1. Introduction

The matter of instability has been a serious challenge that attracted the attention of many scientists in nonlinear sciences [1–3]. So far, it has been subject of many scientific discussions, based on frequency level of waves, quality of wave guide structures and transmission with few losses [4,5]. Moreover, the wave propagation phenomenon has been studied in optical fibers with ultrashort pulses, and the consistency of the instability was also observed. A methodology was proposed to solve this matter [6], the equilibrium between the dispersion and nonlinearity of a certain level till the needed balance [2]. In this work, we suggest an adaptive control [6] of the instability with the parameters of the system and the balance control if necessary to achieve the study aim. Therefore, the generalized cubic-quintic nonparaxial chiral nonlinear Schrödinger equation is derived from the theory of Beltrami-Maxwell formalism to catch up the stability of robust waves traveling in micro-structures with fast motion, and that are capable to face nonlinear phenomena responsible of the instability of pulses in wave guide structures [7]. This approach is a framework to improve the ultrafast propagation with robust waves related to special rogue waves [8] in micro-structures. Rogue waves were found useful and benefit in other systems although their powerful destructive nature in ocean [9,10], this according to their remarkable properties of localization in both space and time [10], their robustness to nonlinear effects [11], their appearance and disappearance in short-time delay [12]. In optics, rogue wave events have been studied with a diversity of analytical and numerical methods through the standard nonlinear Schrödinger (NLS) equation. More specifically, a variety of NLS models were used to investigate on the predictability of rogue wave events whose occurrence have been observed in Bose-Einstein condensates [11,13,14], laser-plasma interactions [15], superfluids [16], in the atmosphere [17],…, etc. The enlightenment on the properties and controllability of different types of rogue waves such as the Peregrine soliton [18], Kuznetsov-Ma soliton [19] and Akhmediev breathers [20] were made possible through a nonlinear management with several nonlinear models from scalar to vectorial ones, [21,22], describing rogue waves in physical systems.

In optical communication, the miniaturization of devices in multiplexed systems can generate non-negligible nonlinear effects such as the optical activity and nonparaxial phenomena in optical systems. Hence, our investigation on robustness of rogue waves in a chiral medium, where we assume that the control of chiral level can help to define the sufficient quantity of optical activity, necessary for the good performance of devices, their better configurations, as well as the types of spontaneous waves, robust to nonlinear effects and compatible for the fast propagation in optical systems. To ensure the stability of robust wave propagating in wave guides, an adaptive control is applied to improve the ultrafast propagation of pulses, this, by choosing adequately the parameters that will insure the fast propagation of rogue waves and their robustness to nonlinear effects.

The paper is organized as follows. In Section 2 the generalized cubic-quintic nonparaxial chiral nonlinear Schrödinger equation is derived. In Section 3 the modulation instability is investigated to find the existence condition of robust waves with ideal shape useful for the ultrafast propagation. In Section 4 the rogue wave solutions are constructed and their stability during the propagation in wave guides filed with chiral material is exhibited . In Section 5 the robustness of rogue wave against nonlinear effects is exhibited through the analytical and numerical methods. Section 6 is devoted to the summary of main results and comments.

2. Derivation of the cubic-quintic nonparaxial chiral nonlinear Schrödinger equation

The concept of chirality, known as optical activity in optics, is the ability to rotate plane polarized light, either to the right or left-hand side called, respectively dextrorotatory and levorotatory [23]. This phenomenon is governed by two main effects, the optical rotatory dispersion (ORD) and circular dichroism (CD). Both have been used in the literature as optical characterization techniques of molecules [24]. The optical activity is extremely microscopic and can not occur naturally and should therefore be constructed in the form of artificial composite materials, which can then be considered at appropriate wavelengths to be an effectively chiral medium [25]. Chiral materials are different from ordinary dielectric or magnetic materials through the constitutive relations

Under the same assumption, the Beltrami-Maxwell formalism taken in the presence of current density ($\vec J=\sigma \vec E$) and charge density (${\rho _V}$) is given by [27]

Here, the main idea is to improve the ultrafast propagation of pulses in wave guides with the help of robust waves that are capable to keep their shapes and amplitude under a chiral distribution in the medium. To manage the perturbations of rogue waves in the above medium, one investigates the stability of robust waves to improve it whenever weak through an adaptive control or balance control. The adaptive control is achivable through the level of optical activity in the medium and also through a meticulous choice of parameters of the model that are perfectly sensible and determinist to a sudden change as nonparaxial parameter ($d$). To achieve this aim under a theoretical approach, one derives a new model, based on the above theory of Beltrami-Maxwell formalism. Substituting Eq. (3) into Eq. (4), one derives the wave equation as follows

