Dispersion Managed Generation of Peregrine Solitons and Kuznetsov-Ma Breather in an Optical Fiber
Dipti Kanika Mahato, A. Govindarajan, M. Lakshmanan, Amarendra K. Sarma
DDispersion Managed Generation of Peregrine Solitons and Kuznetsov-MaBreather in an Optical Fiber
Dipti Kanika Mahato a , A. Govindarajan b , M. Lakshmanan b , Amarendra K. Sarma a, ∗ a Department of Physics, Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India. b Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli - 620 024, India
Abstract
Optical rogue waves and its variants have been studied quite extensively in the context of op-tical fiber in recent years. It has been realized that dispersion management in optical fiber isexperimentally much more feasible compared to its nonlinear counterpart. In this work, we reportKuznetsov-Ma (KM)-like breathers from the first three orders of rational solutions of the nonlinearSchr¨odinger equation with periodic modulation of the dispersion coefficient along the fiber axis.The breather dynamics are then controlled by proper choice of modulating parameters. Addi-tionally, the evolution of new one-peak and two-peak breather-like solutions has been displayedcorresponding to the second-order rational solution. Direct numerical simulations based on mod-ulational instability has also been executed which agree well with the analytical results, therebymaking the proposed system more feasible for experimental realization.
Keywords:
Peregrine solitons, periodic dispersion management, rogue waves, similaritytransformation, pseudo-spectral method
1. Introduction
Since the publication of the remarkable study on optical rogue waves in 2007 [1] and laterthe experimental demonstration of Peregine solitons in nonlinear optical fiber [2], the study ofoptical rogue waves exploded in various contexts [3–9] in the last one decade. The theoreticalunderstanding of optical rogue wave is provided by the so-called nonlinear Schr¨odinger equation(NLSE).The study of NLSE has earned a lot of attention over the past few decades, due to its widerange of applications in numerous branches of science [10–12]. Since its earliest usage in thestudy of deep water wave propagation in oceanography [13, 14], the self-focussing NLSE has beenexploited extensively in different branches of physics, including nonlinear phenomena in optics[15, 16], Bose-Einstein condensate [17–21], plasma physics [22–24] and optomechanics [25]. Here,the NLSE, which is under the investigation, refers to the self-focussing NLSE which has been usedto study high amplitude extreme event phenomena such as rogue waves in optical fibers.The advantage of using the NLSE is that it belongs to a class of completely integrable nonlin-ear evolution equations and also admits an infinite number of exact solutions [26]. To elucidate, it ∗ Corresponding author
Email addresses: [email protected] (Dipti Kanika Mahato), [email protected] (A. Govindarajan), [email protected] (M. Lakshmanan), [email protected] (Amarendra K. Sarma)
Preprint submitted to Phys. Lett. A January 1, 2021 a r X i v : . [ phy s i c s . op ti c s ] D ec xhibits envelope solitons on zero background [27] and Kuznetsov-Ma (KM) breather [28], Akhme-diev breathers [29] and Peregrine solitons (PSs) [30, 31] on finite background. Note that KMbreathers are spatially localized solutions which breathe temporally while Akhmediev breathersare temporally localized solutions which breathe spatially and the Peregrine soliton is localizedin both spatial and temporal co-ordinates. Each of these finite background solutions is specificlimiting cases of a more general first-order two-parameter periodic solution with two periods, onealong the spatial axis and the other along the temporal axis. A detailed relation between thesefirst-order solutions corresponding to the lowest-order solutions of the NLSE can be found in Ref.