Engineering Optical Rogue Waves and Breathers in a Coupled Nonlinear Schrödinger System with Four-Wave Mixing Effect
EEngineering Optical Rogue Waves and Breathers in a CoupledNonlinear Schr ¨odinger System with Four-Wave Mixing E ff ect K. Sakkaravarthi a,b, ∗ , R. Babu Mareeswaran c , T. Kanna d, ∗ a Department of Physics, National Institute of Technology, Tiruchirappalli – 620 015, Tamil Nadu, India b Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli – 620 024, India c PG and Research Department of Physics, PSG College of Arts and Science, Coimbatore–641 014, India d Nonlinear Waves Research Laboratory, PG and Research Department of Physics,Bishop Heber College (A ffi liated to Bharathidasan University), Tiruchirapalli–620 017, Tamil Nadu, India Abstract
We consider a coherently coupled nonlinear Schr¨odinger equation with modulated self-phase mod-ulation, cross-phase modulation, and four-wave mixing nonlinearities and varying refractive indexin anisotropic graded index nonlinear medium. By identifying an appropriate similarity transforma-tion, we obtain a general localized wave solution and investigate their dynamics with a proper set ofmodulated nonlinearities. In particular, our study reveals di ff erent manifestations of localized wavessuch as stable solitons, Akhmediev breathers, Ma breathers, and rogue waves of bright, bright-dark,and dark-dark type and explores their manipulation mechanism with suitably engineered nonlinearityparameters. We have provided a categorical analysis with adequate graphical demonstrations. Keywords:
Coherently coupled nonlinear Schr¨odinger equation; Similarity transformation;Solitons; Akhmediev breather; Ma breather; Rogue waves.————————————————————————————————————————
Journal Reference:
Physica Scripta (2020) 095202. https://doi.org/10.1088/1402-4896/aba664
1. Introduction
Dynamics of nonlinear systems features several interesting phenomena that find multifaceted ap-plications in di ff erent fields of science, engineering, and technology through a systematic understand-ing of the respective mathematical models. The existence of various nonlinear coherent structuresassociated with such models is of considerable physical significance [1]. Among several types oflocalized nonlinear waves, solitons receive profound interest due to their remarkable stability natureand intriguing collision dynamics. For the past few decades, studies on optical / atomic solitons haveshown promising results in theoretical aspects and in their corresponding experimental realizationthrough which a number of phenomena have been demonstrated in various fields [2]. Apart from thesolitons, the amusing nonlinear coherent structure, namely rogue waves that appear from nowhereand disappear without a trace [3], have attracted significant attention during the past ten years orso from theoretical and experimental perspectives [4–7]. These are because of their more frequentappearance in deep sea and due to their experimental observation in information transfer through op-tical fibers, atomic condensates, and di ff erent dynamical systems in recent years. These rogue waves(or freak / monster / killer / extreme / abnormal waves) were first observed in deep ocean water extreme ∗ Corresponding authors.Email address: [email protected]; babu nld@redi ff mail.com; kanna [email protected] Preprint submitted to
Physica Scripta a r X i v : . [ n li n . PS ] A ug mplitude waves with a significant height that is a few times higher than the average wave crests[8]. It is di ffi cult to conduct experiments in the deep ocean to understand them due to the dangerousconditions. However, observations from the replicated laboratory experiments and cross-disciplinaryinvestigations (for example, in optical and atomic systems and so on) can be adopted for realizingtheir complete dynamical evolution as well as prediction mechanism. Such rogue waves are a specialcase of breathers that are another type of coherent structure oscillating periodically in space or timeand display periodic variation in their amplitude during propagation [8]. These breathers have alsoreceived significant importance in recent years. Theoretically, these nonlinear coherent structurescan be well described by the ubiquitous nonlinear Schr¨odinger (NLS) equation [3], its variants andsimilar models. Though they are initially observed as extreme waves in the oceans, now a days theyappear in a wider range of fields such as fiber optics as extreme amplitude pulses, optical cavities ashigh amplitudes, laser outputs as extreme pulses, Bose-Einstein condensates (BEC) as high atomicconcentration, etc. [8, 9].Beyond these localized waves, several types of nonlinear waves arise in various nonlinear dy-namical models under di ff erent circumstances. We do not pursue / discuss them here considering theobjective and length. Indeed multicomponent nonlinear waves show promising characteristics thantheir single-component / scalar counterparts and hence coupled nonlinear systems are under continu-ous exploration. It is clear from the literature that there exist plenty of multicomponent mathemati-cal models, especially di ff erential / di ff erence equations, describing the dynamics of various physical,chemical, biological, engineering, and even medical systems and finance. Appropriate analysis ofthese models provides an understanding of the underlying system. Those analyses can be done byconstructing explicit solutions using / developing analytical (semi-analytical) methods, which may bepossible for certain classes of equations, namely integrable models. However, most of the natural sys-tems are non-integrable, forming another category, namely, exactly non-solvable models that requirestrong computational tools in addition to some symmetry and approximate methods [10].Keeping the above perspectives in mind, in this work, we report the characteristic dynamics ofcertain localized waves in an inhomogeneous nonlinear optical model [2, 10]. To be