Considering the optical field $\vec E$ represented by the right-(R) (positive sign (+)) and left-hand (L) (negative sign (-)) polarization in the z direction as

3. Instability and existence condition of robust waves with ideal shape favorable in ultrafast propagation

The goal to achieve is to provide a framework useful for an adaptive control of nonlinear effects in wave guide structures. As assumption, we consider a section of wave guide highly filled with chiral materials along the core. In addition, this wave guide should favor the wave propagation with few losses that can be corrected by an adaptive control and should be compatible to spontaneous wave propagation, means, waves with short-life time as rogue waves or rogons. Moreover, the wave propagating in the wave guides should be robust against perturbations. In consequence, the prototypes of rogue waves are of special interest due to their spontaneous behavior. Their disappearance and reappearance with the same shape and amplitude have been shown in the literature [10,28]. One denotes three main prototypes of spontaneous waves with finite backgrounds. The Peregrine soliton (PS) known as usual rogue wave due to its localization in both space and time [18], and unusual rogue waves, that are, the Kuznetsov-Ma soliton (KM), localized in temporal dimension with periodicity along the propagation direction [29,30] and finally, the Akhmediev Breathers (ABs), localized along the propagation direction with periodicity in temporal dimension [19]. So far, other derivative prototypes such as the chirp PS and three sisters were found in the literature [31,32]. To find the appropriate prototype of rogue waves for ultrafast propagation, one rewrites the model derived and expressed in Eq. (6) in an appropriate form. For the better understanding of the calculation and simplification of Eq. (5) into Eq. (7) and Eq. (8), see the Appendix A from Eq. (A3) to Eq. (A25) of our previous work [33] where a similar derivation was done with the same terms, except the quintic ones. So doing, we arrive at

In Eq. (8), the coefficient $d$ is the nonparaxial parameter, $P$ the GVD, $\gamma$, the TOD, $\mu$, the gain/loss term, ${C_2}$ and ${\Omega _3}$ are the third-and fifth-order nonlinearity coefficients, respectively. The coefficient $D$ is the linear birefringence, ${\eta _3}$, the differential gain/loss term, ${\alpha _3}$ and ${C_3}$ have the physical sense of SS related to the cubic and quintic nonlinearity, respectively. The parameter ${\sigma _3}$, is the walk-off coefficient. In Eq. (11), $\alpha$ is the attenuation coefficient that influences the gain/loss term ($\mu$) and differential gain/loss term (${\eta _3}$). We mentioned that the vectorial model derived and written in the scalar form for simplification as presented in Eq. (8), can be developed into two equations when considering the plus and minus signs ($\pm$) of the coefficients $\mu$, ${C_2}$, ${\Omega _3}$, $D$, ${\eta _3}$, ${\alpha _3}$, ${C_3}$ and ${\sigma _3}$ (see the expressions given in Eq. (10)). Thus, ${\psi _-}$ is the LCP component of the wave and ${\psi _+}$ represents the RCP component of the wave. The next step consists to do an investigation of the modulation instability (MI) to find the existence condition of rogue waves through the model given in Eq. (8).

To achieve that, we consider the background under the propagation of a plane wave solution in the form

Relation (13) gives the conditions to have an exact plane wave solution with constant background. The first equation allows to obtain the RCP $({k_{0 + }})$ and LCP $({k_{0 - }})$ components of the wave number ${k_0}$ as follow

It can be seen from Eq. (15), regarding the expressions of ${\mu }$ and ${\eta _3}$ in relation (10) that the attenuation parameter is vanished, leaving the total influence of the gain/loss and differential gain/loss terms on the frequency. The components of the wave number $({k_{0 + }}, {k_{0 - }})$ depend on the cubic and quintic nonlinearity, respectively of the Kerr and SS effects, then, to the second-and third-order dispersion, and also on the linear birefringence and walk-off effect. Hence, the richness of the wave number depending on nine parameters of the system compare to the frequency that depend only on two parameters. This information is capital in the choice of the variable of the dispersion relation. Therefore, the investigation of the MI on the amplitude allows to introduce a small perturbation $\chi \left ( {\xi,\tau } \right )$ such as