[26]. The NLSE also admits another spatio-temporally localized solution as the limiting case ofAkhmediev breather when its spatial period is considered to be infinite [32, 33]. This solutionhaving the same expression as the Peregrine soliton, is known as the first-order rational solitonsolution in the form of ratio of two polynomials both being functions of space and time. Alongsimilar lines, higher-order rational solutions were also obtained from the nonlinear superpositionof multiple first-order rational solutions [34–39].The higher-order rational solutions of the constant coefficient NLSE, Peregrine soliton, Akhme-diev breather and KM breather have been studied analytically in the context of rogue wave gen-eration. It is important to note that a rogue wave is a spatiotemporally localized single wavewith high amplitude which emerges suddenly and dies out rapidly without leaving any trace tofollow. In case of an inhomogeneous Kerr nonlinear medium, such as optical fiber, the variation insystem parameters (dispersion coefficient, nonlinearity and gain or loss) needs to be incorporatedby employing the variable coefficient NLSE (Vc-NLSE). Similarity transformation has extensivelybeen used to construct analytical solutions of the Vc-NLSE connecting the solutions of constantcoefficient NLSE [40]. Over the years utilizing this method, several theoretical studies have beenreported on rogue wave dynamics [41–46]. Also, the study of breathers and rational solutions ofdifferent Vc-NLSE have revealed several new features, which include nonlinear tunnelling effectin periodically distributed system and exponentially dispersion decreasing fiber [47] and Peregrinecomb as multiple compression point in the amplitude in periodically modulated fibers [48].Recently it has been reported that periodic modulation of nonlinearity coefficient along thetransverse axis leads to evolution of Akhmediev-like breathers [44]. Also, a previous report revealedhow KM breathers and first and second order rational solutions evolve under periodic modulationof both dispersion and nonlinearity coefficients [42]. Zhong et al. have shown that under suchcondition, KM breathers propagate in a periodically modulated background with three peaks inone breathing unit, while the rational solutions maintain their standard features. In this work, bymanipulating only the dispersion coefficient periodically along spatial axis, we obtain Kuznetsov-Ma (KM)-like breathers analytically from the first three orders of rational solutions of Vc-NLSE forthe first time. The well-known similarity transformation has been used to solve the Vc-NLSE fromthe seed solutions of the standard NLSE. It should be noted that, we have chosen only rationalsolutions and observed their evolutions under periodic dispersion profile. Also, it has been observedthat the evolution of such KM-like breathers does not depend on the order of the rational solutions.Additionally, two new different peak-dynamics corresponding to the second-order rational solutionhave been displayed by controlling the free parameters under the periodic dispersion profile. Also,direct numerical results have been shown by employing pseudo-spectral methods which confirmthe analytical findings. One of the major motivations for focussing on the dispersion managementrather than the nonlinear one is due to the experimental feasibility of the earlier. While there are anumber of experimental reports on successful modulation of dispersion management [49–52], there2s no experimental vindication of nonlinearity management in the context of nonlinear fiber opticsalbeit a few of the former were executed in Bose-Einstein condensate[53, 54].The paper is organized as follows. Section 2 provides appropriate model of (1+1)D Vc-NLSEwith the analytical method to obtain the solutions. In Section 3, we explore rational solutiondynamics and the evolution of the first, second and third-order rational solutions as controlled-breathers in the case of Vc-NLSE under periodic dispersion coefficient. Section 4 illustrates thenumerical result. Finally, Section 5 provides an overall conclusion.
2. The Model and Similarity Transformation Technique
The pulse propagation in a Kerr nonlinear medium, say an optical fiber, is best described bythe so-called nonlinear Schr¨odinger equation with variable coefficients, which reads as i ∂u∂z + β ( z )2 ∂ u∂x + χ ( z ) | u | u = 0 . (1)where u ( z, x ) denotes the complex envelope of the optical field, z and x , respectively, are the prop-agation distance along the medium and the retarded time. Also, the parameters β ( z ) and χ ( z ) arethe group velocity dispersion (GVD) and the nonlinear (self-phase modulation) coefficients, respec-tively. Motivated by previous studies [42–44], we utilize the well-known similarity transformationin order to solve Eq. (1), which is written as u ( z, x ) = A ( z ) V ( T, X ) e iB ( z,x ) , (2)As it is generally known, this similarity transformation reduces Eq. (1) to the known standardNLSE i ∂V∂T + 12 ∂ V∂X + | V | V = 0 , (3)where V ( T, X ) represents the complex