precise, we takethe following coherently coupled nonlinear Schr¨odinger (CCNLS) equation in dimensionless form: i ∂ q ∂ z − ∂ q ∂ t − σ ( z ) (cid:16) | q | + | q | (cid:17) q − σ ( z ) q q ∗ + Ω ( z , t ) q = , (1a) i ∂ q ∂ z − ∂ q ∂ t − σ ( z ) (cid:16) | q | + | q | (cid:17) q − σ ( z ) q q ∗ + Ω ( z , t ) q = , (1b)where the derivatives with respect to z and t represent the propagation direction and transverse co-ordinate, respectively. The above equation is referred as coherently coupled nonlinear Schr ¨odingerequation with varying nonlinearity σ ( z ) governs the co-propagation of two complex envelope opticalmodes q j ( z , t ) , j = , , in a medium like optical fiber with modulated parabolic refractive indexprofile Ω = µ ( z ) t for the linear part of the refractive index. To be precise, one can express thee ff ective refractive index of the optical system as n ( z , t ) = n + n µ ( z ) + n σ ( z ) I ( z , t ), where n de-notes constant refractive index of the medium while n and n represent the modulated linear andnonlinear (Kerr type) contribution of refractive index, and I is the total intensity of the optical wave22]. In one way, the above type of model can appear in cubical nonlinear (Kerr) medium with gradedrefractive index [6, 7]. Also, equation (1) contains coherent coupling nonlinearity resulting fromfour-wave mixing e ff ect in addition to the standard incoherent nonlinearities self-phase modulation(SPM) and cross-phase modulation (XPM). The above equation is non-integrable while its homoge-neous version turns out to be integrable for which and for several similar models various nonlinearwave solutions including solitons, periodic waves, breathers, rogue waves, etc. have been obtainedby using various methods [11–24] and one version is also referred as pair-transition-coupled NLSmodel [21–24]. Further, there exist several studies on certain scalar inhomogeneous models and theirrelated coupled systems exploring the dynamics of di ff erent nonliner waves in both one- and higher-dimensions, to mention a few [25–30]. However, the present inhomogeneous system requires anauthoritative investigation, which motivated us to proceed with the analysis for a category of local-ized structures in this report. Though localized waves arise in di ff erent contexts from hydrodynamicsto atomic condensates, control strategies of such waves to understand their transitions are still dif-ficult / less (particularly, engineering the extreme waves) due to limited experimental implementationand a few tools are being developed in recent years [31–36]. Thus, we devote ourselves to manipu-lating such localized nonlinear waves through explicit solutions with inhomogeneity parameters andwill concentrate on exploring their dynamical features.We provide the essential mathematical tool of similarity transformation briefly along with theestimated nonlinearities in Sec. 2. We construct the inhomogeneous localized nonlinear wave solu-tion in Sec. 3 and discuss the impact of chosen modulated nonlinearities in each type of nonlinearwaves and the importance of available control parameters in Sec. 4, where we have also given certainpossible future directions along this study. Finally, we summarize the important outcomes in Sec. 5.
2. Similarity Transformation & Nonlinearity Management
Among various methodologies to solve nonlinear models, similarity transformation occupies aprominent place due to its simple adaptability / applicability in executing the objectives. To beginwith, we consider the following form of solution to Eq. (1): q j ( z , t ) = (cid:15) (cid:112) σ ( z ) Q j ( Z ( z ) , T ( z , t )) exp[ i ξ ( z , t )] , j = , , (2)where (cid:15) √ σ ( z ) determines the amplitude, while ξ ( z , t ) is the phase and T ( z , t ) and Z ( z ) are the simi-larity variables, the explicit form of all these variables has to be determined in a systematic way. Onthe substitution of Eq. (2), the model (1) reduces to a set of coupled equations of Q j ( Z , T ) in the form iQ Z − Q TT − ( | Q | + | Q | ) Q − Q Q ∗ = , (3a) iQ Z − Q TT − (2 | Q | + | Q | ) Q − Q Q ∗ = , (3b)3long with a set of constraint relations in terms of varying nonlinearity and arbitrary constants (cid:15) and (cid:15) as given below. ξ ( z , t ) = σ z σ t + (cid:15) (cid:15) σ t + (cid:15) (cid:15) (cid:90) σ dz , (4a) T ( z , t ) = (cid:15) (cid:32) σ t + (cid:15) (cid:15) (cid:90) σ dz (cid:33) , (4b) Z ( z ) = (cid:15) (cid:90) σ dz , (4c) µ ( z ) = σ zz σ − σ z σ , (4d)Equation (3) is nothing but an integrable CCNLS type model and its solutions will indirectly give thecorresponding solutions of (1) through Eqs. (2) and (4). As mentioned in the introduction, studies ondi ff erent nonlinear waves (especially solitons, breathers, rogue waves) of Eq. (3) and its variants areabundant including an interesting energy-switching collisions of single-double-hump solitons [12]and higher order rogue waves [16–19, 23, 24].In the present study, we focus on an interesting solution structure supporting multi-wave evolutionand will demonstrate the impact of varying nonlinearity of di ff erent forms that drive the nature of therefractive index of a harmonic oscillator. The nonlinearity modulations considered in this work are(i) periodic, (ii) step-like, and (iii) bell-type nonlinearities, respectively in the form σ ( z ) = b + b sin( b z + b ) , (5a) σ ( z ) = b + b tanh( b z + b ) , (5b) σ ( z ) = b + b sech( b z + b ) , (5c)where b , b , b , and b are arbitrary real constants. Further, the mutual dependency of the refractiveindex profile on this nonlinearity can be given in an explicit form through Eq. (4d). For illustrativepurpose, we have shown their modulation with respect to z in Fig. 1 and their impact on the designatednonlinear waves will be analyzed categorically.(a) (b) (c) Figure 1: Nature of three di ff erent nonlinearities σ ( z ) (a) and respective modulation in refractive index profile µ ( z ) (b)and Ω ( z , t ) (c) for b = . b = . b = .
