Manipulation of vector solitons in a system of inhomogeneous coherently coupled nonlinear Schrödinger models with variable nonlinearities
MManipulation of vector solitons in a system of inhomogeneouscoherently coupled nonlinear Schr ¨odinger modelswith variable nonlinearities
R. Babu Mareeswaran a , K. Sakkaravarthi b,c, ∗ , T. Kanna d, ∗ a Department of Physics, PSG College of Arts and Science, Coimbatore–641 014, India b Department of Physics, National Institute of Technology, Tiruchirappalli – 620 015, Tamil Nadu, India c Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli – 620 024, 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 investigate non-autonomous solitons in a general coherently coupled nonlinear Schr¨odinger (CC-NLS) system with temporally modulated nonlinearities and with an external harmonic oscillator po-tential. This general CCNLS system encompasses three distinct types of CCNLS equations that de-scribe the dynamics of beam propagation in an inhomogeneous Kerr-like nonlinear optical mediumfor di ff erent choices of nonlinear polarizations owing to the anisotropy of the medium. We identifya generalized similarity transformation to relate the considered model into the standard integrablehomogeneous coupled nonlinear evolution equations with constant nonlinearities, accompanied bya constraint relation expressed in the form of the Riccati equation. With the help of a non-standardHirota’s bilinearization method and exact soliton solutions, we explore the impact of varying non-linearities and refractive index in the propagation and collisions analytically by reverse engineering.Interestingly, we show the emergence of several modulated solitonic phenomena such as periodic os-cillation, amplification, compression, tunneling / cross-over, excitons, as well as their combined e ff ectin the single-soliton propagation and two-soliton collisions with appropriate forms of nonlinearity.Notably, we identify a tool to transform the nature of soliton collisions with certain type of inho-mogeneous nonlinearities. The results could be of significant interest to the studies on managementof nonlinear waves in the contexts like nonlinear optics and can also be extended to Bose-Einsteincondensates and super-fluids. Keywords:
Nonlinear optics; Inhomogeneous system; Coupled nonlinear Schr¨odinger equations;Vector solitons; Variable nonlinearity; Soliton management.————————————————————————————————————————
Reference:
Journal of Physics A: Mathematical and Theoretical (2020). https://doi.org/10.1088/1751-8121/abae3f
1. Introduction
Nonlinear wave phenomena have a deep physical and mathematical interest as they arise in awide landscape ranging from science to engineering and technology such as nonlinear optics, fluiddynamics, plasma physics, lattice dynamics, and Bose-Einstein condensates (BECs) [1–4]. In non-linear dynamical systems, introducing inhomogeneous and non-autonomous nonlinearities showcasedistinct dynamical behaviour of nonlinear waves that find applications in optical communication, ∗ Corresponding authors.Email address: babu nld@redi ff mail.com; [email protected]; kanna [email protected] Preprint submitted to
Journal of Physics A: Mathematical and Theoretical a r X i v : . [ n li n . PS ] A ug ater waves, and Bose-Einstein Condensates (BECs) [5–11]. Along this direction, theoretical andexperimental investigations have been carried out during the past few decades. Recently, the spatiallymodulated Kerr nonlinearity is observed experimentally in nonlinear optics [7, 8]. In the context ofBECs, the spatial (or) time modulated nonlinearity is achieved through the Feshbach resonance mech-anism with a non-uniform magnetic field [12]. Various reports on these types of inhomogeneous andnon-autonomous systems are available in the literature.Among several types of nonlinear waves, solitons found ever increasing interest due to their re-markable stability and intriguing collision dynamics. Further, they have multifaceted applicationsin almost all areas of science and technology [13]. Solitons emerging from nonlinear Schr ¨odinger(NLS) type equation with temporally varying / distributed