envelope of the optical field whose expression is known; T ( z ) is the dimensionless propagation distance and X ( z, x ) denotes the similarity variable whichneeds to be determined. Also, A ( z ) and B ( z, x ), both being real, are the amplitude and the phasefunction, respectively. Therefore, substitution of Eq. (2) into Eq. (1) will connect its solutionto the exact known solution of Eq. (3), provided following set of relations and partial derivativeequations (PDEs) are satisfied: AT z = 1 , (4a) β AX x = 1 , (4b) χA = 1 , (4c) B z + β B x = 0 , (4d) A z A + β B xx = 0 , (4e) β A X xx = 0 , (4f) B x = − β X z X x . (4g)3n the above equations, subscripts denote partial derivatives with respect to z or x . While solv-ing Eqs. (4), it turns out that the variable coefficients of Eq. (1) spontaneously emerge in theparameters of similarity transformation as follows, T ( z ) = (cid:90) z β ( s ) w ( s ) ds, (5a) w ( z ) = w β ( z ) χ ( z ) , (5b) X ( z, x ) = xw ( z ) + θ ( z ) , (5c) A ( z ) = 1 w ( z ) (cid:115) β ( z ) χ ( z ) , (5d) B ( z, x ) = w z β w x B ( z ) , (5e)with a condition, β w zz = β z w z , (6)where w ( z ) is the width of the rational soliton solution, w being the initial width. Also, θ ( z ) and B ( z ) are real integration constants which are chosen to be θ ( z ) = 0, w = 1, B ( z ) = 1 for therest of the calculations. Finally, assembling all the solutions from Eqs. (5), the exact solution ofEq. (1) can now be obtained as given below u n ( z, x ) = 1 w ( z ) (cid:115) β ( z ) χ ( z ) V n ( T, X ) e i (cid:0) wzβ w x +1 (cid:1) . (7)where n denotes the order of the solution.As previously described, the NLSE (Eq. (3)) has several exact analytical solutions describ-ing different physical phenomena. We will specifically deal with the first three orders of rationalsolutions obtained by the well-known Darboux transformation [55, 56]. A well-established classifi-cation of these rational solutions is given by a complex parameter s j , where j is a positive integer, j = 1 , , ...n .There is no s j parameter in the standard first-order rational solution, while there is one s parameter in the second-order solution defined as s = a + ib . The second-order solution showstwo different kinds of characteristics, type [0] when s = 0 ( a = b = 0) and type [1] when s (cid:54) = 0[57]. Similarly, there exist two parameters s = a + ib and s = c + id in the third-order rationalsolution. The third-order solutions are denoted as type [0,0] when s = s = 0. For different s , s values they are classified as type [0,1], [1,0] and [1,1] [58]. These parameters a , b , c and d areknown as the ‘ free parameters ’.On the basis of the work done in Refs. [35, 58], we use the first,second and the third-order rational solutions, V = (cid:20) − iT )1 + 4 X + 4 T (cid:21) e iT , (8) V = (cid:20) − D D (cid:21) e iT , (9)4 = (cid:20) − (cid:80) j =0 H j X j (cid:80) j =0 F j X j (cid:21) e iT (10)and study the intensity distribution of Eq. (7) choosing a specific functional form of β ( z ) and χ ( z ) for different s and s parameters.
3. Dynamics of rational solutions for periodic dispersion profile
In this section, we choose the dispersion coefficient to be periodic of the form, β ( z ) = 1 + σ cos ( ωz ). Here, σ is the amplitude part of the modulation with − < σ < ω (cid:54) = 0 isthe spatial frequency. The corresponding pulse width, w ( z ), the amplitude function, A ( z ), andthe similarity variable, X ( z, x ), are given as follows: w ( z ) = σ cos( ωz )1+ σ , A ( z ) = σ √ σ cos( ωz ) , X ( z, x ) = x (1+ σ )1+ σ cos( ωz ) .We consider that there is no nonlinear management in the system, i.e., χ ( z ) = 1. It is worthwhile to note that such dispersion profiles could easily be generated in a fiberdrawing process in experiments. In fact, numerous experimental and theoretical studies have beenreported in the past where the dispersion coefficient is perturbed periodically, quasi-periodicallyor even randomly [59–64].With these variables and Eqs. (7), (8), (9) and (12), we illustrate the intensity distribution ofthe first three orders of rational solutions for a periodically modulated group velocity dispersion.It is well-known that the standard first-order rational solution (Peregrine soliton) shows a single (a) (b)(c) (d)Figure 1: Intensity distribution of the first-order rational solution (or Peregrine soliton). (a) Standard first-ordersolution without modulation. Controlled KM-like breather with: (b) ω = 1, σ = 0 .
