Peregrine Soliton and Akhmediev Breathers in a Chameleon Electrical Transmission Line

We analyze the particular behavior exhibited by a chaotic waves field containing Peregrine soliton and Akhmediev breathers. This behavior can be assimilated to a tree with “roots of propagation” which propagate randomly. Besides, this strange phenomenon can be called “tree structures”. So, we present the collapse of dark and bright solitons in order to build up the above mentioned chaotic waves field. The investigation is done in a particular nonlinear transmission line called chameleon nonlinear transmission line. Thus, we show that this line acts as a bandpass filter at low frequencies and the impact of distance, frequency and dimensionless capacitor are also presented. In addition, the chameleon’s behavior is due to the fact that without modifying the appearance structure, it can present alternatively purely rightor left-handed transmission line. This line is different to the composite one.


Introduction
Metamaterials are materials which have both the permeability ( µ ) and the permittivity ( ε ) parameters are set negative at the same frequency [1] [2]. This kind of materials is often called double-negative material or left-handed metamaterials [1] [2]. This class of materials has negative refractive index. So, they The paper is organized as follows. In Section 2, we present the logarithmic nonlinearity for the capacitance, the voltage propagation equations and the nonlinear Schrödinger equation model. The coefficients of this last equation are plotted in order to improve the comprehension of strange phenomena studied.
Thereafter, we apply the collective coordinates technique in order to obtain the collective coordinate equations of motion. In Section 3, we apply numerical experiments in order to investigate numerically the collective coordinates and present the results. The outcomes are summarized in Section 4.

Preliminaries
The model under consideration represents a modulable nonlinear electrical transmission line where elementary cell is illustrated in Figure 1 [29] [30].
Each unit cell, such as the n th one, contains a linear inductor 1 L in parallel with a linear capacitor 1 C in the series branch and a linear inductor 2 L in parallel with a nonlinear capacitor ( ) n C V in the shunt branch. Here we assume that the logarithmic nonlinearity for the capacitance is given by [32] [33]: where 0 V and 0 C take constant values. Applying Kirchhoff's laws to the circuit model, we can obtain the following voltage propagation equations [30]:  . Equation (2) shows that an additional dispersion coefficient r C is considered on the line. Thereafter, the numerical simulations will consider the following parameters [23] [30]: The capacitor 1 C is considered as a free parameter with physically acceptance value [30]. According to the investigations done in [30], the parameter r C will impose the behavior of the transmission line. So, they show that when we have the right-handed behavior, but for ≈ . This situation justifies the name "Chameleon transmission line" because the line changes its behavior but does not modify its external aspect.

Theoretical Model for Electrical Transmission Line
The nonlinear Schrödinger equation inspired from [30], but reformulated in terms of slowly varying envelope of the electric field ( ) , A Z τ as follows [34]: where ( ) , A Z τ is the slowly varying envelope of the electric field at position and at time 2 t τ ε = [30]. Here ε is a positive and small parame- where the wave number k is taken in the Brillouin zone. This dispersion relation admits two cutoff frequencies at The group velocity is given by [30]: The group velocity g V plays a key role in the nature of waves.

Collective Coordinate's Theory
The collective coordinate's technique is a great method of characterization of a

Conventional Gaussian Ansatz Function
For our variational analysis, the desired form of the Gaussian ansatz function f is given by [37] [38]: Equations (

Initial Conditions at the Beginning of the Propagation
The initial conditions at the beginning of the propagation are the same compared to those used in [35] [36] [37] [38]. For the analysis of our optical system the wave number is taken as

Introduction of Right-Handed Propagation
The right-handed behavior occurs on the line when 0 0.691 r C ≤ < [30]. Consequently, two cases will be investigated known as  ].

Akhmediev-Peregrine Waves Field Generation at Low
Frequencies The first considered case is 0 r C = at low frequencies 0 0.78 rad s ω < < for × . This situation reveals that a strange phenomenon acts in the system. So, the bright soliton has lost its stability. Besides, the residual field energy increases and approaches 100 percent. This information suggests an increase of internal distortions as depicted in Figure 5(a). The strong negative nonlinearity continues to act on the weak negative dispersion in order to induce the fragmentation of the bright soliton into several grains of activity [43]. Hence, once grains activated, the spatial inhomogeneity acts as a nonlocal coupling that provides a coherent build-up of an extreme event. These monster events continue to undergo distortions originating from the strong negative nonlinearity. Besides, this trouble situation provokes the multiplication of strong harmonic waves with several residual wave motions at adjacent sideband frequencies [16] [22] [43]. This situation induces the modulation of residual waves. Hence, this Journal of Applied Mathematics and Physics perturbed situation provokes the generation of a chaotic waves field containing one Akhmediev-Peregrine soliton [44] (left) and several Akhmediev breathers (right) [15] [43], as depicted in Figure 6(a).