75, and b = − . . Localized Wave Solution In order to analyze the influence and respective applications of nonlinearity variation in the system(1), we need to construct its solution. For this purpose, we adopt the solution methodology proposedby Zhao et al for another version of the CCNLS model in Ref. [21] where Darboux transformationmethod was employed. It is well-known that the Darboux transformation is one of the e ffi cientanalytical tools to obtain a variety of nonlinear wave solutions of equations which admit Lax pair.We have computed the exact solution for the present CCNLS equation (1) through an appropriatescaling and the derived similarity transformation (2). An important reason for considering this typeof solution is by virtue of only two arbitrary parameters rich dynamical features can be unearthedappropriately.Without going into much details, by following Ref. [21] and the similarity transformation givenabove in Sec. 2, we obtain a general localized wave solution to equation (1) which can be written inan explicit form as q = (cid:15) √ σ (cid:32) aP P ∗ | P | + | P | + s e is Z (cid:33) e i ξ , (6a) q = (cid:15) √ σ aP P ∗ | P | + | P | e i ξ , (6b)where P = τ (cid:16) √ a + τ Ψ − √ a − τ Ψ (cid:17) , P = τ (cid:16) √ a + τ Ψ − √ a − τ Ψ (cid:17) e is Z , (6c) Ψ = exp (cid:104) τ T − i (cid:16) s + a τ (cid:17) Z (cid:105) , Ψ = exp (cid:104) − τ T − i (cid:16) s − a τ (cid:17) Z (cid:105) , τ = √ a − s (cid:44) . (6d)Further, the form of Z ( z ), T ( z , t ) and ξ ( z , t ) appearing in the above solution is as given in Eq. (4).Thus our solution has four arbitrary real parameters (cid:15) , (cid:15) , s and a in addition to the freedom toadopt any nonlinearity function. Especially, the latter two arbitrary real parameters s and a willplay a crucial role in manipulating the above solution into di ff erent varieties of nonlinear waves suchas solitons, breathers, and rogue waves. Compared to many localized wave solutions, this is moreinteresting because of its multifaceted nature merely with two arbitrary parameters. For example,a single arbitrary real parameter in a scalar NLS solution results in the formation of breathers androgue waves [36, 37], but they are of bright type only. However, the present solution (6) is muchmore general and provides bright, gray, dark, bright-dark, gray-dark, and dark-dark type breathersand rogue waves in addition to the standard bright solitons. In the forthcoming sections, we willexplain and discuss all such solutions as well as their modulation e ff ects.