coe ffi cients (dispersion / nonlinearity), so-called non-autonomous soliton, found important advancements in the context of optical communica-tion systems. Nowadays, these non-autonomous solitons play an important role in optical fibre system(see [14–19] and reference therein). In the pioneering works [20, 21], the concept of non-autonomoussoliton was first introduced within the framework of the NLS system with variable dispersion andnonlinear coe ffi cients and it has been show that the amplitude, widths, velocity and central positionof soliton was completely a ff ected by varying management parameters (dispersion and nonlinearity).The bright and dark spatial self-similar solitons in graded-index fiber with linear refractive indexhave been investigated [22]. In ref. [23, 24], authors have studied the dynamics of bright soliton in adispersion managed erbium doped inhomogeneous fibre with gain / loss. Especially, the existence ofbright-dark dispersion managed soliton with randomly varying birefringence has been investigated[25]. The dynamics of spatio-temporal light bullets in three-dimensional nonlocal NLS system withvariable coe ffi cients have also been studied [26] and it shows that intensity, width, phase and the chirpof light bullets are strongly modified by dispersion and nonlinearity coe ffi cients. Specifically, thespatio-temporal multi-soliton solutions with and without continuous wave backgrounds in the NLSequation with variable dispersion and nonlinearity coe ffi cients have been obtained in [27]. Moreover,the snake-like nonautonomous solitons in planar grating waveguides have been investigated [28, 29]and reported the control of the shape preserved soliton’s motion on the graded-index waveguides. Re-cently, the shape of the dissipative dispersion-managed solitons in optical fiber systems with lumpedamplification have been studied both experimentally and numerically [30]. Apart from the solitons,analysis on the exact periodic traveling wave and soliton pair (bright-dark and dark-dark) solutions ofthe coupled NLS equations with harmonic potential and variable coe ffi cients has been reported [31].The e ff ects of the periodically modulated nonlinearity on the soliton propagation and interaction ina dispersion-managed birefringence system is also of much importance [32–34]. Analytical vectornon-autonomous soliton solutions for the coupled NLS with spatially modulated coe ffi cients and co-herent coupling were studied [35]. Particularly, the phase dynamics of bright and dark solitons invariable coe ffi cient coupled NLS equation was reported [36]. More recently, the evolution and stabil-ity of bright vector soliton in coupled Ginzburg-Landau equation with variable coe ffi cients has beenstudied [37]. This clearly indicates that study of non-autonomous solitons is of much importance notonly in one dimensional NLS type systems but also in higher dimension and dissipative nonlinearsystems as well as multi-component optical systems such as multi-mode fibers.2ased on the above interesting works, in this work, we give a detailed analysis of soliton man-agement in a general coupled NLS type system with variable Kerr nonlinearity and linear parabolicrefractive index profile arising in the context of polarized light beam propagation in low birefringentnonlinear optical media. Here the relative phase factors of the co-propagating electric fields lead to theonset of fourwave mixing e ff ects. This coupling arises naturally in weakly anisotropic or birefringentmedia. The following system of coherently coupled nonlinear Schr¨odinger (CCNLS) equations (indimensionless form) [1, 38] describes such type of beam propagation in Kerr-like nonlinear medium: i ∂ A ∂ z + (cid:32) δ ∂ ∂ x + υ ( x , z ) (cid:33) A + γ ( z )[ σ | A | + σ | A | ] A − δ γ ( z ) A A ∗ = , (1a) i ∂ A ∂ z + (cid:32) δ ∂ ∂ x + υ ( x , z ) (cid:33) A + γ ( z )[ σ | A | + σ | A | ] A − δ γ ( z ) A A ∗ = , (1b)where A j ( j = ,