Tree Like-Structures at Low Frequencies
The observation of 2D full equation of this perturbed waves field shows the fragmentation of the bright soliton as depicted in Figure 6(c). Besides, the collapse of this wave leads to a strange phenomenon. In fact, we observe a strange structure similar to a tree with the multiple roots [25]. Those roots propagate randomly in the optical system as seen in Figure 6(c). Then, this structure can be assimilated to a tree with many roots which can be called roots of propagation [25]. have the researches concerning specific tree structures called Christmas tree [25]. Consequently, this expanding structure, which is called Christmas tree appears to emerge past the formation of the original Peregrine peak [25]. As the structure expands, progressively at the peak emergence times more localized peaks arise. Furthermore, this strange tree structure has been also investigated in Kundu-Eckhaus equation by Bayindir [27]. Indeed, it has been assimilated to a chaotic waves field induced by modulation instability [27]. Otherwise, a similar situation has also been presented in optical field by Dudley et al. [28]. In fact, tree structure is assimilated to signatures of analytic nonlinear schrödinger equation solutions in chaotic modulation instability. Hence, the structure obtained is similar to a density map [28]. Then, according to previous investigations [25] [27], each rogue event has a particular signature corresponding to a specific tree structure. So, we present here the particular signature of a chaotic waves field containing Akhmediev-Peregrine waves, Peregrine waves and Akhmediev wave trains.

Soliton Stability at High Frequencies
We consider 0 r C = and the propagation at high frequencies is obtained for 0.78 rad s ω > , the bright soliton gains in stability as depicted in Figure 6(b) where the robustness of the pulse is presented. In fact, the strong negative nonlinearity and the weak negative dispersion are completely compensated [24], and have built up a robust wave. Despite the action of perturbations depicted in Figure 5(b), the stability of bright soliton is totally restored at high frequency as shown in Figure 6(d) by the signature of the stable soliton. Moreover, similar soliton light pulse stability was recently investigated in ultracold bosonic seas by Charalampidis [25].

Second Case of Right-Handed Propagation
The second case of right-handed behavior of the electrical line is considered for 0.3 r C = [30]. When the frequency is considered such as 0.15 rad s ω = the second-order dispersion coefficient has a weak negative value −10 24 ps 2 •m −1 and the cubic-nonlinearity presents a strong positive value 1.5 × 10 −3 W −1 •m −1 as depicted in Figure 2(b) and Figure 2(d). The product of these two effects is 2 0 Θ ϒ < as illustrated in Figure 3(b). Hence, at low frequencies the dark soliton propagates. The soliton light pulse regains its stability as seen in Figure 7(a).
Besides, it appears that the weak negative value of dispersion is completely compensated the strong positive value of the nonlinearity in order to build-up the dark soliton depicted in Figure 7(a). This stability is maintained at high frequencies as illustrated in Figure 7

Introduction of Left-Handed Propagation
The left-handed behavior occurs when 0.691 r C > . Three cases will be investigated  Figure 9(b). Indeed, as previously observed the chaotic waves field presents Akhmediev-Peregrine freak wave and Akhmediev waves trains as depicted in Figure 9(b). The signature of this chaotic waves field is also illustrated in Figure 9(d). However, the aspect of this chaotic waves field is practically identical to that obtained at low frequencies when right-handed propagation was considered as depicted in Figure 9(d) and Figure   6(c). So, the significant increase of the length of propagation from 3 × 10 −24 m to 10 −6 m induces a considerable decrease of soliton peak power from 2.5 × 10 26 W to 7 W.  The propagation is not favorable at high frequencies for left-handed behavior of the line. More so, the behavior of the line has changed. When right-handed behavior occurs the soliton light pulse propagates at high frequencies and it is destroyed at low frequencies. However, when left-handed behavior arrives the soliton light pulse propagates very well at low frequencies and it is totally destroyed at high frequencies. Hence, all these results suggest that the soliton light pulse propagates in opposite sense when the line changes its behavior from right-handed to left-handed. This result allows us to rename this transmission line as chameleon transmission line since its behavior changes without modify its physical aspect. This information is identical to that recently in [30].

