lines correspond, respectively, to stable and unstable numerical solutions. From bottom to top: $V_d=1,2,3,4$, and 5 V.

Figure 5

Existence islands for $n$-peak breathers. The white line denotes the threshold where the homogeneous state becomes unstable (above the line). Black areas correspond to stable solutions and orange (gray) areas to unstable solutions. (a) One-peak breather (the homogeneous state threshold is located at the top border of the areas), (b) a family of two-peak breather, and (c) a family of three-peak breathers. All data correspond to numerical simulations.

Figure 6

Breather profiles (left column) and corresponding Floquet multiplier spectra (right column) for $\tau =0$, $V_d=4$ V, and $f=243.6$ kHz. The top (bottom) panels correspond to the unstable (stable) one-site (intersite) breather. All data correspond to numerical simulations.

Figure 7

Dependence of the Floquet multipliers on the frequency for $N=32$ cells and $V_d=1.5$ V. The top (bottom) row of panels corresponds to the onsite (intersite) breather. The left and right column of panels depicts, respectively, the argument and magnitude of the Floquet multipliers. The horizontal (red) line in the spectra depicts the stability threshold. All data correspond to numerical simulations.

Figure 8

Same as Fig. 7 but for $V_d=2.5$ V.

Figure 9

Same as Fig. 7 but for $V_d=4$ V.

Figure 10

The ILM can exist at any site of the lattice; here $f=253$ kHz and $V_d=4$ V. We demonstrate that the ILM can be made to jump to the neighboring site upon the temporary creation of an impurity there. The left-most profile (centered at site 16) turns into the dotted (red) trace when an external inductor is placed in the direct vicinity of the $L_2$ inductor at site 17. Upon removal of this external inductor, the ILM persists at site 17 (solid black trace). This process can be repeated to move the ILM further to the right, as shown. All data correspond to experiments.

Figure 11

Existence regions of one-peaked (top panel) and a family of three-peaked (bottom panel) breathers obtained numerically (black areas correspond to stable solutions and orange/gray one to unstable solutions) as compared to experimental data identifying the range of observations of the corresponding type of states (circles). The theoretical data are displaced by a $+$7 kHz frequency offset.

Figure 12

Experimental ($•$) and numerical ($\circ$) one-, two-, and three-peak breather profiles for different driver voltage $V_d$ and frequencies values. The left (right) column of panels corresponds to a driver strength $V_d=2$ V ($V_d=4$ V). Each panel correspond to the following experimental ($f_{\mathrm{expt}}$) and numerical ($f_{\mathrm{num}}$) driving frequencies in kHz: (a) $(f_{\mathrm{expt}},f_{\mathrm{num}})=(275,268)$, (b) $(f_{\mathrm{expt}},f_{\mathrm{num}})=(276,269)$, (c) $(f_{\mathrm{expt}},f_{\mathrm{num}})=(280,273)$, (d) $(f_{\mathrm{expt}},f_{\mathrm{num}})=(248,244)$, (e) $(f_{\mathrm{expt}},f_{\mathrm{num}})=(254,247)$, and (f) $(f_{\mathrm{expt}},f_{\mathrm{num}})=(249,245)$.