2) are the slowly varying envelopes of the electric fields associated with two or-thogonally polarized components. In Eq. (1), x and z are the transverse and longitudinal coordinates,respectively, while the asterisk ( ∗ ) denotes the complex conjugation and δ represents the group veloc-ity dispersion coe ffi cient [39]. Further, γ ( z ) is the variable nonlinear parameter, the z -dependence ofwhich stems from the inhomogeneity of the optical medium [5, 7] and υ ( x , z ) is the graded refractiveindex profile in the form [40]. Eq. (1) contains phase-independent incoherent nonlinearities givenby σ i j , i , j = ,
2, where σ and σ represent the strengths of self-phase modulation (SPM), while σ and σ denote the strengths of cross-phase modulation (XPM). Additionally, the four-wave mix-ing (FWM) nonlinearity strength is represented by the coe ffi cients δ and δ of the phase-dependentcoherent coupling. Here one can have a di ff erent consideration with varying dispersion e ff ect or dis-persion management δ ( z ) as well as spatio-temporal modulated nonlinearities γ ( x , z ) in the CCNLSsystem (1), which deserve a separate intensive investigation and shall be reported shortly. For a spe-cial set of parameters ( σ = σ = δ j =
0) along with constant nonlinearities and in the absenceof graded refractive index, the system (1) reduces to the standard scalar NLS equation for the twoseparate modes A and A , namely iA j , t + δ A j , xx + γ | A j | A j = , j = ,
2, whose various nonlinearwave solutions including solitons, breathers, rogue waves, periodic waves under di ff erent physicalsituations have been extensively studied. Further, when the FWM nonlinearity is absent, the solitondynamics for focusing and mixed type nonlinear Schr¨odinger modles were investigated with explicitsoliton solutions and asymptotic analysis exploring the energy-sharing and elastic collisions of brightsolitons [41–45].Moreover, it is important to mention that Eq. (1) is a general version of inhomogeneous CCNLSsystem. Interestingly, it results into six di ff erent models for appropriate choices of dispersion andnonlinearity coe ffi cients. The inhomogeneous model (1) with constant nonlinearities and refractiveindex describes beam propagation in uniform / homogeneous Kerr-like nonlinear medium reads as i ∂ A ∂ z + δ ∂ A ∂ x + γ (cid:16) σ | A | + σ | A | (cid:17) A − δ γ A A ∗ = , (2a) i ∂ A ∂ z + δ ∂ A ∂ x + γ (cid:16) σ | A | + σ | A | (cid:17) A − δ γ A A ∗ = . (2b)3he above Eq. (2) is non-integrable in general and shall pass the integrability test only for certainchoices of coe ffi cients [46]. As mentioned earlier, the above generalized two-component CCNLSequation shall take di ff erent versions for various choices of nonlinearity and dispersion coe ffi cients,as given below.Model (i) [47]: δ = − , σ = σ = − , σ = σ = − , δ = δ = , (3)Model (ii) [48]: δ = , σ = σ = , σ = σ = , δ = δ = , (4)Model (iii) [50]: δ = − , σ = σ = − , σ = σ = − , δ = δ = − , (5)Model (iv) [49]: δ = , σ = − σ = , σ = − σ = − , δ = − δ = , (6)Model (v) [51]: δ = , σ = σ = − , σ = σ = − , δ = δ = , (7)Model (vi) [51]: δ = , σ = σ = , σ = σ = , δ = δ = − . (8)Among the above six versions, models (v) and (vi) correspond to the choices (7) and (8) are equiv-alent to (3) when x → ix and z → − z , respectively, while the model (iii) arising for the choice(5) can be reduced to model (ii) resulting for (4) when z → − z . Further, based on the impact ofnonlinearities, models resulting for the choices (3), (7) and (8) shall be defined as CCNLS systemswith positive coherent coupling, while that of the choices (4) and (5) are as CCNLS systems withnegative coherent coupling, and the choice (6) is designated to the CCNLS system with mixed typenonlinearities. Thus, e ff ectively there exist only three distinct models (3), (4), and (6) representingthe coherent propagation of two orthogonally polarized modes featuring di ff erent SPM, XPM, andfour-wave mixing nonlinearities. The exact bright soliton solutions of these models (3), (4), and(6) were constructed by employing a non-standard way of Hirota’s bilinearization method in Refs.