4. Dynamics of Localized Waves under Inhomogeneous Nonlinearities
As pointed out above, the obtained inhomogeneous wave solution (6) comprises various typesof localized structures ranging from the standard solitons to Akhmediev as well as Kuznetsov-Matype breathers and rogue waves. One can rewrite the general solutions (6) in terms of hyperbolicand periodic functions and can derive explicit expression to each nonlinear wave structures which5re given below for di ff erent possible combinations of these two arbitrary parameters ‘ s ’ and ‘ a ’.Through this investigation, based on the nature of τ and s / a defined by the appropriate choices of ‘ s ’and ‘ a ’ parameters, we have categorized the resulting wave patterns and summarized them in Table1. We investigate the consequences of modulated nonlinearities in each of them in this section one byone, except for the singular structures case (viii). In cases (vi) and (vii), the values of τ are non-zero,but very small and close to zero ( τ (cid:44) − << τ << q component comprises either dark or gray type profile for which the lowest intensity of the dip is zerofor the former while non-zero in the latter. Importantly, these nonlinear localized structures can beobtained even for a fixed “ a ” with suitably changing “ s ” value, such that this “ s ” shall be referredas “switching parameter” as it switches among solitons, Akhmediev and Ma breathers, and roguewaves. Table 1: Formation of di ff erent nonlinear wave structures based on the s and a parameters. Case a s τ s / a q q Wave nature(i) ± + ± + − ± Bright Dark Akhmediev Breathers(iii) ± − − ∓
Dark Dark Akhmediev Breathers(iv) ± ± + +
Bright Dark Ma Breathers(v) ± ∓ + − Dark Dark Ma Breathers(vi) ± ± ± + Bright Dark Rogue waves(vii) ± ∓ ± −
Dark Dark Rogue waves(viii) ± ± a ∗ ∗ ∗ Singular structures
A clear analysis of solution (6) reveals that both components q and q will have the same typeof stable localized waves, which are nothing but stationary bright solitons. This is possible when theswitching parameter is absent ( s =
0) which leads to τ >
0, and it can be written in a convenienthyperbolic form as q = q = a (cid:15) √ σ sech (cid:34) a (cid:15) (cid:32) σ t + (cid:15) (cid:15) (cid:90) σ dz (cid:33)(cid:35) e i ( ξ − a Z ) . (7)As a special case of this solution, first, we briefly discuss the homogeneous case with constant nonlin-earity σ ( z ) = constant. If we check out such a homogeneous soliton solution, it is evident that thereis not much freedom in its dynamics except the amplitude and width controlled by a . On increasingthe ‘ a ’ parameter, soliton amplitude increases (directly proportional) while its width decreases as itis inversely proportional to a , but do not propagate and remains stationary over time. This is onelimitation of the obtained solution as it neither shows any e ff ect of nonlinear coherent coupling northe multi-component nature of the system, which usually portrays a rich dynamics and collisions. Forexample, the general soliton solution directly constructed with another method namely Hirota bilin-earization method revealed a variety of soliton structures ranging from single-hump to double-hump6nd flat-top profiles [12] that results due to the contribution from four-wave mixing nonlinearity ofthe system (1). However, the present soliton solution can be considered as a special case of that one[12] with the same dynamics on both components, namely a degeneracy state ( q and q are identicalor simply q = q ). For completeness, we have depicted such stable degenerate solitons in Fig. 2(a).Thus, in some sense, the present switching parameter s can be associated / equivalent with / to the auxil-iary function given in Ref. [12]. Here, the entire spectrum of wave structures and their dynamics canbe controlled by a single parameter ‘ s ’ (relative to ‘ a ’ ). Based on the dynamics, it can be associatedwith phase-dependent coherent nonlinearity of the system which shows promising energy-switchingbehaviour [12]. For a more detailed understanding of the general bright solitons and their collisions,one can refer [12]. Importantly, the ramifications of varying nonlinearities in such solitons and theircollisions shall also be the next immediate assignment along this direction.As the main objective of this work is on the inhomogeneity, we come back to the general solution(7) and consider the modulated nonlinearities given in Eq. (5). The periodic nonlinearity modulation σ ( z ) transforms the stable soliton into a breather, which oscillates in the amplitude and along the z direction. However, there is no significant change along t , which can be visualized from Fig.2(b). This can be viewed as Ma type breather oscillating along propagation direction but with zerobackground in contrary to the standard Ma breathers, which usually appear on a non-zero constantbackground. In addition to a , (cid:15) and (cid:15) values, one can control the amplitude, width and velocity ofthese breathing solitons by tuning the arbitrary parameters b , b , b , and b .(a) (b) (c) (d) Figure 2: Dynamics of stationary solitons in inhomogeneous optical fiber for s = a = . σ , (c) kink-like σ , and (d) bell-type σ nonlinearities given in (5), showing the (a) stable soliton, (b) periodicbreather, (c) amplification with compression, and (d) rogue-wave type exciton, respectively. Here the nonlinearity param-eters are chosen as (a) b = . b = b = .
0, (b) b = . b = . b = .
2, (c) b = . b = . b = .
2, and(d) b = . b = . b = . (cid:15) = . (cid:15) = .
