Discrete breathers in an electric lattice with an impurity: Birth, interaction, and death

We have simulated aspects of intrinsic localized modes or discrete breathers in a modulated lumped transmission line with nonlinear varactors and a defect unit cell. As the inductance or capacitance of this cell is increased, a transition from instability to stability takes place. Namely, there exist threshold values of the inductance or capacitance of a lattice impurity for a breather to be able to attach to. A resistive defect can also anchor a breather. Moreover, by either gradually lowering all the source resistances, or else increasing the modulation frequency, multiple secondary ILMs can be spontaneously generated at host sites (with only a single inductive or capacitive defect). Further, if two impurities are subcritically spaced (the separation increasing with the amplitude of the modulation voltage), a breather can pop up midway, with no breathers at the impurity sites themselves. Finally, an ILM can pull closer its neighbors on both sides, only to perish once these ILMs have gotten sufficiently close. To our knowledge, these effects have not been reported for any discrete nonlinear system.

Figure 1

(a) Circular lumped band-pass transmission line; 32 such cells, with nonlinear varactors $C\left(V_N\right)$ fed by harmonic voltages $V_d\left(t\right)=V_{d}sin\left(2\pi f_{d}t\right)$ are connected by inductances $L_1$. The larger dot represents a defect unit cell. (b) In this “general” unit cell, the inductance $L_N$ is given by $L_2$ in all unit cells except the cell $N=18$, where it has some increment ${\delta }_{L_2}$ as defined following Eq. (2). Our parameter values are $L_1=680\mu \mathrm{H}$, $L_2=330\mu \mathrm{H}$, $R=10\mathrm{k}\mathrm{\Omega }$, $R_l=20\mathrm{k}\mathrm{\Omega }$, $C_0=788$ pF (Diode varactor NTE 618), $V_d=3$ V.

Figure 2

(a) Maximum and minimum varactor voltages $V_{18}$, $V_{19}(=V_{17})$, and $V_{20}(=V_{16})$ of the perturbed unit cell 18, its nearest neighbors 17 and 19, and next nearest neighbors 16 and 20 as a function of the percentage increase of the inductance $L_2$. Calculated at a time $t=250/f_d$ after switch-on (at $t=0$) of the modulation source. The arrow indicates a transition from instability to stability at ${\delta }_{L_2}$ equal to 0.23% of $L_2$. The parameters are listed in the caption of Fig. 1 and $f_d=268$ kHz. The inset zooms in on the region of very small change of ${\delta }_L$. The arrow in the inset indicates a transition from unperturbed to unstable oscillations for ${\delta }_{L_2}$ equal to $-0.007\%$ of $L_2$. (b) Maximum and minimum voltages $V_N$ in each of the 32 unit cells of the TL for an increase of the inductance $L_2$ in cell 18 greater than 0.23%. This breather has a width of about seven cells.

Figure 3

Maximum and minimum varactor voltages $V_N$ in the 32 unit cells of the TL as a function of the source resistance $R$. For $R=10\mathrm{k}\mathrm{\Omega }$ the same ILM is obtained as in Fig. 2, all the parameters being identical (including the inductive impurity in cell 18). The more $R$ is reduced (in all the cells) the more breathers are produced—up to a total of five. Note that all but one of these are located at sites without impurities ($\text{N}\ne 18$). The animation Fig. A of the Supplemental Material [20] shows how the breather structure changes upon continuous variation of $R$.

Figure 4

Spontaneous creation of ILMs with increase of the modulation frequency $f_d$, from one to two to three ILMs in (a), then a fourth ILM in (b), and finally five ILMs in (c). The values of the voltages $V_N$ are indicated by their color, according to the side bar. Parameters as in Fig. 1. In Fig. B of the Supplemental Material [20] we visualize the creation of these breathers, including transitions and changes in line shape that they undergo in the interval 267.5–358.0 kHz.

Figure 5

Two identical inductive impurities can anchor two identical breathers if their separation is at least nine cells. If the separation is eight or less cells, only a single breather is created midway between the impurities. Parameters as in Fig. 1 and $f_d=268$ kHz. See Fig. C of the Supplemental Material [20] for animation of this process.

Figure 6

Development of ILMs as function of number of periods $f_{d}t$, with $f_d=272.5$ kHz, after the modulation sources have been switched on. The breather at $N=2$ attracts the two neighboring breathers until a point of closest approach. This attraction is, however, “fatal,” for the breather at $N=2$ “perishes” in the process. Parameters as in Fig. 1. This process is animated in Fig. D of the Supplemental Material [20].