[47], [48], and [49], respectively, along with a detailed study on the propagation and collision dy-namics of these solitons. Particularly, the solitons were classified as coherently coupled solitons andincoherently coupled solitons based on the presence and absence of the phase-dependent (four-wavemixing) nonlinearity, respectively, exhibiting single-hump, double-hump and flat-top profiles. Fur-ther, an interesting energy-switching collision of bright coherent-incoherent solitons was investigatedin addition to their energy-sharing and elastic collisions [47–49].On the other hand, equations similar to (3-8) can also arise in the context of BECs governing thedynamics of spinor condensates when the spin-mixing nonlinearity plays a crucial role. For example,soliton solutions and their interactions of an autonomous and non-autonomous spin-1 condensatesystem usually referred as three-coupled Gross-Pitaevskii equations were obtained in Refs. [52–57] and have been classified as interesting ferromagenetic and polar solitons, based on the e ff ect ofspin-mixing nonlinearity. This three-component system reduces to the two-component CCNLS typesystem when we consider pseudo-spinors and is referred as degenerate CCNLS system [51]. A variantdegenerate coherently coupled spin system similar to Eq. (7-8) has been discussed in autonomousand non-autonomous settings [51]. The study unraveled various coherent structures through linearsuperposition and by varying nonlinearities and external potential.Motivated by the propagation and collision dynamics of solitons in homogeneous optical media[47–49] as well as the investigation of non-autonomous solitons in BECs [57], in this work, we fo-4us our investigation on the generalized CCNLS system (1) with varying nonlinearity and refractiveindex profile. The objectives are to construct exact soliton solutions with inhomogeneity and vary-ing nonlinearity coe ffi cients, which can be implemented through an appropriate similarity or lenstype transformation. Further, the propagation as well as collision dynamics of such inhomogeneoussolitons will be studied extensively with appropriate analysis and graphical demonstrations.The remaining part of this work is arranged in the following manner: The conversion of inho-mogeneous 2-CCNLS system (1) into a homogeneous version (2) with a similarity transformation isgiven in Sec. 2. along with the importance of considered varying nonlinearities. Section 3 consistsof the propagation dynamics of inhomogeneous solitons which enacts the manipulation mechanismof optical solitons through nonlinear optical fibers / communication systems. Further, various typesof inhomogeneous soliton collisions are presented in Sec. 4 with categorical analysis. Further, thepossibility and occurrence of inhomogeneous soliton bound states are discussed in Sec. 5. The finalsection 6 is devoted to summarize the important results along with certain future perspectives.
2. Transformation to the Integrable Homogeneous CCNLS Model
Solving inhomogeneous nonlinear models is comparatively di ffi cult, but quite possible, than thatof their homogeneous (constant parameter) counterparts. However, we require explicit solutions fora complete understanding of the considered system. This can be accomplished by two broad routes:directly solving the equations by retaining the variable coe ffi cients and transforming the equation intoa convenient model that can be exactly solved with various analytical methods.In this section, we adopt the second route by implementing a similarity transformation to extractexplicit solutions for Eq. (1). For this purpose, we apply the following similarity transformation toEq. (1): A j ( x , z ) = ρ ( z ) Q j ( X ( x , z ) , Z ( z )) exp[ i ζ ( x , z )] , j = , . (9)where ρ ( z ) is the amplitude, while ζ ( x , z ) is the phase and X ( x , z ) and Z ( z ) are the similarity variables,the explicit form of all these variables has to be determined. The above transformation (9) reducesEq. (1) into the following homogeneous CCNLS equation: iQ , Z + δ Q , XX + ( σ | Q | + σ | Q | ) Q − δ Q Q ∗ = , (10a) iQ , Z + δ Q , XX + ( σ | Q | + σ | Q | ) Q − δ Q Q ∗ = . (10b)The only di ff erence between Eqs. (2) and (10) is that the constant nonlinearity coe ffi cient γ takes afixed value of γ = ff erential equations(PDEs) for these unknown functions as given below. X xx = , X z + δ X x ζ x = , δζ x + ζ z − F ( z ) x = , (11a) Z z − ρ ( z ) γ ( z ) = , ρ z + δρ ( z ) ζ xx = , X x − ρ ( z ) γ ( z ) = . (11b)5olving the above PDEs successively, we find the following expressions with auxiliary real arbitraryconstants (cid:15) and (cid:15) : ζ ( x , z ) = − δ ddz (ln γ ) x + (cid:15) (cid:15) γ x − δ(cid:15) (cid:15) (cid:90) γ dz , (12a) X ( x , z ) = (cid:15) (cid:32) γ x − δ(cid:15) (cid:15) (cid:90) γ dz (cid:33) , (12b) Z ( z ) = (cid:15) (cid:90) γ dz , (12c) ρ ( z ) = (cid:15) (cid:112) γ ( z ) , (12d) F ( z ) = δ (cid:32) γ z γ − γ zz γ (cid:33) . (12e)Note that the external potential and nonlinearity shall not be independent, either one can be arbitraryfunction of z , while the other admits suitable form through Eq. (12e). This condition can also berewritten in a more convenient form of Riccati equation Y z − Y + δ F ( z ) =
0, where Y = γ z /γ . Now,with the identified explicit similarity transformation (9) and corresponding variables (12), one caneasily construct exact nonlinear wave solutions of inhomogeneous model (1) when we are able toprovide the respective solutions for constant parameter CCNLS equation (10).The above mentioned similarity transformation and the varying nonlinearities can be adopted forany inhomogeneous nonlinear systems, in the present case for all versions of the CCNLS models.Hereafter, we explore their impact in the propagation and collision dynamics of bright solitons ofthese CCNLS systems by constructing their explicit solutions obtained by using a non-standard typeof Hirota’s bilinearization method. We refrain from presenting the detailed procedure here and onecan refer to [47–49] for the systematic construction as well as the analysis on homogeneous solitons.Especially, we consider the three distinct versions (3), (4), and (6) one by one.
3. Inhomogeneous bright one-soliton
In order to achieve the first objective, understanding the role of varying nonlinearities on solitonpropagation, we construct explicit soliton solutions by adopting the Hirota’s bilinearization methodwith an auxiliary function to homogeneous form (2) [47–49] and deduce solutions of the inhomoge-neous equations (1) under investigation, using (9). First of all, the bilinearizing transformation andgeneralized bilinear forms of CCNLS equation (2) can be written as,Bilinear transformation ⇒ Q = GF , Q = HF , (13a)Bilinear equations ⇒ ( iD Z + δ D X ) G · F = δ S G ∗ , (13b)( iD Z + δ D X ) H · F = δ S H ∗ , (13c) δ D X F · F = | G | + δ | H | ) , (13d) S · F = G + δ H , (13e)6here G , H and S are complex functions, while F is a real function and D represents the standardHirota derivative [58] of the respective independent variables Z and X . The above bilinear forms areapplicable to the general CCNLS model (1) which includes all versions given by (3-8) for respectivechoices. When we apply the bilinearizing transformation (13a) to the three distinct versions of CC-NLS models (3), (4) and (6), we shall get three di ff erent sets of bilinear forms. By combing thosethree forms, we have written in the above form (13) in a conenient way. By following the standardprocedure, we can construct soliton solutions after expanding the dependent functions as power seriesand then by recursively solving the resultant ordinary di ff erential equations arising at di ff erent ordersof expansion parameters [47–49].The generalized bright one-soliton solution of the inhomogeneous CCNLS system (1), especially,to the three distinct CCNLS Eqs. (3), (4), and (6), can be obtained as A j ( x , z ) = (cid:15) √ γ (cid:16) α ( j )1 e η + e η + η ∗ + δ ( j )11 (cid:17) + + e η + η ∗ + R + e η + η ∗ + (cid:15) ( j )11 e i ζ ( x , z ) , j = , , (14a)where e δ (1)11 = α ( j ) ∗ S k + k ∗ ) , e δ (2)11 = δδ α (2) ∗ S k + k ∗ ) , e R = | α (1)1 | + δ | α (2)1 | ( k + k ∗ ) , e (cid:15) = | S | ( k + k ∗ ) , (14b) η = k ( X + i δ k Z ) , X = (cid:15) (cid:32) γ x − δ(cid:15) (cid:15) (cid:90) γ dz (cid:33) , Z = (cid:15) (cid:90) γ dz . (14c)The above soliton solution is obtained with an auxiliary function S = S e η , which plays an im-portant role in the classification of the respective solitons as coherently- and