Here, ${A_1}$ and ${A_2}$ are the complex constant amplitudes of the perturbation. $K$ is the wave number and $\Omega$, the frequency of the perturbed background. The letter $cc$ represents the complex conjugate of the quantity it carries. Considering the perturbation of the background, the substitution of Eq. (16) into Eq. (8) gives the following system

The components of the perturbed frequency issue from the imaginary part take the form

The system given in Eq. (19) admits the plane wave solutions with the perturbed wave number $K$ and frequency $\Omega$ if the dispersion relation is

The dispersion relation has the reduced form of the quartic equation and is known as the biquadratic equation. Using the Cardan’s solution for quartic equation, Eq. (22) is transformed to a cubic equation

Using the Cardan’s solution, the cubic equation is transformed to the reduced form

As we can see, the roots ${Z_1},{Z_2},\ldots,{Z_6}$ are the solutions of the cubic equation and by the way, the intermediate solution of the dispersion relation. The dispersion relation can be transformed as follows

Therefore, the four roots of the wave number are expressed as follows

More specifically, the instability is obtained when the discriminant of the quartic equation is negative. Thus, by setting $\Omega = 0$, this implies that ${\lambda _1} = {\lambda _2} = \lambda$. Therefore, the MI condition becomes

Here, we consider as the existence domain of MI, the gain band where $G\left ( \Omega \right ) = - \operatorname {Im} \left ( K \right ) > 0$ and as the grow rate or explosive rate, the area where perturbations grow exponentially as $\exp \left ( {G\left ( \Omega \right )\;\xi } \right )$ at the expense of the pump. As the cubic nonlinearity parameter ${C_2}\,>\,0$ with $-{C_2} \,<\, 0$, in relation (8), we are working under the defocusing regime in which rogue waves exist whenever the MI is present, mostly when the baseband MI is present. We should keep in mind that the existence domain of the MI can be represented for all the four roots of the dispersion relation when considering all the six solutions of ${\phi _0}$ given by ${\phi _0} = {Z_1},{Z_2},\ldots,{Z_6}$. Here, we mainly focus on the solutions for the case, ${\phi _0} = {Z_1}$. So doing, Figure 1 reveals the MI maps $\left ( {\Omega,d} \right )$ and $\left ( {\Omega,a} \right )$ with sensible parameters of the system, that are, the nonparaxial parameter $d$ and the amplitude $a$ of the background, respectively. The same conclusion is obtained when operating with the maps $\left ( {\Omega,{\omega _0}} \right )$ and $\left ( {\Omega,{T_C}} \right )$ with other sensible parameters as the frequency of the background ${\omega _0}$, and and optical activity or chiral parameter $Tc$ that are not represented here. The presence of baseband MI is well-observed in Fig. 1(a), confirming the assumption of the presence of rogue waves that mostly appear whenever the baseband MI is present. These maps exhibit the influence of nonparaxial coefficient on the existence domain of MI. View the shapes of the existence domain of MI through the maps ($\Omega$ and $d$), ($\Omega$ and $a$), we can conclude that the assumption done on the nonparaxial parameter $d$, and the amplitude of the background $a$ are effectively sensitive parameters and can be chosen for an adaptive control of MI in the system. The same conclusion it also applied for the frequency ${\omega _0}$ and the chiral coefficient $Tc$. Thus, any perturbation susceptible to affect the stability of rogue waves during their propagation, can therefore be managed by an adjustment of the above sensitive parameters of the medium to pick up the ultrafast propagation. Now that the baseband MI that coincides with the existence condition of rogue waves can be localized on different maps, let us find the rational solutions related to rogue waves via the model derived earlier to select the ones that are more adequate for the fast propagation, based on their shape and robustness under perturbation.

 

Fig. 1. Gain maps: $\left ( {\Omega,d} \right )$ in Fig. 1(a) and $\left ( {\Omega,a} \right )$ in Fig. 1(b) with ${\phi _0} = {Z_1}$

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4. Construction of rogue wave solution of the cubic-quintic nonparaxial chiral nonlinear Schrödinger equation

Among the variety of rogue wave solutions that are known in the literature, the Peregrine soliton (PS) is of special interest according to the exceptional characteristic of its both localization in space and time, comparatively to other rogue wave prototypes that are either localized in the propagation direction with periodicity in the transverse direction (Akhmediev Breathers (ABs)) or localized in temporal dimension with periodicity along the propagation direction (Kuznetsov–Ma (KM)). This additional property has many applications in medical sciences. Hence, the importance to test the ability of PS under perturbations. Thus, let find the rogue wave solutions of the cubic-quintic nonparaxial chiral nonlinear Schrödinger equation given in Eq. (8). That equation can be rewritten as follows