1; (c) ω = 1, σ = 0 . ω = 2, σ = 0 . z -axis as shown in Fig. 1(b). Such a temporally breathing solution has strong resemblancewith the KM breather, and thus it can be termed as ‘ controlled KM-like breather ’. Here, thebreather and the background have different frequencies in z , thus producing a beating betweenthe breather solution and the background. The parameter, σ , controls the background amplitudeand the breather peak power see Figs. 1(b) and 1(c) , while the spatial frequency ( ω ) controls thebreathing frequency of the KM-like breather and the periodicity of the finite background as shownin Fig. 1(d). (a) (b)(c) (d)Figure 2: Intensity distribution of the second-order rational solution. For a = 20: (a) Standard second-order type [1]solution and (b) Controlled type [1] breather, both symmetrical about x-axis. For b = 20: (c) Standard second-ordertype [1] solution, (d) Controlled type [1] breather, both symmetrical about z-axis. Modulating parameters chosen: σ = 0 . ω = 1. Similar controllable KM-like breather can be obtained in the case of second-order rationalsolution as well. As per the previously mentioned classification of the standard second-orderrational solution which is based on s parameter [58], type [0] solution possesses single peak. Onthe other hand, type [1] solution possesses three peaks (triplets) distributed in triangular shape.These triplets in second-order solution are symmetric about the x -axis when a (cid:54) = 0, b = 0 asobserved in Fig. 2(a) and about the z -axis when a = 0, b (cid:54) = 0. Considering specific values of σ = 0 . ω = 1, it has been observed that both type [0] and type [1] solutions show breathingfeatures under periodic dispersion. Here, we have shown the intensity evolution and contour plotof modulated type [1] solution in Figs. 2(c) and 2(d). This kind of solution could be termedas ‘ controlled type [1] breather ’. Considering the triangular peak distribution in Figs. 2(a) as oneunit, in each such periodic unit of the type [1] breather, the triplets are symmetric about the x -axis6hen a (cid:54) = 0, b = 0 (see Fig. 2(b)). On the other hand, the triplets are symmetric about the z -axiswhen a = 0, b (cid:54) = 0. The distance between the peaks gets increased with the increase in the valueof either a or b . Thus, type [1] breather maintains the characteristics of type [1] rational solutionwhile breathing.Unlike the first and the second-order rational solutions, the characterization of the basic third-order rational solution requires two parameters, s and s as already pointed out. From the previousresearches, we know that the standard third-order solution shows four different features, namely,the one peak or type [0,0] [35], the triangular cascade or type [1,0] [39, 65], the pentagram structureor type [0,1] [39, 65] and the claw-structure or type [1,1] [37, 58]. The intensity distribution ofthe standard third-order type [1,0] and type [1,1] solutions are shown in Fig. 3(a) with s = 100, s = 0 ( a = 100, b = 0, c = d = 0) and in Fig. 3(e) with s = 31, s = 500 ( a = 31, b = 0, c = 500, d = 0), respectively. The standard type [1,0] solution corresponds to a unit of triangular structure,which is basically six first-order rational solutions of the same amplitude arranged in a triangularshape [Fig. 3(a)].On the other hand, intensity distribution of type [1,1] solution displays clawlike structure whichcontains three first-order and one second-order solutions with their highest amplitude [Fig. 3(e)].Now, considering σ = 0 . ω = 0 .
5, it has been observed that, for s = 100, s = 0, theunit of triangular structure is getting repeated periodically along the z -axis [Fig. 3(c)]. Here, theextreme-end peaks of two consecutive units (at negative x -axis) combine together and appear withlower amplitude in comparison with the other peaks. For better understanding, the correspondingcontour plot of controlled type [1,0] breather has been displayed in Fig. 3(d). For the claw-likestructure, considering s = 31, s = 500 and σ = 0 . ω = 0 .