05, and b = . Moving further to the case of step-like nonlinearity σ = b + b tanh( b z + b ) (5b), we observean interesting phenomenon called amplification accompanied with a compression which can alsobe visualized as selective amplification through cascaded compression. This is a single-step steadyincrease (decrease) in the amplitude (width) of the soliton, see Fig. 2(c). A reverse transition canalso be achieved by suppressing the energy / intensity of a pulse / beam along with a suitable increasein bandwidth when b <
0. Also, by controlling the nonlinear phase factor b we can shift the criticaltransition point along ‘ z ’ based on the requirement.The bell type (‘sech’) nonlinearity σ ( z ) induces a localized change in the available wave of anycharacteristics. This can be broadly of two types; the first one has localized excitation or exciton for-7ation in the absence of background of the bell nonlinearity ( b = b . It can be furtherdivided into two types, namely the tunneling-through a high potential barrier (barrier penetration for b >
0) and cross-over of a potential well ( b < z with extended tails along t as shownin Fig. 2(d). Such a localized modulation resembles the collision scenario of two oppositely movingbright solitons, which produces a maximum intensity in the collision regime. Additionally, in thecase of tunneling and cross-over processes, the solitons appear stable before and after the barrier / wellwithout any other change. At the potential barrier / well, they induce a localized hump / dip and we re-frain from giving its graphical demonstration here. Beyond the limited functionalities of the presentsolitons, the impression of nonlinearities in other localized structures is much more interesting, asshown in the forthcoming discussions. Beyond the standard solitons discussed above, solution (6) exhibit periodically oscillating struc-tures called breathers for s (cid:44)
0. One such entity is Akhmediev breathers, which is nothing but anonlinear wave localized along the propagation direction z and periodic along the transverse direc-tion t . This arises for the choice a , s (cid:44) τ < q = (cid:15) √ σ ( a − s ) cosh[4 ab Z ] + iab sinh[4 ab Z ] a cos[2 b T ] − s cosh[4 ab Z ] e i ( ξ − s Z ) , (8a) q = (cid:15) √ σ a cosh[4 ab Z ] + iab sinh[4 ab Z ] − sa cos[2 b T ] a cos[2 b T ] − s cosh[4 ab Z ] e i ( ξ − s Z ) , (8b)where b = | a − s | , ξ , T and Z take the form of as given in Eq. (4). Interestingly, as mentionedin cases (ii) and (iii) of the Table 1, one can obtain bright, gray and dark type Akhmediev breatherswith respect to di ff erent choices of a and s parameters. Further note that the q and q componentsadmit single-hump and double-well (or double-dip) breathing patterns, respectively. Also, their in-tensity / amplitude shall be tuned by varying these a and s in addition to the transformation parameters.The obtained Akhmediev breathers can be controlled with the help of varying nonlinearities andit modulates their identities like localization, periodicity, amplitude, width / thickness of the beam,etc. Particularly, with a periodically varying nonlinearity σ ( z ), its localization is broken and nowbecomes as doubly-periodic breathers, which means periodic in both z and t . To be precise, in the q component, the single-hump excitations split / deform into soliton lattices on either side of the centraldip and this form periodically manifest along z indefinitely with substantially lesser amplitude. Onthe other hand, in q component, the double-well with central maximum intensity spread out bybreaking the localization and also become periodic along ‘ z ’ as shown in Fig. 3(b). Here the importantramification of periodic nonlinearity in Akhmediev breathers is on the localization and amplitude.Also, the period of oscillation and amplitude can be altered by tuning the arbitrary b j as well as (cid:15) j parameters.Contrary to the periodic nonlinearity, the kink-like (tan-hyperbolic) nonlinearity σ (5b) does not8a) (b) (c) (d) Figure 3: Dynamics of bright-dark Akhmediev breathers for s = a = − .
75 with (a) constant nonlinearity and theirtransformation due to (b) periodic, (c) kink-like, and (d) bell-type nonlinearities revealing (b) localization-broken doubly-periodic breathers, (c) escalated background amplitude time-periodic breathers, and (d) localization retaining centrallysymmetric breathers, respectively. The parameters are chosen as (a) b = . b = b = b = .
0, (b) b = . b = . b = .
75 & b = .
02; (c) b = . b = . b = . b = .
02, and (d) b = . b = − . b = .
5, & b = .
02 with other values fixed as (cid:15) = .
25, and (cid:15) = . alter / break the localization. Still, the modulation preserves the localization in z with a centrally-symmetric deformation and their background energies are uniformly escalated like a step function inboth components. Also, unlike in the case of solitons, here the kink nonlinearity does not induce anyamplification or compression of the localized excitations, and we have shown such a kink modulatedbreather in Fig. 3(c).Another kind of hyperbolic nonlinearity σ = b + b sech( b z + b ) having bell-type localizedform also not a ff ect the localization of the Akhmediev breather in the present system. However, italters the nature of periodic structures from uniform breathing to a centrosymmetric double-peakedbreather, as shown in Fig. 3(d) for one class of nonlinearity parameters b j (cid:44)
0. For a di ff erentset of parameters ( b =
0) one can witness the algebraically localized rogue-wave-type excitonswith sustaining side-band tails along z . Additionally, the single-hump and double-dip nature of theAkhmediev breathers are substantially preserved. Opposite to the Akhmediev breathers given in the previous part, On the other hand, for τ >
0, onecan have another form of breathing structure from the general wave solution (6). They are periodicalong the propagation direction z and localized in the transverse coordinate t and are usually referredas Ma breather or Kuznetsov-Ma soliton. In a mathematical sense for this choice, Eq. (6) reduces to9he following simple form: q = (cid:15) √ σ ( s − a ) cos[4 ab Z ] + iab sin[4 ab Z ] s cos[4 ab Z ] − a cosh[2 b T ] e i ( ξ − s Z ) , (9a) q = (cid:15) √ σ sa cosh[2 b T ] − a cos[4 ab Z ] + iab sin[4 ab Z ] s cos[4 ab Z ] − a cosh[2 b T ] e i ( ξ − s Z ) , (9b)where the form of ξ , T , and Z are as defined in Eq. (4) while b = | a − s | . Similar to the Akhmedievbreathers, the Ma breathers also exhibit both bright-dark and dark-dark type single-hump / single-welland double-well localized structures for appropriate choices of s and a as shown in Table 1. Still,the localization and periodic directions get exchanged here. For illustrative purposes, we portraybright-dark Ma breathers in Fig. 4(a) for the constant nonlinearity σ ( z ) = σ = b + b sin( b z + b ) modulates the background ofthe breathers by introducing oscillations and redistributes their peak intensity. The newly developedbackground oscillations continuously decrease and will vanish away as t → ±∞ in the q component.However, in the q component, those background oscillations are stable and sustain even when t →±∞ due to the non-zero initial intensity in dark or gray breathers as evidenced from Fig. 4(b). Anotherfeature is that the peak intensity oscillations repeat periodically along the propagation direction z . Asusual, the arbitrary parameters b , b , b , b , (cid:15) and (cid:15) do help in manipulating the obtained Mabreathers. (a) (b) (c) (d) Figure 4: Dynamics of modulated Ma breathers for s = .
