**5. Modulation instability**

Modulation instability (MI) is a universally known phenomenon-affecting continuum. It reflects the ability of a weakly perturbed wave to undergo very strong modulations that finally break down in a stable wave [6–8] train. Historically, the MI studied in hydrodynamic fields [9–11] and subsequently the phenomenon observed in other media of propagation of the waves such as electrical transmission lines nonlinear [12–15] optical guide. The MI can occur in space or in time domain. Benjamin and Feir were the pioneers of the study of the MI. In 1967, they studied the evolution in time of MI and demonstrated both theoretically and experimentally that a uniform continuous wave train could be unstable to the disruptions that modulate its envelope [7, 8]. The impact of the work of Benjamin and Feir, often called instability of Benjamin-Feir, is mainly in hydrodynamics.

#### **5.1. Criterion of modulation instability in the electrical bi-inductance line**

In a previous work, Pelap [1] conducted a theoretical study of a power nonlinear transmission line bi-inductance. He first showed that the wave propagation in the line governed by an equation of type Ginzburg Landau complex (GLC) then sought solutions of the equation in discrete semi-approximation. The approach was different from that adopted by Lange and Newell for hydrodynamic fields [16] to establish a criterion of instability for a plane wave propagating through a nonlinear bi-inductance linear and weakly dissipative inductance. The wave is unstable under modulation if the pseudo-product (Eq. (3)) is positive that is:

$$P\_r Q\_r + P\_\perp Q\_\parallel \ge 0\tag{3}$$

where *Pr* , *Pi* , *Qr* , and *Qi* are, respectively, the real and imaginary coefficients of dispersion terms *P* and of nonlinearity *Q* .

If the pseudo-product (Eq. (3)) is positive, it means then that the wave is unstable under the modulation and the system will be the seat of an MI.

We study, for our electrical nonlinear bi-inductance transmission line, the evolution of the pseudo-product in HF mode (**Figure 8**) and the BF mode (**Figure 9**).

The signal can then undergo a significant distortion as it penetrates into the line. We present a characteristic effect related to the effects of nonlinearity of the signals (in **Figure 5**). It is obvious that the 144th cell wave become very asymmetrical, as phases of the various points of the

**Figure 7.** Destruction of the sine wave kept after crossing a section of electrical nonlinear line aft 288th cells by a sinusoidal signal of frequency *f* = 20 kHz, amplitude Vsignal = 1.5 V, and a polarization of the line voltage V0 = 1.5 V.

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In fact, everything happens as if parts of the wave with large amplitudes are moving faster than low amplitude. The wave starts to break down; it shows more and more overtones, thus reflecting the complexity of the shape and explains why nonlinearity leads to a spreading of

Ultimately, the wave front flattens completely; called shock wave similar to the phenomenon observed in the aerodynamic field when a mobile starts to move at a speed greater than the

Modulation instability (MI) is a universally known phenomenon-affecting continuum. It reflects the ability of a weakly perturbed wave to undergo very strong modulations that finally break down in a stable wave [6–8] train. Historically, the MI studied in hydrodynamic fields [9–11] and subsequently the phenomenon observed in other media of propagation of the waves such as electrical transmission lines nonlinear [12–15] optical guide. The MI can occur in space or in time domain. Benjamin and Feir were the pioneers of the study of the MI. In 1967, they studied the evolution in time of MI and demonstrated both theoretically and

speed of sound. At this moment, the wave breaks (**Figure 7**).

wave velocity are different.

**5. Modulation instability**

the wave (**Figure 6**).

Knowledge of the values of the critical wave in the HF mode allows us to clarify if areas of the curve dispersion of the wave are stable under the modulation or not (**Figure 10**). Thus, we are building a decision tool that allows us in our different investigations to determine the frequency of the signals that we send in the line for the observation of specific phenomena properly.

**Figure 8.** Shape of the pseudo-product in BF mode for the inductance bi line. The value of the wave number is critical in this mode which marks the separation between the area of stability, and the zone of instability is *kcl* = 1.23305 rad./cel.

**Figure 9.** Changing of the pseudo-product in HF mode for the inductance bi line. The value of the wave number critical separation between the area of stability and the zone of instability in this mode is *kch* = 1.07595 rad./cel.

**Figure 10.** Delimitation of modulation areas of stability and instability of the electric nonlinear bimodal transmission line based on the criterion of modulation instability.