5, the periodic claw-like structure type[1,1] breather is observed under periodic modulation [Figs. 3(g) and 3(h)]. It can be observed thatfor specific values of modulating parameters σ and ω , we get KM-like breathers in the first threeorders of rational solutions. Thus, we can interpret that it is also possible to generate and controlKM-like breathers from higher-order rational solutions with simple periodic dispersion profile. In this section, we briefly explore some unique aspects of the second-order rational solutionfor periodically varying dispersion profile. In the case of the constant coefficient NLSE solution,for s (cid:54) = 0, if the value of real or imaginary part of s parameter ( a or b ) is increased, the peakseparation gets increased for both the cases [58]. But under periodically varying dispersion profile,for a fixed value of the modulating parameters σ and ω , the breather dynamics show a completelydifferent characteristics depending on the value of the real or imaginary part of s parameter.To explain further, for specific modulating parameters, σ = 0 . ω = 0 .
6, when the realpart of the s parameter, i.e. a is increased under periodically modulated dispersion, the distancebetween the triplets of each unit is found to increase at first (see Figs. 4(a) and 4(b)). Butafter a certain a value, the controlled three-peak type [1] breather transforms into a single-peaktype [1] breather with a lower amplitude. The appearance of the single-peak type [1] breather isshown at a specific value a = 1500 as presented in Fig. 4(c). This controlled single-peak type [1]breather appears similar to the controlled KM-like breather shown in Fig. 1(b), having its originshifted to the positive x -axis. The reason behind the disappearance of the other two peaks may beattributed to the high a value in comparison with the modulating period (2 π / ω ) of the background.The higher a value leads to a larger peak separation in each triplet unit of the controlled type [1]breather. In the presence of lower modulating period and larger peak separation (controlled by7 a) (b)(c) (d)(e) (f)(g) (h)Figure 3: Intensity distribution and contour plots for third-order rational solution. For s = 100, s = 0: (a),(b)Triangular cascade structure of the standard third-order type [1,0] solution and (c),(d) Controlled type [1,0] breather.For s = 31, s = 500: (e),(f) Claw-like structure of the standard third-order type [1,1] solution and (g),(h) Controlledtype [1,1] breather. Modulating parameters chosen: σ = 0 . ω = 0 . a) (b) (c)Figure 4: Intensity distribution for controlled type [1] breathers with σ = 0 . ω = 0 . a (cid:54) = 0, b = 0. For: (a) a = 50; (b) a = 100; (c) a = 1500. a ), the other two peaks in each triplet unit stop breathing. When a > x -axis. (a) (b) (c)Figure 5: Intensity distribution for controlled type [1] breathers with σ = 0 . ω = 0 . a = 0, b (cid:54) = 0.For: (a) b = 50; (b) b = 100; (c) b = 1500. On the other hand, for the same modulating parameters σ = 0 . ω = 0 .
6, when theimaginary part of s parameter, i.e. b is increased, the distance between the triplets within eachunit is again found to increase as given in Figs. 5(a) and 5(b). But unlike the previous case, at aparticular value b = 1500, the transformation from three-peak type [1] breather into two-peak type[1] breather can be observed (see Fig. 5(c)). The breathing period of the third peak in each unitis less than the peak separation distance (controlled by b ). In this case, another feature has alsobeen observed that each unit gets shifted along the positive z -axis as shown in Fig. 5(c). When b > z -axis. It iswell-known that the standard second-order rational solutions (type [0] and type [1]) do not possesstwo peaks, but by regulating the free parameters b under periodic dispersion profile, we obtaintwo-peak type [1] breather solution. Thus, if the value of modulating period of finite background(controlled by spatial frequency in periodic dispersion profile) is less than the peak separationvalue (controlled by free parameters), one can obtain a new breather dynamics corresponding tothe higher-order rational solutions. 9 igure 6: Numerical MI dynamics showing a sea of pulses for the case of standard NLS systems. The upper panelsindicate the formation of AB-like states (indicated by a rectangle box drawn by dashed white line in (b)) inducedby a small amplitude of perturbation ( (cid:15) = 0 .
01) while the bottom panels (c) show a comparison of a single analyticPS and a localized state extracted from the density map at z = 4 . β ( z ) = χ ( z ) = 1 and Ω = 1.