This model verifies the condition of controllability of higher-order NLS equation [35,36] throughout the coefficients ${\alpha _3}$ and ${C_3}$, that are, respectively, the SS related to the cubic nonlinearity and quintic nonlinearity. It can be seen that the model is non integrable and it is of crucial interest to reduce Eq. (36) into an integrable model. Based on the well-known integrability conditions of the Hirota equation [37] and on our previous work [37], the integrability of the model is satisfied for $d = - 10\nu$, $P = 1/2$, ${C_2} = - \sigma$, ${\alpha _3} = -6\nu \sigma$, $\gamma = 6\nu \sigma$, ${r_1} = {r_2} = 1$, $\mu = D = - 3\nu$, ${\eta _3} = {\sigma _3} = - 3\nu$, ${\Omega _3} = - 5\nu \sigma /2$, ${C_3} = -3\nu \sigma$. Therefore, if we set $\nu = 0$, Eq. (36) is reduced to the standard NLS equation. Let consider ${r_1} = {r_2} = 1$, to find the integrability constraints of the model through the similarity reduction method, that is, the envelope field in the form [14]

The first step is to find the integrability conditions of the parameters related to the envelope field among which, the amplitude of the wave, $A(\xi )$, the effective propagation distance, $Z(\xi )$, the similitude variable, ${T\left ( {\xi,\tau } \right )}$, the complex field, $V\left [ {Z(\xi ), T\left ( {\xi,\tau } \right )} \right ]$, and the phase of the wave ${\rho \left ( {\xi,\tau } \right )}$. The substitution of Eq. (37) into Eq. (36) gives a system of two partial differential equations with constant coefficients

This integrable model can generate a rogue wave solution if $\sigma = 1$. Thus, for $V\left [{Z(\xi ),T(\xi,\tau )} \right ]$ satisfying the relation (41), the similarity reduction method gives the above differential equations

For ${C_2} = - 1$, and ${\alpha _3} = - 2$, a particular solution of the phase of the envelope field is given by $\rho \left ( {\xi,\tau } \right ) = - \tau + {\rho _0}\left ( \xi \right )$, where ${\rho _0}(\xi )$ should be defined as well as ${T_0}\left ( \xi \right )$. The substitution of known relations into Eq. (43) gives

For $P = 1/2$, the particular solution of the arbitrary function ${{T_0}(\xi )}$ takes the form ${T_0}\left ( \xi \right ) = - 4\sqrt 2 \xi$. The substitution of variables and parameters into Eq. (43) shows that the first derivative of the phase ${\rho _{0\xi }}$ is a constant. This imply that the arbitrary function of the phase will take the form ${\rho _0}\left ( \xi \right ) = C\xi$, where $C$ is the unknown constant that is deduced from Eq. (51) as follow

For $\mu = {\eta _3} = -1$, $D = {\sigma _3} = -1$ and $d = \frac {{ - 5{\alpha _3}}}{{3{C_2}}} = - \frac {{10}}{3}$, the simplified form of ${\rho _0}\left ( \xi \right )$ becomes ${\rho _0}\left ( \xi \right ) = \frac {1}{2}\xi$. From Eq. (47), it can be seen that the SS coefficients related to the cubic nonlinearity, ${\alpha _3} = - \gamma$. Then, from Eq. (48), the SS coefficients related to the quintic nonlinearity is ${C_3} = \frac {{ - \alpha _3^2}}{{\gamma T_1^2(\xi )}}$. From Eq. (53), the quintic nonlinearity coefficient is given by

In view of great success of the Peregrine soliton in the modeling of some realistic problems, one investigates on the construction of rogue wave solutions. Based on the methodology applied on the infinite hierarchy of NLS equation which has a well-known rogue wave solutions in the focusing regime $(\sigma = 1)$, the finding of PS with the Darboux-dressing method is on special interest. Therefore, the complex field $V\left [ {Z(\xi ),T(\xi,\tau )} \right ]$ that is valid for $\sigma = 1$ in rogue wave finding is written as [38].