#### **5.2. Observation of MI in the nonlinear bimodal transmission line**

We observe modulation instability in the bi-inductance line in the BF mode. To do this, on the one hand, we must choose a signal whose frequency is in the region of modulation instability such as provided for in the calculations, and on the other hand, to introduce amplitude that is strong enough to initiate the disruption that will trigger the IM of the wave. To do this, we have chosen a sinusoidal signal whose frequency was *f* = 491.5 kHz for

**Figure 12.** Details of signal in BF mode in the nonlinear bi-inductance transmission line for a signal of frequency *f* = 491.5 kHz and amplitude VM = 3.6 V observed at the 138th cell. Note that wave plane of entry is modulated in

**Figure 11.** Observation of MI in the BF mode in the nonlinear bimodal transmission line for a signal of frequency

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*f* = 491.5 kHz, and amplitude VM = 3.6 V observed at the 138th cell.

amplitude (*M* = 0.18 modulation rate).

**Figure 11.** Observation of MI in the BF mode in the nonlinear bimodal transmission line for a signal of frequency *f* = 491.5 kHz, and amplitude VM = 3.6 V observed at the 138th cell.

**Figure 12.** Details of signal in BF mode in the nonlinear bi-inductance transmission line for a signal of frequency *f* = 491.5 kHz and amplitude VM = 3.6 V observed at the 138th cell. Note that wave plane of entry is modulated in amplitude (*M* = 0.18 modulation rate).

instability such as provided for in the calculations, and on the other hand, to introduce amplitude that is strong enough to initiate the disruption that will trigger the IM of the wave. To do this, we have chosen a sinusoidal signal whose frequency was *f* = 491.5 kHz for

**5.2. Observation of MI in the nonlinear bimodal transmission line**

based on the criterion of modulation instability.

We observe modulation instability in the bi-inductance line in the BF mode. To do this, on the one hand, we must choose a signal whose frequency is in the region of modulation

**Figure 10.** Delimitation of modulation areas of stability and instability of the electric nonlinear bimodal transmission line

**Figure 9.** Changing of the pseudo-product in HF mode for the inductance bi line. The value of the wave number critical

separation between the area of stability and the zone of instability in this mode is *kch* = 1.07595 rad./cel.

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the most complex. They found that all the energy which was initially excited almost returned to fundamental mode. This phenomenon was later called recurrence Fermi-Pasta-Ulam (FPU). This experience was important for two reasons—first, it highlighted the complexity of

The electrical nonlinear bimodal transmission line presents a level of complexity compared to its counterpart mono inductance as far as it has two modes of propagations, the HF and BF

In BF mode, we introduce a sine wave of frequency *f* = 475 KHz in the line with an amplitude Vsignal = 1.5 V, the polarization of the line voltage is V0 = 1.5 V, and we collect the signal level

the collected signal presents deformations of stochastic appearance that initially will grow as it sinks into the line to finally find the sinusoidal shape of the signal of departure to the 22nd cell (**Figure 14**). This phenomenon observed in the electrical nonlinear bimodal transmission

**Figure 14.** FPU Recurrence observed in electrical nonlinear transmission line in BF mode on inductance L2, for a signal

of frequency *f* = 475 KHz and Vsignal = 1.5 V and V0 = 1.5 V as voltage polarization of the line.

inductors are located on cells of even order). We see that

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nonlinear systems; second, it demonstrated the power of the complex systems.

mode. We propose to conduct a study of the recurrence of FPU in each mode.

(*L*<sup>2</sup>

line is known as the Fermi-Pasta-Ulam (FPU) recurrence.

**electrical nonlinear bi-inductance line**

*6.2.1. The FPU recurrence in the BF mode*

of the inductance of each cell *L*<sup>2</sup>

**6.2. Experimental observation of the Fermi-Pasta-Ulam recurrence in a transmission** 

**Figure 13.** Observation of modulation instability in HF mode in the transmission of nonlinear bi-inductance line for a signal of frequency *f* = 897 KHz and amplitude VM = 3.6 V observed at 130th cell.

amplitude VM = 3.6 V. We observe in **Figure 11**, the signals are collected at the level of the 138th cell.

Zooming on the recording of the signal shows that plane wave, which is injected at the entrance to the line, shows the MI in amplitude whose rate is 0.18 (**Figure 12**).

Modulation instability was also observed in HF mode. In this case, we have chosen a signal whose frequency is in the region of modulation instability such as provided for by the calculations in HF mode and which has a magnitude large enough to initiate the disruption that will trigger the MI of the wave. To do this, we have chosen a sinusoidal signal whose frequency is *f* = 897 KHz and amplitude VM = 3.6 V. We observe in **Figure 13**, the signals collected at the level of the 130th cell.