75 and a = . b = . b = b = b = .
0, (b) b = . b = . b = .
75 & b = .
02; (c) b = . b = . b = .
75 & b = .
02, and(d) b = . b = . b = .
5, & b = .
02 with other values fixed as (cid:15) = .
25, and (cid:15) = . Next, the kink-like nonlinearity acts as a simple intensity amplifier with cascaded compression10n Ma breathers and is likely to be similar to the behaviour observed in inhomogeneous solitons(7). Here the breathing structures reemerge after a step-amplification from an initial stable profilewithout any change in their localization t and retain their periodic oscillation of intensity along z in both components for bright as well as dark Ma breathers, refer Fig. 4(c). Here the period ofbreathing oscillations and amplitude of peaks / dips get modulated and they can be tuned by using b j ,and (cid:15) j parameters. The third type of nonlinearity, bell-type σ , displays all the three e ff ects in theMa breathers as well starting from the tunneling ( b > b <
0) and localized exci-ton formation ( b =
0) based on the b j , j = , , , ff ect in Fig. 4(d), which preserves the uniform breathing patterns with a mod-ified / increased period of oscillations. It induces a localized maximum intensity at the barrier in bothcomponents, while it generates long-lasting sideband tails only in the dark q mode. Similarly, thecross-over and exciton formation can also be observed, which also preserves the nature of breatherswith a substantial change in the amplitudes. Rogue waves are doubly-localized (both in space and time or both in propagation and transversedirections) structures with extreme amplitudes over a continuous / constant background. It is evidentfrom solution (6) that the present CCNLS system (1) with modulated nonlinearity admits both brightand dark rogue waves of single-hump / single-dip and double-dip profiles for a particular choice ofparameters ‘ | s | ≈ | a | ’ as shown in the Table 1 and as demonstrated in Fig. 5. One can deduce anexplicit mathematical form for rogue waves from either Akhmediev breathers (8) or Ma breathers (9)with a condition that τ → b → ff erence in the present solutionis that the rogue waves do not have side-band tails, which makes it look like standard lumps / wells.Also, the amplitude / depth of the bright / dark rogue wave shall be controlled by the parameter s .(a) (b) (c) (d) Figure 5: Dynamics of (a-b) bright-dark rogue waves for s = .
75 and (c-d) dark-dark rogue waves for s = − .
75 forconstant nonlinearity σ = a = . As the name suggests, the periodically varying nonlinearity of the form σ = b + b sin( b z + b )breaks the doubly localized rogue waves into periodic one akin to breathers. Especially, the zero-background bright rogue wave admits a train of peaks with high amplitude in the center and decreasesin either direction along z . Along the t direction, initially, there appear small-amplitude tails whichfurther decrease and finally vanish away. Instead of a single-hump, the nonlinearity modulates it toexhibit multi-periodic A-shaped humps and vanishing thereafter. On the other hand, in dark roguewave ( q component), it induces periodic W -shaped structures around the center. As z increases, the11 -shape becomes a constant amplitude and then periodic stable waves as shown in Fig. 6(a). In oneway, this may be similar to a periodic modulation of the Ma breathers, but in the earlier case, it isperiodically repeating, and here it is not so. Further, by tuning the parameters a , s , b j , and (cid:15) j , theperiodicity, amplitude, and width shall be manipulated appropriately.(a) (b) (c) (d) Figure 6: Dynamics (a) A-shaped & M-shaped periodic wave trains, (b) localized amplification with compression, (c)tunneling, (d) exciton with side band formation of bright-dark rogue waves due to modulated (a) periodic, (b) kink-like,and (c-d) bell-type nonlinearities for s = .