Impact of Cr on Tree Structures
We consider the case of right-handed propagation where 0 r C = at low frequencies. So, we obtain the tree structure depicted in Figure 6(c). This tree structure corresponds to rogue events signature where Akhmediev-Peregrine waves and Akhmediev waves trains appeared. At low frequencies, the rogue events which appear on the rogue signature depicted in Figure 6(c) are completely cancelled when 0.3 r C = as seen in Figure 7(b).

Modified Chaotic Waves Field
This figure corresponds to the signature of the stable soliton light pulse. So, the Journal of Applied Mathematics and Physics transmission line acts as bandpass filter at low frequencies. At high frequencies, when 1 r C = the left-handed behavior occurs and the rogue events signature is restored as seen in Figure 9(d). It is clearly observed that the number of big black points was increased from three to four as seen in Figure 6(c) and Figure  9(d). This result suggests that the radiations have increased in the system and they have provoked the multiplication of rogue events when 25 Mrad s ω = . If the frequency is maintained and r C increases from 1 to 1.5 the number of black big points decreases from four to three as depicted in Figure 9(d) and Figure  10(a). This situation denotes that the radiations have decreased in the system. So, the number of Akhmediev waves trains have decreased. If 10 r C = the radiations significantly increase leading to a strong perturbed chaotic waves field depicted in Figure 10(b). Besides, this perturbed system exhibits a strange tree structure illustrated in Figure 10(c). The precedent chaotic waves has changed into a strong perturbed chaotic waves field. The new field exhibits three Akhmediev-Peregrine rogue waves corresponding to the most colored points as seen in Figure 10(c). There are also some Peregrine waves represented by the least colored points. Otherwise, some Akhmediev waves trains are also represented.

Modified Tree Structure
In addition, when we pass from right-handed behavior ( 0 r C = ) to left-handed one ( 1 r C = ) the peak power decreases from 2.5 × 10 26 W to 7 W as seen in   Figure 6(c) and Figure 9(d). This brutal decrease of pulse peak power is not only due to the increase of the distance, but also due to a strange phenomenon.
This strange phenomenon responsible to the strong decrease of peak power is called absorption. This phenomenon is normal because absorption is always observed in left-handed materials [45]. Besides, it has been demonstrated that in a realistic metamaterial system, absorption is unavoidable [45] [46] [47] [48]. In fact, some researchers such as Popov and Shalaev [45] sustain that absorption can be counteracted by optical amplification. Moreover, for left-handed behavior when r C increases from 1 to 10 the peak power also increases from 7 W to 25 W as depicted in Figure 9(d) and Figure 10  25 Mrad s ω = . Consequently, the soliton peak power significantly decreases from 6×10 23 W to 7 W as depicted in Figure 9(a) and Figure   9(b). It clearly appears that the influence of frequency on peak power strongly dependant on the value of dimensionless capacitor r C in the case of left-handed behavior.

Conclusion
In summary, we have studied an electromagnetic wave propagation when second-order dispersion and cubic-nonlinearity effects come into play. Those ef- . The numerical analysis leads to the following numerical outcomes. Firstly, we have verified that the line supports dark and bright soliton solutions. Thereafter, we also have verified that the line can exhibit chameleon's behavior. Indeed, we have shown that for right-handed behavior ( 0 r C = ), we obtain a chaotic waves field containing Akhmediev-Peregrine freak wave and Akhmediev freak waves trains, at low frequencies. However, this situation has been improved at high frequencies where a stable soliton has been obtained. Secondly, the second case of right-handed behavior ( 0.3 r C = ) has shown that the line acts as a bandpass filter at low frequencies and has maintained the robustness of the soliton at high frequencies. Thirdly, the left-handed behavior ( 1 r C = ) has restored the chaotic waves field at high frequencies and a stable soliton at low frequencies. Fourthly, we have presented the key role played by the dimensionless capacitor r C on the line. One, it has modified the aspect of rogue events signature and has induced the multiplication of rogue events. Two, it has introduced right-or left-handed behavior on the transmission line. Three, it also has modified the pulse peak power and has provoked the inversion of sense of waves propagation on the line.