The envelope field found is the rational solution related to rogue waves under its general form given by

This solution given in relation (66) is a particular solution whereas the solution given in relation (65) is more general to describe the fast propagation of ultrashort pulses in nonparaxial chiral media. Let reminds that the PS (66) found here is a compact form of a vectorial solution with two components, the LCP component represented by ${\psi _-}$ and the RCP component represented by ${\psi _+}$. More specifically, the vectorial status of this solution can help to examine the dynamical behavior of the LCP and RCP components of the waves under different aspects. At first glance, one can investigate the self-character of each component under a diversity of possibilities of focusing and defocusing interactions of the SPM nonlinearity. Afterwards, this solution offers a possibility to handle the control of nonparaxial effect responsible of the fast propagation, the linear goup velocity, the GVD, the TOD, the attenuation, the SPM, the SS and the linear and differential gain/loss parameters. Finally, this solution can be used to manage the characteristic properties of the device for it to shelter the transmission of categories of pulses. Therefore, one can assume that this vectorial PS solution has a multitask function and can provide a more convenient and controlled environment for further applications in optics and medical diagnostic analysis. Now, let examines the robustness of PS against perturbations.

5. Robust nature of rogue waves against perturbations

To illustrate the robustness of rogue waves against perturbations, one focuses the attention on the usual rogue wave, that is, the Peregrin Soliton (PS) constructed in section 4 and expressed in (65). To achieve this aim, two sensitive parameters of the system, that are, the frequency (${\omega _0}$) and the amplitude of the background (a) are of special interest. The variation of these parameters in the particular solution (66) exhibits through the analytical simulations presented in Figs. 2, 3 and 4, different single PS whose the forms and the shapes remain unchanged with a slightly change in size when the amplitude of the background increases as observed in Figs. 2, 3 and 4. It can be seen that the structures are identical one another, obeying to the properties of PS, that are based at the center of coordinates with one peak surrounding by two holes. For a range of frequency ${\omega _0}$ starting from $0.05$ to $10$, the same profiles are observed with a weak difference of the amplitude of the PS. Hence the evidence of stability of PS to these sensitive parameters. However, the perturbation of the amplitude of the background does not affect the structure of the PS even for hight values of the frequency as observed in Figs. 3 and 4. Hence, the robustness of the PS to nonlinear perturbations. A similar behavior is observed through a numerical simulation where profiles are represented in Figs. 5 and 6. These solutions are obtained with a pseudo-spectral method namely, difference-differential equation method [39]. The discrete Fourier transform is used to evaluate the spatial derivative of the model. Hence, ${\psi _{\xi \xi }}$ and ${\psi _\xi }$ are expressed into finite difference formulae with errors of second-order. The substitution of these derivatives into Eq. (8) allows to generate the difference-differential equation where we set $\psi =u$ to arrive at

 

Fig. 2. Representation of PS in the focusing regime for ${C_2}=-1$, ${\sigma }=1$, ${\omega }=10$ and for $a=1$, and $a=0.5$,

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Fig. 3. Rogue waves representation for ${C_2}=-1$, ${\omega _0}=10$, $a=1$, ${\sigma }=1$, and for ${A_0} = a{\left ( {\frac {{ - 3T_1^2(\xi )\gamma }}{{{\alpha _3}}} + 0.5} \right )^{1/2}}$ and ${A_0} = a{\left ( {\frac {{ - 3T_1^2(\xi )\gamma }}{{{\alpha _3}}} + 3} \right )^{1/2}}$.

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Fig. 4. Rogue waves representation for ${C_2}=-1$, ${\omega _0}=10$, ${A_0} = a{\left ( {\frac {{ - 3T_1^2(\xi )\gamma }}{{{\alpha _3}}} + 3} \right )^{1/2}}$, ${\sigma }=1$, and for $a=2$ and $a=3$.

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Fig. 5. Representation of rogue wave components (LCP, RCP) where ${\omega _0}=0.005$ with ${k_0} = \sigma '{a^2} + 1/2{\omega _0}$, $a=0.5$ and ${T_c}=0.8$.

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Fig. 6. 2D representation of rogue wave components (LCP, RCP) where ${\omega _0}=0.005$, with ${k_0} = - 1/2d$, $a=0.5$ and ${T_c}=0.1$.