75 and a = . b = . b = . b = .
5; (b) b = . b = .
5, & b = .
25; (c) b = . b = .
25, & b = .
5, and (d) b = . b = .
5, & b = . b = . (cid:15) = .
25 and (cid:15) = . The kink-like nonlinearity σ = b + b tanh( b z + b ) given by Eq. (5b) leads to initial widen-ing of the doubly-localized rogue waves with lesser amplitude and focuses it around the switch-ing / amplification regime leading to enhanced intensity in both components. Then both localizedwave intensity as well as the background energy got increased and remain stable throughout withoutany emergence of periodic structures, as portrayed in Fig. 6(b). It is very clear that the localiza-tion is preserved after the amplification of both bright and dark rogue waves. Compared to this kinknonlinearity, the bell-type nonlinearity shows more promising evolution which includes the alreadyintroduced tunneling Fig. 6(c) and exciton formation Fig. 6(d) in addition to the cross-over e ff ect. Asthese e ff ects are well described in the previous cases, here we refrain from repeating the same discus-sion. In a general picture, it seems that the modulation due to nonlinearities in the Ma breathers andthe rogue waves similar. However, more careful analysis brings out the di ff erences between them.Especially, the nature of modulated waveforms is di ff erent in both cases except for the respectivelocalization preservation. Further, these two localized waves show a completely di ff erent pattern ofmodulations compared to that of the Akhmediev breathers. The above discussions have provided a deeper insight into the role of various modulated nonlin-earities and their significance on the nonlinear coherent structures in the system (1). To be specific,12hese modulated nonlinearities that can be tuned suitably by b j parameters influence the stationarysolitons as well as bright, dark / gray, bright / gray-dark / gray type Akhmediev breathers, Ma breathers,and rogue waves. Still, some interesting hidden factors influencing these coherent structures are yetto be addressed. An important point to be noted here is that the significance of the two arbitraryparameters (cid:15) and (cid:15) available under the similarity transformation. Particularly, (cid:15) provides the pos-sibility of determining the inclination / angle of the localized structures without altering any of theirother identities. But, (cid:15) parameter supports the change of inclination at a very small level comparedto that of (cid:15) . However, the change in (cid:15) highly a ff ects the amplitude and helps in making the wave toa desirable width, which is inversely proportional to the amplitude. So, one can readily transform therogue waves to a wider (or narrower) one with smaller (or larger) amplitude based on the requirement.For illustrative purposes, we have shown the change in the inclination of bright-dark rogue waves inFig. 7(a-c). A similar property can be observed in the Akhmediev breathers, which remain localizedin z and induces inclination / angle of the single-hump / double-well structures, as evidenced from Fig.7(d). (a) (b) (c) (d) Figure 7: Change in the inclination / angle of single-hump bright and double-well dark rogue waves for di ff erent (cid:15) and (cid:15) parameters. Their values are (a) (cid:15) = .
25 and (cid:15) = − .
75, (b) (cid:15) = .
25 and (cid:15) = .
75, and (c) (cid:15) = .
35 and (cid:15) = . (cid:15) = .
25 and (cid:15) = − .
75 with other parameters as a = . s = − . b = b j = , j = , ,
3, but their localizationremain undisturbed.
Interestingly, by utilizing these (cid:15) j parameters, we can transform the stationary solitons to the trav-eling (left or right moving) solitons with appropriate amplification of intensity accompanied by com-pression / expansion and they influence the central position and velocity of the associated solitons too.We have depicted such propagating degenerate solitons in Fig. 8 for further insight. Further, from the t -localized Ma breathers, one can also obtain a set of generalized breathers that are neither localizedin propagation direction not transverse direction, and they propagate with specific velocity definedby the parameters. To be precise, localization-breaking ( t -localized to non-localized) breathers are13chieved by (cid:15) and (cid:15) parameters, which we have demonstrated such breather phenomenon in Fig. 9for a better understanding. Apart from the above, the explored features such as velocity-controlledsolitons and breathers, changing the angle of the peak / hole intensities rogue waves and Akhmedievbreathers, intensity as well as periodicity tailoring in solitons and Ma breathers, shall be e ff ectivelyemployed in the respective type of above discussed modulated nonlinearities by tuning (cid:15) and (cid:15) .(a) (b) (c) Figure 8: Traveling degenerate bright solitons for di ff erent choices of (cid:15) j parameters with s = a = . b = b = b = b =
0. (a) Left moving solitons for (cid:15) = .
25 and (cid:15) = .
75, (b) Right moving solitons for (cid:15) = .
25 and (cid:15) = − .
75, and (c) Intensity increasing and compressed soliton with velocity change for (cid:15) = .
35 with (cid:15) = . (a) (b) (c) Figure 9: Transforming bright-dark type Ma breathers into traveling general (non-localized) breathers for di ff erent choicesof (cid:15) j parameters. (a) Left moving breathers for (cid:15) = .
