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From Solitons to Rogue Waves in Nonlinear Left-Handed Metamaterials

In the present work, we explore soliton and rogue-like wave solutions in the transmission line analogue of a nonlinear left-handed metamaterial. The nonlinearity is expressed through a voltagedependent and symmetric capacitance motivated by the recently developed ferroelectric barium strontium titanate (BST) thin film capacitor designs. We develop both the corresponding nonlinear dynamical lattice, as well as its reduction via a multiple scales expansion to a nonlinear Schr\"odinger (NLS) model for the envelope of a given carrier wave. The reduced model can feature either a focusing or a defocusing nonlinearity depending on the frequency (wavenumber) of the carrier. We then consider the robustness of different types of solitary waves of the reduced model within the original nonlinear left-handed medium. We find that both bright and dark solitons persist in a suitable parametric regime, where the reduction to the NLS is valid. Additionally, for suitable initial conditions, we observe a rogue wave type of behavior, that differs significantly from the classic Peregrine rogue wave evolution, including most notably the breakup of a single Peregrine-like pattern into solutions with multiple wave peaks. Finally, we touch upon the behavior of generalized members of the family of the Peregrine solitons, namely Akhmediev breathers and Kuznetsov-Ma solitons, and explore how these evolve in the left-handed transmission line.


I. INTRODUCTION
Over the past few years, the study of metamaterials, i.e., artificially engineered structures exhibiting electromagnetic (EM) properties not commonly observed in nature, has seen an explosion of interest [1][2][3][4]. An especially intriguing aspect of these metamaterials is their so-called left-handed (LH) nature, which features simultaneously negative effective permittivity and permeability µ, i.e., the relevant signs of these quantities are opposite to those of conventional right-handed (RH) media. The resulting difference between these two scenarios is that in the LH (RH) regime, the energy and the wave fronts of the EM waves propagate in opposite (same) directions, giving rise to backward-(forward-) propagating waves. Consequently, these left handed metamaterials (LHM) can exhibit negative refraction at microwave [5,6] or optical frequencies [7].
Apart from a classical EM approach involving the study of an effective medium, which can naturally be used to study such metamaterial media [8], transmission line (TL) theory constitutes a convenient framework to analyze their evolutionary dynamics. A TL-based analysis relies on the connection of the EM properties of the medium ( and µ) with the electric elements of the TL unit cell, namely, the serial and shunt impedance [2]. Equivalent TL models have been used to describe periodic lattices of prototypical magnetic and electric metamaterial structures in the form of split ring resonators (SRRs) and complementary split ring resonators (CSRRs) [9,10]. In this context, each of the SRRs/CSRRs can be analyzed in the form of a corresponding LC circuit, while the whole metamaterial is an array of such circuits, with the coupling between the elements being modeled by a mutual inductance / capacitance. The serial and shunt impedance are directly related to the actual properties of these structures.
In addition to the more standard case of linear LHMs, the study of nonlinear LHMs has been receiving increased attention [11]. Here, the EM properties -such as and µ (or, equivalently, the serial and shunt impedance at the TL level) -depend on the intensity of the EM field (equivalently at the TL level, e.g., on the voltage). Practical proposals for the experimental realization of such features involve embedding an array of wires and SRRs into a nonlinear dielectric [12,13], or the insertion of diodes (varactors) into resonant conductive elements, such as the SRRs [14][15][16]. The interplay of strong dispersion exhibited by left handed transmission lines with the nonlinear voltage dependence of the group velocity results in unusual dynamical behavior. In this theme, the extensive theoretical studies have led to numerous experimental realizations of features such as pulse propagation [17], envelope soliton formation [18] and the emergence of bright [19] or dark [20] solitons; see also Ref. [21], and the more recent work [22] for soliton generation in active metamaterials. A relevant -but earlier -review of experimental studies can be found in Ref. [23].