incoherently-coupledsolitons, and it takes the form S = ( α (1)1 ) + δ ( α (2)1 ) . (14d)The above solution is applicable to all the three inhomogeneous CCNLS models (1) correspondingto (3), (4), and (6) with appropriate choices of dispersion ( δ ), incoherent nonlinearities ( σ i j ), andcoherent nonlinearity ( δ j ). From the above general one-soliton solution (14), one can easily understand that the choice S = e δ (1)11 and e (cid:15) vanish. This results into a simple form corresponding to that of Manakov typesolitons without the contribution from coherent nonlinearity, which can be designated as inhomoge-neous incoherently coupled solitons (IICSs). Such type of inhomogeneous soliton can be casted in astandard hyperbolic form as A j ( x , z ) = B j sech( η R + R / e i ( ζ + η I ) , j = , , (15)where B j = α ( j )1 (cid:15) (cid:112) γ ( z ) e − R / , η R = k R ( X − δ k I Z ), η I = k I X + δ ( k I − k R ) Z , and other parameters ζ , X and Z are as given in Eq. (14). This IICS admits well-known bell-type / symmetric-single-hump7rofile with certain amplitude B j , width (proportional to inverse of amplitude), and velocity 2 δ k I ofpropagation represented by the above form. Here we should note that the above IICS is possible forthe models (3) and (4), while it results in singular solutions in system (6) as the restriction S = In addition to the above special / restricted IICS (15), the general bright one-soliton solution (14)usually contains the contribution from both coherent and incoherent nonlinearities ( S (cid:44) A j ( x , z ) = C j (cid:32) cos( P j ) cosh ( Q ) + i sin( P j ) sinh( Q )4cosh ( Q ) + L (cid:33) e i ( ζ + η I ) , j = , , (16)where C j = (cid:15) √ γ e lj + δ ( j )11 − (cid:15) , P j = e δ ( j )11 I − ljI , l j = ln( α ( j )1 ), Q = η R + (cid:15) , L = e ( R − (cid:15) ) − η R = k R ( X − δ k I Z ), η I = k I X + δ ( k R − k I ). Here X and Z are nonlinearity dependent coordinates asgiven in Eq. (14). When we analyze the above form, it is clear that ICCSs admit di ff erent profilesranging from double-hump and flat-top structures including an asymmetric single-hump as well (butnot in exact sech form) for P j (cid:44) S (cid:44) B j and C j in j -th mode respectively are strongly influenced by the varying nonlinearity γ ( z ). Additionally, thevelocity, position and phase of these solitons are also altered by nonlinearity γ ( z ), which are not at allpossible in homogeneous solitons where they depend only on the wave vectors k . So, by properlychoosing the arbitrary nonlinearity parameter, one can engineer the resultant solitons, so that theycan be utilized for a wider range of applications. Here, we concentrate on some simple solitonmanagement mechanisms such as amplification, compression, oscillation, and tunneling of solitonswith appropriate forms of nonlinearities. As mentioned below Eq. (12), the nonlinearity γ ( z ) and varying refractive index profile F ( z ) (ul-timately v ( x , z )) are mutually dependent. So, throughout our study, we consider a set of interestingnonlinearity functions and explain how they play crucial role in the evolution of solitons in the con-sidered system. Here, we choose the following forms of nonlinearity function as Jacobian elliptictype function (correspond to soliton lattices) and exponential type: γ ( z ) = γ + γ sn( z , m ) , < m < , (17a) γ ( z ) = γ + γ cn( z , m ) , < m < , (17b) γ ( z ) = γ + γ exp( γ z ) , (17c)where γ , γ , and γ are arbitrary real constants. From the above nonlinearities, one can obtainexplicit form of F ( z ) from Eq. (12e) which show the nature of varying refractive index profile v = x /
2. We portray the variation of these nonlinearities γ ( z ) and potential function F ( z ) with respect to‘ z ’ in Fig. 1. Especially, for di ff erent values of elliptic modulus parameter m in the nonlinearities, theysmoothly transfer from a periodic profiles (for m =
0) to a step-like and localized-hump structures(for m =
1) in turn alters the refractive index profile too as depicted in the right panel of Fig. 1. Suchsmooth function of nonlinearities are reasonable candidates for experimental implementation [59].