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The explicit algorithm in the discretized domain is defined in Eq. (67) with the conditions of implementation of the index $n$ in Eq. (68). The transverse differential operators ${\partial ^2}/\partial {\tau ^2}$, ${\partial ^3}/\partial {\tau ^3}$ and $\partial /\partial \tau$ are computed through the fast Fourier transforms (FFTs) method. This numerical approach is in accordance with the flexibility and rapid convergence, in the modeling of nonparaxial NLS models [39]. Under this approach, the PS is used as initial condition to simulate the dynamical behavior of ultrafast pulses related to robust waves issue from the model, judged compatible to investigate the auto-amplification of pulses in a chiral core, on the one hand, and to improve the fast propagation of robust waves with negligeable losses, on the other hand. So doing, rogue wave profiles are illustrated in Figs. 5 and 6 for the following expressions of the initial wave number ${k_0} = - 1/2d$ and ${k_0} = \sigma '{a^2} + 1/2{\omega _0}$ with the seeding solution bellow, taken as initial condition [38]

The parameters $K'$, $K''$, $K'''$, ${\mu _0}$, ${\varepsilon _2}$, ${\varepsilon _4}$, $\nu$, and $\sigma$ are chosen randomly for the computation. The coordinates ($\xi$, $\tau$) are bounded in the interval $\left [ { - 5,5} \right ]$. The spectral parameter $k$ is defined for a short length ($L= 10$) with $N=32$ iterations. The implementation of $n$ in Eq. (67) is done for 102 iterations in the propagation direction $\xi$ with $\Delta \xi = 0.05$. The initial condition used (Eq. (69)) introduces a single pulse in the waveguide filled with chiral material that generates an optical activity in the medium. In the literature, it is well-known that fibers with chiral core allow the propagation of two modes that are, the LCP (levorotatory: ${\psi _ - }$) and RCP (dextrorotatory: ${\psi _ + }$ ) components of the waves. Thus, the rotatory property of the chiral wave guide involves the degeneration of lobes of butterfly structures of the PS, exhibited in Fig. 5 for high value of chiral parameter (${T_c}=0.8$). This degeneration can involve a collision as the one observe in Fig. 5(a). Moreover, it has been noticed that a single wave introduce in a fiber can also split itself into two or four components of waves, depending on the number of wave field (i.e. ${\psi _1},\;{\psi _2}$ or ${\psi _ + },\;{\psi _ - }$ ) taken into account in the NLS model. These behaviors are known as phenomena of two-wave mixing observed in physical system described with scalar NLS model, and four wave mixing observed with vectorial NLS model. The model derived in Eq. (8) is in fact, a vectorial model with two components (LCP and RCP) but was written in a scalar form for simplification reasons. We noticed that the robustness of rogue waves increases with the increase of chiral parameter. Moreover, the increase of the optical activity favors the smoothness of the wave shape and involves the good stability of the robust waves during the ultrafast propagation. One denotes that the decrease of chiral parameter (${T_c}=0.1$), favors the degeneration of dark PS and the occurrence of usual PS at the center of coordinates as depicted in Fig. 6. This phenomenon is due to a transfer of energy from the vicinity to the center of coordinates. Therefore, one can conclude that, a good manipulation of chirality in chiral materials or biological systems can help to direct the energy in a particular area of physical systems. This theoretical framework can be used for further experimental investigation in medical science. More specifically, in the processes of transcription and replication of DNA, where the PS is used as inhibitory factor for local and short excitation of DNA strands.

6. Conclusion

The dynamics of ultrafast pulses related to robust waves is characterized by the cubic-quintic nonparaxial chiral NLS equation. This model is derived in this work to ensure the total reflexion of ultrafast pulses in optical systems due to the induced optical activity and the artificial one that adjust or correct the polarization in the system. This model improves the dynamics of ultrafast propagation of rogue waves in chiral wave guides, in the sense that it verifies the controllability of the system that exhibits the direct implications of nonlinearities, nonparaxiality, optical activity and more specifically, the robustness of rogue waves. Through the baseband MI theory, the range of frequencies susceptible to generate a rogue wave phenomenon can be determined. Through the integrability conditions of the model and symmetry reduction method, the solution of PS was constructed. The richness and the multitask function of the PS has been elucidated. It has been noticed that a such solution can offer a more convenient and controlled environment for further applications in optics and medical diagnostic analysis. The variation of chiral parameter, frequency and amplitude of the seeding solution has been used to enlighten the robustness of PS against perturbation. The prospective results are convincing through the numerical and analytical representations, showing the improvement of ultrafast propagation through the adjustment of optical activity, the frequency and the amplitude of the background. Therefore, a good management of the chirality in chiral materials or biological systems can help to direct the energy in a particular area of physical system. This theoretical framework can be used for experimental investigation in medical science. More specifically, in the processes of transcription and replication where the PS is used as inhibitory factor for local and short excitation of DNA strands. In addition, this work provides a novel approach to improve the ultrafast propagation and good transmission of PS in biological systems, as well as further consideration of total control of light in life-science and industry.