In our present considerations, we study a nonlinear left handed transmission line. In part, we are motivated specifically by the recent development of strongly nonlinear and voltage symmetric barium strontium titanate (BST) thin film capacitors [24]. We thus consider a nonlinear LHM through a TL approach, which exhibits the symmetric capacitance-voltage dependence. Our aim is to investigate the properties of the nonlinear waveforms that arise and are robustly sustained by this LHM. To gain theoretical insight into this, we utilize a multiscale expansion method that reduces the model, in a self-consistent fashion (up to cubic order in a suitable amplitude parameter), to a nonlinear Schrödinger (NLS) equation [25][26][27]. We identify regimes, depending on the frequency of the carrier wave, where the NLS equation is focusing or defocusing. The prototypical soliton solutions of this model, namely the bright and the dark soliton, are found to be robustly preserved by the transmission line dynamics. However, a more ambitious goal of the present study is to examine whether rogue wave (RW) patterns, such as the Peregrine soliton (PS) of the focusing NLS equation, can emerge in the LH transmission line. The only work that we are aware of connecting these two themes (LHMs and RWs) is that of [28], which focuses on a rather qualitative comparison for very short propagation distances. Here, we actually engineer initial data that, at the NLS level, would lead to a PS profile. As a result of the dynamics, we observe both similarities with and differences from what we expect at the NLS level. We discuss these at some length and the impact that the intrinsic features of the LHM system have on the potential emergence and form of the Peregrine-like structure. We do not restrict our considerations to solely this rogue wave pattern; rather, we extend them to additional members of the relevant family of solutions, including the spatially periodic Akhmediev breathers (ABs) and the temporally periodic Kuznetsov-Ma solitons (KMs).
Our presentation is structured as follows. In section II we present the model, discuss the nature of the nonlinearity and explain the reduction to the NLS setting. In section III, we present prototypical numerical results not only for the bright and dark solitons but also, more importantly, for Peregrine-like solitons/rogue waves and related (periodic in space or time) patterns. Finally, in section IV, we summarize our findings and present our conclusions.

II. THE MODEL AND ITS ANALYTICAL CONSIDERATION
2 capacitance-voltage dependence. Our aim is to investigate the properties of the nonlinear waveforms that arise and are robustly sustained by this LHM. To gain theoretical insight into this, we utilize a multiscale expansion method that reduces the model, in a self-consistent fashion (up to cubic order in a suitable amplitude parameter), to a nonlinear Schrödinger (NLS) equation [25][26][27]. We identify regimes, depending on the frequency of the carrier wave, where the NLS equation is focusing or defocusing. The prototypical soliton solutions of this model, namely the bright and the dark soliton, are found to be robustly preserved by the transmission line dynamics. However, a more ambitious goal of the present study is to examine whether rogue wave (RW) patterns, such as the Peregrine soliton (PS) of the focusing NLS equation, can emerge in the LH transmission line. The only work that we are aware of connecting these two themes (LHMs and RWs) is that of [28], which focuses on a rather qualitative comparison for very short propagation distances. Here, we actually engineer initial data that, at the NLS level, would lead to a PS profile. As a result of the dynamics, we observe both similarities with and differences from what we expect at the NLS level. We discuss these at some length and the impact that the intrinsic features of the LHM system have on the potential emergence and form of the Peregrine-like structure. We do not restrict our considerations to solely this rogue wave pattern; rather, we extend them to additional members of the relevant family of solutions, including the spatially periodic Akhmediev breathers (ABs) and the temporally periodic Kuznetsov-Ma solitons (KMs).