25 and (cid:15) = .
75, and (b) Right moving breathers for (cid:15) = . (cid:15) = − .
75, (c) Right moving general breathers with Intensity / oscillation increase and compression with a smallchange in the velocity for (cid:15) = .
35 with (cid:15) = − .
75. Other parameters are chosen as a = . s = . b = b = b = b = As a future study, the present investigation shall be extended to general solitons and their col-lisions with appropriate modulated nonlinearities which is reported recently [38]. Further, one canalso study the dynamics of various nonlinear coherent structures, their co-existence and dynamics indispersion as well as nonlinearity management in various systems, for example [39, 40], by incorpo-rating gain / loss too. Possible outcomes will be reported separately with categorical analysis.14 . Conclusions In this work, we have considered a coherently coupled nonlinear Schr¨odinger system consistingof modulated self-phase modulation, cross-phase modulation, and four-wave mixing nonlinearitieswith a varying refractive index. We have obtained a general two-parameter localized wave solutionby using an appropriate similarity transformation and following the Ref. [21]. By adopting threedi ff erent types of varying nonlinearities, namely periodic, kink-like, and bell-type functions, we haveexplored the manipulation mechanism of the resultant nonlinear wave structures. Notably, we foundthat through proper engineering of parameters, the present solution manifests into various forms oflocalized waves ranging from stable degenerate solitons to Akhmediev breathers, Ma breathers, anddoubly-localized rogue waves of the bright, gray and dark type. Additionally, the modulated nonlin-earities revealed various phenomena such as periodically oscillating waves (from localized to periodicwaves), amplification of localized waves with cascaded compression, tunneling as well as cross-overof the excited barrier / well, and localized exciton formation with sideband tails. Further, the con-trolling mechanism of individual identities of each localized waves like velocity, amplitude, centralposition, and orientation / angle / inclination, in addition to localization-breaking features resulting ingeneral breathers from Ma breathers are possible with the considered type of varying nonlinearityparameters. The results presented in this work will be applicable to the studies on engineering local-ized waves like solitons, breathers, and rogue waves as well as to various experimental investigationson the controlling mechanism of rogue waves in optical systems, cavity soliton based fiber lasers,intracavity and pump waves in polarization controlled waveguides, atomic condensates, deep wateroceanic waves, and other related coherent wave systems that can be implemented with nonlinearitymanagement. Acknowledgement : The work of KS was supported by Department of Science and Technology -Science and Engineering Research Board (DST-SERB), Govt. of India, sponsored National Post-Doctoral Fellowship (File No. PDF / / / / References [1] Whitham G B 1974
Linear and Nonlinear Waves (Wiley, New York).[2] Kivshar Y S and Agrawal G P 2003
Optical Solitons: From Fibers to Photonic Crystals (Academic Press, SanDiego).[3] Akhmediev N, Ankiewicz A and Taki M 2009 Waves that appear from nowhere and disappear without a trace,
Phys.Lett. A
Nature
JOSA B Phys. Rev. Lett. Nature Phys. Rogue and Shock Waves in Nonlinear Dispersive Media (Springer, NewYork).[9] Dudley J M, Genty G, Mussot A, Chabchoub A and Dias F 2019 Rogue waves and analogies in optics and oceanog-raphy
Nature Rev. Soliton Management in Periodic Systems (Springer-Verlag, New York).[11] Park Q-H and Shin H J 1999 Painlev´e analysis of the coupled nonlinear Schr¨odinger equation for polarized opticalwaves in an isotropic medium
Phys. Rev. E
12] Kanna T, Vijayajayanthi M and Lakshmanan M 2010 Coherently coupled bright optical solitons and their collisions
J. Phys. A: Math. Theor. J. Phys. A: Math. Theor. J. Math. Phys. Phys. Lett. A
EPL
Nonlinear Dyn . Phys. Scr. Nonlinear Dyn . Appl. Math. Lett.
Commun. Nonlinear Sci. Numer. Simul. Phys. Rev. E Appl. Math. Lett. Proc. R. Soc. A ff ect Int. J. Mod. Phys. B Optik
Results Phys. + ffi cients in a graded-Index waveguide Commun. Theor. Phys. Z. Naturforsch. A ffi cients Appl. Math. Comput.
Nat. Commun. Phys. Rev. X ff oli A, Randoux S, Suret P and Onorato M 2018 Spontaneous emergence of roguewaves in partially coherent waves: A quantitative experimental comparison between hydrodynamics and optics Phys. Rev. E Proc. R. Soc. A
Adv. Photonics Sci. Rep. Phys. Lett. A
J. Phys. A: Math. Theor. https: // doi.org / / / abae3f[39] Rao J, Porsezian K, Kanna T, Cheng Y and He J 2019 Vector rogue waves in integrable M-coupled nonlinearSchr¨odinger equations Phys. Scr. Nonlinearity4090.