4. Modulational instability analysis: Direct numerical outcomes
It is a well-established fact that modulational instability is a precursor for the formation ofrogue waves as the latter is a phenomenon of extreme events triggered by the maximum unstablemodes [66]. Indeed, in realistic physical settings including optical fiber and sea waves the planewave is perturbed by a quantum noise and subsequently the formation of extreme waves (roguewaves) is traced at a particular propagation distance. It is to be noted that since the roguewaves are generated due to perturbative fields, they are highly unstable which in turn results toan instantaneous life time to the former. Hence considering the analytical solutions as the seedsolutions in the numerical evolution are unrealistic ones. One of the standard ways to simulatethese analytical solutions is to rely on the noise induced MI where the the plane wave experiencesan instability due to the perturbation [67]. In this process, the spectral sidebands of unstablemodes undergo an exponential growth and eventually lead to a more complex dynamics whichcould be mapped to the well-known four-wave mixing process. During this complex dynamics,there occurs a cyclic energy exchange between multiple spectral modes due to an inelastic collision[63]. Besides, it has been suggested that the occurrence of MI is closely related to the phenomenon10 igure 7: Dynamics of KM-like breathers by a train of PSs along the evolution co-ordinate is shown in upper panels.In bottom panels, (c) comparison of the localized state of PS extracted from the density map with the analytic KMbreathers and (d) shows the growth rate against the small perturbation. Here, the parameters are the same as usedin Fig. 1(b) with Ω = 1 .of Fermi–Pasta–Ulam recurrence in optics since the MI, in the long run, results in a train ofultra-short pulses with a same repetition rate in its chaotic MI map consisting of multiple high-amplitude waves which can show a signature of extreme events [68]. To observe the realtime roguewaves investigated in the proposed system of periodically varying group dispersion, we solve Eq.(1) by the standard pseudo-spectral method in
Python programming language with appropriateboundary conditions including 1024 Fourier points [69]. To this end, we assume the following planewave solution with an infinitesimal parameter (cid:15) , which is to be accountable for the perturbation. u (0 , x ) = u (1 + (cid:15) cos(Ω x )) . (11)where Ω is the (temporal) frequency of the applied field. Here the values of CW amplitude andthe seed parameter are, respectively, chosen to be u = 1 .
21 and (cid:15) = 0 .
01 unless otherwise stated.Prior to analyzing the generation of rogue waves in the inhomogeneous system, we first producethe results for the constant NLSE by considering the periodic GVD parameter as β ( z ) = 1 bykeeping the nonlinear parameter as χ ( z ) = 1. As shown in Fig. 6(a), Akhmediev-like breathers(ABs) are formed after a finite distance ( z >
3) (see Fig. 6(b)) before which the non-zero CWbackground does not experience any instability. Note that these ABs are generated by an arrayof Peregrine solitons (refer to the pulse marked with a white rectangle box in Fig. 6(b) and the11ne dimensional plot drawn in Fig. 6(c)) confined in temporal axis ( x ) and these results well agreewith the earlier analytical predictions shown in Fig. 1(a). While making this conclusion, it mustbe kept in mind that both the amplitude and phase of the nonlinear structure are needed to arriveat a reasonable inference in order to prove the typical signatures of PSs [70]. In Fig. 6(d), we haveshown the phase portraits of both the analytical solution shown in Fig. 1(a) and the numericallyextracted PSs (for instance, see Fig. 6(b)) at the propagation distance z = 4 .
5. It is obvious thatthe outcomes driven by the modulational instability of plane wave induced by the perturbationagree well with the measured intensity and phase of the nonlinear wave packets as were done inRefs. [34, 71]We now investigate the dynamics of Peregrine solitons in the case of inhomogeneous systemswith the periodic profile ( β ( z ) = 1 + σ cos( ωz )). By retaining the same parameters used in Fig.1(b) with σ = 0 . x ) to capture the formation of KM- like breathers and the correspondingramifications shown in Figs. 7(a) and 7(b) corroborate well with the analytical calculations depictedin Fig. 1(b). In support of this validation, a comparison of the two dimensional intensity plot ofFig. 1 (b) with the evolution of the plane wave along the propagation direction is shown inFig. 7(c). It is evident from Fig. 7(c) that the obtained numerical KM- like breathers are aclear manifestation of periodic management of dispersion parameter. In addition, the growthrate against the evolution co-ordinate is given in Fig. 7(d), which further supports the presentednumerical findings adequately. Although the first-order rational solutions can be traced after anumber of numerical experiments pertaining to the modulational instability caused by the smallperturbation, it is highly cumbersome to predict the existence of higher-order Peregrine solitons[71]. Nevertheless, we have presented the dynamics of third-order rational solutions in Fig. 8with the system parameters chosen as σ = 0 .