Our presentation is structured as follows. In section II we present the model, discuss the nature of the nonlinearity and explain the reduction to the NLS setting. In section III, we present prototypical numerical results not only for the bright and dark solitons but also, more importantly, for Peregrine-like solitons/rogue waves and related (periodic in space or time) patterns. Finally, in section IV, we summarize our findings and present our conclusions. Following Ref. [21], we consider the transmission line framework of Fig. 1 in order to model a left handed TL metamaterial. Assuming that the diode can be emulated by a nonlinear, voltage-dependent capacitance (see below), we can employ Kirchhoff's voltage and current laws for the unit-cell circuit of Fig. 1, and derive (see details in Ref. [21]) the following differential-difference equation for the unknown voltage V n in the parallel branch at the n-th site of the lattice:

II. THE MODEL AND ITS ANALYTICAL CONSIDERATION
Here, U n is the voltage across the nonlinear capacitance C L that emulates the in-line BST capacitor (notice that U n = V n−1 − V n ), C R is the linear shunt capacitance, while L L represents the inductive elements, connected to the ground; notice that subscripts L and R denote the LH and RH elements of the unit cell circuit. A key feature considered herein is the symmetric (contrary to what was the case in Ref. [21]) dependence of the capacitive element on the voltage V of the form: motivated by results in strongly nonlinear voltage symmetric capacitance for molecular beam epitaxy grown BST thin films [24]. For the voltages, we will assume for simplicity a Taylor expansion and an accurate to O(U 4 ) functional form provided in the second expression of Eq. (2). Here C 0 is the zero bias capacitance and V 0 is the "2:1" voltage, i.e., C(V 0 ) = C 0 /2. We assume the the fringing capacitance of [24] is vanishing herein. Following Ref. [21], we consider the transmission line framework of Fig. 1 in order to model a left handed TL metamaterial. Assuming that the diode can be emulated by a nonlinear, voltage-dependent capacitance (see below), we can employ Kirchhoff's voltage and current laws for the unit-cell circuit of Fig. 1, and derive (see details in Ref. [21]) the following differential-difference equation for the unknown voltage V n in the parallel branch at the n-th site of the lattice: Here, U n is the voltage across the nonlinear capacitance C L that emulates the in-line BST capacitor (notice that U n = V n−1 − V n ), C R is the linear shunt capacitance, while L L represents the inductive elements, connected to the ground; notice that subscripts L and R denote the LH and RH elements of the unit cell circuit. A key feature considered herein is the symmetric (contrary to what was the case in Ref. [21]) dependence of the capacitive element on the voltage V of the form: motivated by results in strongly nonlinear voltage symmetric capacitance for molecular beam epitaxy grown BST thin films [24]. For the voltages, we will assume for simplicity a Taylor expansion and an accurate to O(U 4 ) functional form provided in the second expression of Eq. (2). Here C 0 is the zero bias capacitance and V 0 is the "2:1" voltage, i.e., C(V 0 ) = C 0 /2. We assume the the fringing capacitance of [24] is vanishing herein.
As a result of this expansion, Eq. (1) becomes: where a = (16/9)(C 0 /V 2 0 ). Next, measuring time in units of 1/ω 0 = √ L L C 0 and voltage in units of 3V 0 /4, we express Eq. (3) in the following dimensionless form: To obtain an analytical handle on the nonlinear waveforms that the model of Eq. (4) may possess, we will employ a quasi-continuum approximation [29]. In particular, we consider waveforms characterized by a discrete carrier and a slowly-varying continuum pulse-like envelope, and thus seek solutions of Eq. (4) of the form: where V are unknown envelope functions, depending on the slow variables: with v g being the group velocity, as can be found self-consistently from the linear dispersion relation (see below). Finally, ω and k denote the carrier's frequency and wavenumber, respectively, and is a formal small parameter.
Here we should notice that the above ansatz implies that we are assuming a carrier wave of effective linear propagation, as will be more transparent in what follows. This carrier wave is modulated by a slow envelope that encompasses the nonlinear dynamics of the model. This slow envelope is expected to be governed, as we will see in what follows, by the NLS model. This expansion is a small-amplitude one (i.e., weakly nonlinear), as the relevant control parameter characterizes the solution amplitude. At the same time, it is a long-wavelength expansion characterizing wide regions of the lattice of size of 1/ and long time scales of the size of 1/ 2 .
We now present the resulting equations from the multiscale expansion order by order.
The first one of these, at O( ), represents the linear dispersion relation of the LHM, which is depicted in the left panel of Fig. 2. Notice that the dispersion relation (7) suggests that there exist two cutoff angular frequencies, namely an upper one, ω max = 1/ √ g ≈ 4.22 (corresponding to k = 0), and a lower one, ω min = 1/ √ g + 4 ≈ 0.5 (corresponding to k = π), for g = 0.056; notice that the lower cutoff frequency is due to discreteness since, evidently, this frequency vanishes in the continuum limit. At the next order, the solvability condition yields the group velocity (the velocity of wavepackets) v g = dω/dk, which is not only distinct from the phase velocity v p = ω/k, but also carries opposite sign, as per the left-handed nature of the medium; this becomes clear by the form of the dispersion relation shown in the left panel of Fig. 2, which features a negative slope. Note that, at the second order, the solvability condition leads to a vanishing contribution V 2 = 0, as is commonly the case in such multiscale expansions.
At the third order, we obtain the NLS equation for V 1 . Its dispersion and nonlinearity coefficients, P and Q respectively, depend on the frequency, but the latter is slaved to the wavenumber through the dispersion relation. Last, but not least, the third-order reduction/decomposition of the solution is also derived.
We now consider some prototypical values of the relevant parameters motivated also in part from the experiments of [21]. For instance, for g = 0.056, we present P Q as function of ω in the right panel of Fig. 2. In the region where the relevant quantity is positive, per the standard general theory of the NLS equation [25][26][27], the dynamics is associated with a self-focusing scenario that should bear structures like bright solitons, but also potentially Peregrine solitons and related waveforms. On the other hand, when P Q < 0, then we are in a self-defocusing regime where, e.g., dark solitons may arise. The analytical availability of these waveforms at the NLS level, as well as the explicit form of the transformation allows us to express these potential solutions in the LHM dynamics. As an aside, we note that for radio-frequency, thin film varactors as in the case of [24], the value of g may be considerably different. However, we have verified in that case too the existence of self-focusing and self-defocusing frequency ranges and the persistence of the solitary wave structures explored below. Hence, we now turn to direct numerical computations to examine the robustness of such states (and the features thereof) in LHMs.

III. NUMERICAL COMPUTATIONS
In the present section, we will explore different waveforms that arise at the level of the NLS model within the realm of the LHM. To do so, we numerically integrate Eq. (4) using a 4th-order Runge-Kutta method and periodic boundary conditions.

A. Bright soliton
It is evident from Fig. 2 that there exists a wide parametric interval of frequencies, for which the effective nonlinearity of the NLS model is self-focusing. In this case, the prototypical structure that it is relevant to explore is the bright soliton. At the level of the leading-order for the voltage, the relevant waveform introduced on the basis of the NLS reduction has the form: where u 0 and c are free O(1) parameters setting the amplitude/inverse width and wavenumber of the soliton, respectively. Utilizing the above expression, and reconstructing the initial condition (of the modulated amplitude wave, within the multiscale expansion) based on Eqs. (5)-(9), we can initialize the nonlinear dynamical lattice of Eq. (4) and observe the resulting evolution presented in Fig. 3 [55]. The dynamics clearly illustrates that for different solutions of varying wavenumbers (and frequencies), as well as amplitudes even up to order O(1), we observe the extremely robust propagation of a bright soliton through the LHM. As expected, the multiscale approximation is more accurate for smaller amplitudes, as observed in the left panel of Fig. 3. On the right panel, for larger amplitude close to O(1), the resulting bright soliton wavepacket tends to have smaller group velocity than the theoretical approximation. In this case, a larger fraction of the energy is lost to dispersive wavepacket radiation. Figure 4 cements the relevant result by illustrating the evolution of the amplitude of the bright soliton (i.e., the maximal absolute value of the voltage) over time. We can see that, although the voltage is modulated by the LH lattice, it is sufficiently robust to be preserved under the long-time evolution. Hence, the bright soliton is an entity able to propagate undistorted over long distances in such transmission line metamaterials.

B. Dark soliton
While in Fig. 2, it can be observed that the interval of frequencies considered is dominated by effective self-focusing dynamics, nevertheless, the quantity P Q can change sign. Hence, it is natural to explore the potential for the formation of dark soliton states, voltage dips on top of a carrier wave (voltage) background. The relevant functional form of the dark soliton solution of Eq. (9) for P Q < 0 reads: where u 0 and K is the background amplitude and wavenumber of the carrier, while B and A set the amplitude ("darkness") and velocity of the soliton respectively (note that A 2 + B 2 = 1). We have once again used Eq. (11) and the long-wavelength multiscale expansion machinery of Eqs. (5)-(9) to construct a suitable initial condition for the dynamical lattice of Eq. (4). The result in the space-time contour plot evolution of the voltage is shown in Fig. 5. In this case too, although the entire background is excited, we can observe that the voltage dip propagates essentially undistorted over a long propagation distance, following the prescribed (through the analysis) group velocity. We now turn to a type of state that has not been explored in this context, to the best of our knowledge, in a systematic, quantitative way in any previous study, namely the Peregrine soliton.