125 and ω = 1. It is very clear to observe theformation of claw-like structure (refer to the PS indicated by a red arrow line in Fig. 8(a)) withthe triplets of Peregrine solitons. We affirm here that, to the best of our knowledge, these are thefirst ever numerical proofs for the existence of Peregrine solitons and KM breathers done in theinhomogeneous systems through the direct simulations induced by the infinitesimal perturbations.Also, it is quite remarkable to note that such perturbation induced nonlinear structures havebeen mapped well with the analytical solutions through the MI simulations instead of simulatingthe former along the evolution co-ordinate by considering it as the seed solution even in thisinhomogeneous system. It is worthwhile to mention that such PSs and their different variantsincluding ABs and KM-like breathers found both analytically and numerically in the present systemcan be characterized and distinguished by some other import tools such as numerical inversescattering transform (IST) and finite gap theory [67, 72–74]. In particular, IST spectra obtainedfrom the eigenvalues of the considered systems stand out to be an accurate tool in characterizingextreme waves (rogue waves) and their mechanisms. Our analytical and numerical findings are alsoexpected to rely on this method and those ramifications shall be published elsewhere. Since theperiodic GVD can easily be fabricated by the state-of-the-art-technology available in this modernera and these novel solutions have already been observed through the numerical experiments, wedo hope that the present study will stimulate experiments in these directions, which may pave anew avenue on rogue waves in the lightwave communication systems.12 igure 8: Evolution of third-order rational waves with type [1,1] induced by the amplitude of perturbation (cid:15) = 0 . σ = 0 . ω = 1 and χ = Ω = 1.
5. Conclusion
In summary, the evolution of first three orders of rational soliton solutions has been investigatedunder the presence of a longitudinally varying periodic dispersion via appropriate similarity trans-formation in the context of optical fiber. It has been shown that it is possible to transform rationalsoliton solution into KM-like breather with judicious choice of dispersion coefficient parameters.By modulating the amplitude part and the spatial frequency of the periodic dispersion profile, onecan realize novel rational solution dynamics. Our study also shows how for periodic dispersion,the second-order rational solution dynamics change depending on free parameters correspondingto the complex s parameter. For specific modulating parameters, different peak-dynamics havebeen observed by regulating the real and imaginary parts of the s parameter. Direct numericalsimulations have also been executed to corroborate the analytical predictions through the modula-tional instability analysis by perturbing the applied field with a small perturbation in the form ofwhite noise. Since in reality modulating dispersion profile is much easier, as evident from numer-ous experimental studies, than modulating nonlinearity or any other inhomogeneous profiles, webelieve our study on the dynamics of rational solutions can be helpful to understand the dynamicsof different types of rogue waves in realtime experiments. Declaration of competing interest
The authors declare that they have no known competing financial interests or personal rela-tionships that could have appeared to influence the work reported in this paper.
Acknowledgement
D.K.M. thanks MHRD, Government of India for financial support through a fellowship. A.K.S.acknowledges financial support from Science and Engineering Research Board (SERB), Governmentof India under MATRICS scheme (Grant No. MTR/2019/000945). A.G. and M.L. acknowledgethe support of Science and Engineering Research Board (DST-SERB) for providing a DistinguishedFellowship project to M.L. (Grant No. SB/DF/O4/2017) in which AG was a Visiting Scientist.A.G. is now supported by University Grants Commission (UGC), Government of India, through aDr. D. S. Kothari Postdoctoral Fellowship (Grant No. F.4-2/2006 (BSR)/PH/19-20/0025).13 eferences [1] D. Solli, C. Ropers, P. Koonath, B. Jalali, Optical rogue waves, Nature 450 (7172) (2007) 1054.[2] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, J. M. Dudley, The Peregrine solitonin nonlinear fibre optics, Nature Physics 6 (10) (2010) 790–795.[3] J. M. Dudley, M. Erkintalo, G. 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