In our problem, given the NLS reduction, we can utilize the Peregrine soliton solution of the focusing NLS model in the form: (here, as before, u 0 is the amplitude of the background carrier wave) to reconstruct the initial condition of a waveform to be introduced in Eq. (4) via Eqs. (5)- (9). We initialize the relevant waveform at a time well before the formation of its maximum and observe its full evolution. We do this both for a smaller amplitude case, where the reduction should be more representative of the true NLS dynamics, as well as for a larger amplitude one. Figure (6) shows a Peregrine soliton example with a small amplitude; = 0.1 is used here. The dynamical evolution illustrates that the number of peaks progressively increases; i.e., while there is the emergence of the fundamental peak associated presumably with the Peregrine soliton, for longer times an evolution somewhat reminiscent of modulational instability and the formation of a more complex pattern consisting of multiple breathing solitary wave entities appears to emerge. It is worthwhile to mention (also in connection with the results that will follow) that the growth towards the formation of the Peregrine soliton is not monotonic (as is expected by the exact solution). Rather, there is a slight interval of amplitude decay before the growth, ultimately leading to the emergence of the extreme event. Figure 7 corresponds to a case of substantially larger initial voltage, where we expect the small amplitude reduction to no longer be valid. This case also illustrates a number of similarities and differences with respect to the original NLS model. In the NLS, a monotonic growth of the "bulge" develops leading to the peak of the Peregrine soliton (which thus seems to "appear out of nowhere and disappear without a trace" [51]). Here, in our lattice, the growth still occurs, yet it involves a decay stage before the growth stage leading to the peak. After the formation of the peak, a somewhat unconventional sequence arises in the time evolution of the maximum. While, that is, we expect decay anew, this decay occurs only briefly, with another growth stage and a sharp (in fact, even sharper than the previous one) peak emerging. The top right panel of Fig. 7 illustrating the space time evolution until t = 3000 sheds light on this feature. In particular, what happens is that the original "wider" waveform splits into two narrower peaks, which evolve rather independently. At the level of the amplitude, further evolution leads to decay and then once again to growth (the latter time developing even higher voltage amplitudes). Once again, the contour plots of the bottom panels for considerably larger times reveal the explanation: in a similar way as the single wave eventually grows and splits into two, the two subsequently proceed to split forming an additional one. This way, the number of wave structures appears to be increasing over time. While this is not consonant with the exact solution of the Peregrine soliton in NLS, we should note that it is reminiscent of an evolution leading to a progressive increase in the number of peaks in the recent work [52]. Furthermore, although it is far more ordered, it carries some of the breathing characteristics of the smaller amplitude case in Fig. 6.
Thus, summarizing our findings, there exist definite similarities between the NLS reduction and the dynamics of the LHM, including the formation of extreme wave patterns. Nevertheless, there are also notable differences, such as the non-monotonic growth, or the breakup of the latter initial profile into multiple waves, which -especially at large amplitudes -seems to be more complex than what may be expected on the basis of the NLS reduction.

D. Akhmediev breathers and Kuznetsov-Ma solitons
As is well-known [31][32][33], the Peregrine soliton can be viewed as a low wavenumber or a low frequency limit of a generalized family of solutions including on the one hand the Akhmediev breathers and on the other Kuznetsov-Ma solitons, respectively. Both these structures are solutions of the focusing NLS equation and can be written in a single form as: where b = 8a(1 − 2a), and K = 2 √ 1 − 2a. For 0 < a < 0.5 the solution is referred to as an Akhmediev breather, with period 2π/K in X. For a > 0.5, K and b become imaginary, thus the solution is periodic with period π/(b|P |) in T and Eq. (13) represents a Kuznetsov-Ma soliton. In the limit of a → 0.5, these periods (spatial and temporal, respectively) approach to ∞ and Eq. (13) has as a limiting case the Peregrine soliton solution of NLS. Given the NLS reduction, we can utilize Eq. (13) with different values of a to reconstruct these types of initial condition of Eq. (4) via Eqs. (5)- (9).
Figure (8) shows the dynamics of the LHM with initial data taken from Eq. (13) for a = 0.1131, in the regime where the solution is anticipated to evolve into an Akhmediev breather. As in the Peregrine examples, our initialization time is before the formation of the maximum amplitude of the Akhmediev breather. We observe for the amplitude that, instead of growing to form the relevant pattern and subsequently decaying to a constant background forever as is prescribed by NLS, it oscillates until the humps become irregular. Such a manifestation is, once again, somewhat reminiscent of modulational instability, and the subsequent formation of more highly localized waveforms. As a complementary simulation, we also used initial data of the LHM with T = 0 of Eq. (13), i.e. at the maximum amplitude of the Akhmediev breather. Then we observe that, besides the group velocity being slightly slower than the theoretical prediction, the amplitude for each individual hump actually oscillates simultaneously for a while until around t = 1400 with a much shorter period in Fig. 9 than in Fig. 8. Eventually, however, in this case too the oscillatory pattern destabilizes and leads to an irregular profile of the energy distribution over the lattice, which also features occasional sharper localization phenomena. Figure 10 shows the dynamical evolution of the LHM with initial data from Eq. (13) at a = 0.9, in the regime of the Kuznetsov-Ma soliton. Since the Kuznetsov-Ma soliton is time periodic, this time we start from T = 0, i.e. at the maximum amplitude of the Kuznetsov-Ma soliton. We observe that the single hump splits into two humps at a very early stage, one with group velocity considerably smaller than the theoretical prediction while the other one with a group velocity larger than the one suggested by the NLS reduction. This procedure keeps cascading for the duration of our numerical integration, in a way once again reminiscent of the pattern formation via modulational instability. In fact, even in the case of the Peregrine soliton, an initialization at the maximum amplitude leads to the observation of similar dynamics with a splitting at an early stage. Besides that, as in the right panel of Fig. 10, we observe an oscillating amplitude, which is similar initially to the Kuznetsov-Ma soliton, until the interactions with the apparently unstable background disrupt its nearly periodic evolution.

IV. CONCLUSIONS AND FUTURE CHALLENGES
In the present work, we have revisited the study of left-handed transmission line metamaterials, motivated by the consideration of strong voltage symmetric nonlinearities demonstrated for epitaxially fabricated BST capacitors. Upon introducing the relevant theoretical model, we have argued that its dispersive character renders it suitable for a carrier-envelope decomposition and an associated multiple scales reduction. This approach, as is customary in such models, leads to a nonlinear Schrödinger (NLS) equation which is a host to a diverse array of coherent waveform structures.
We illustrated that focusing, as well as defocusing nonlinearities can be engineered on the basis of varying the frequency (or wavenumber) of the carrier wave. In the case of effectively self-defocusing nonlinearities, we observed the robust propagation of dark solitons in the system. In a similar way, for focusing nonlinearities, bright solitons were found to be generically robust. What was most interesting, however, was the possibility for producing extreme waveform events, in the form of rogue waves (Peregrine solitons, but also Akhmediev breathers and Kuznetsov-Ma solitons) for such left-handed media. We observed that such events do arise through suitable initial conditions, motivated by the NLS reduction. In particular, these extreme waveforms demonstrated both similarities and differences from the standard Peregrine soliton case, the differences being the non-monotonic growth, as well as the subsequent (to the formation of the peak) emergence of multiple peaks signaling, arguably, the modulational instability of the background. Similarly to the case of the Peregrine soliton, the Akhmediev breather and the Kuznetsov-Ma soliton preserved some of their characteristics such as the approximate spatial or temporal (respectively) periodicity, but at the same time, they also manifested nontrivial perturbations in both space and time, due to the modulational features of their corresponding background.
The results of the present study stimulate numerous further explorations within this general area of the interplay of nonlinearity and left-handed media, especially around the subject of extreme events and rogue waves. At the one dimensional level, it may be well worthwhile to examine more general lattices, potentially also involving right handed parasitic elements (as in Ref. [21]), or more broadly composite left-handed and right-handed element chains as, e.g., in Ref. [53]. A natural question is to what degree Peregrine type patterns may persist in such settings. Another major direction for future investigations is that of exploring the role of dimensionality. In particular, recent studies have explored even experimentally the role of geometry (e.g., square vs. triangular, etc.) in transmission line implementations of two-dimensional lattices [54]. It would be especially relevant to consider left handed such media and particularly the possibility of inducing 1D (or even more intriguingly 2D) extreme events in the latter. Such studies are currently in progress and will be reported in future publications.