Figure 1: Nature of nonlinearity parameter γ ( z ) and F ( z ) for di ff erent choices. (i) tanh: m = m = m = m = γ = .
12 in (17c) with other values as γ = . γ = . By adopting the above types of nonlinearities (17), we investigate the evolution of inhomogeneoussolitons given by Eqs. (15) and (16) in the following part.
Among the considered nonlinearity functions (17), first two forms when m = γ ( z ) = γ + γ sin( z ) and γ ( z ) = γ + γ cos( z ). Mathematically,their implication with respect to z is well known, which is nothing but a periodically oscillating dy-namics with a phase-shift between them. In the present system, they a ff ect the nature of soliton prop-agation such as the amplitude, velocity, and position which become functions of γ ( z ) as evidencedfrom the explicit solutions (15) and (16). We have shown such traveling soliton in Figs. 2 and 3,where one can witness the periodically modulated / oscillating amplitude as well as direction / velocity.To be explicit the variation in the amplitude of IICSs can be represented as B j = α ( j )1 (cid:15) (cid:112) γ ( z ) e − R / ,while that of ICCSs is defined by C j = (cid:15) √ γ exp (cid:16) ( l j + δ ( j )11 − (cid:15) ) / (cid:17) , with other parameters asshown below Eqs. (15-16), which clearly depicts the significance of nonlinearity parameter γ in am-plitude. Further, the central position as well as the velocity of IICSs and ICCSs are described by k R (cid:15) (cid:104) γ x − δ(cid:15) ( (cid:15) (cid:15) + k I ) (cid:82) γ dz (cid:105) + R / k R (cid:15) (cid:104) γ x − δ(cid:15) ( (cid:15) (cid:15) + k I ) (cid:82) γ dz (cid:105) + (cid:15) , respectively.Here the periodic nonlinearity makes the propagation of these localized IICSs / ICCSs resembles thesnake-like pattern and it can also be referred as creeping soliton. Figure 2 depicts the modulation ofIICSs admitting single-hump profile with ‘sine’ and ‘cosine’ type nonlinearities. They have equal-intensity / energy in both components and the main di ff erence between these two nonlinearities is onlythe well-known phase-shift which can be identified from the substantial shift along ’ z ’. Another sig-nificant feature is that these can be manipulated with the available arbitrary parameters γ and γ . Byincreasing the values of γ we can control the “creeping” nature which enhances beating (oscillations)9 = γ + γ sn( z , γ = γ + γ cn( z , Figure 2: Propagation of single-hump IICSs with periodically oscillating intensity and position / velocity with ‘sine’ and‘cosine’ type nonlinearities resulting for m =
0. Both are symmetric to each other except a small shift along ’ z ’. Here theparameters are chosen as δ = (cid:15) = .
5, and (cid:15) = . γ = . γ = . k = + . i , α (1)1 = .
5, and α (2)1 = . i .Figure 3: Propagation of double-hump and flat-top ICCSs with periodically oscillating intensity and position / velocitywith varying nonlinearity γ = γ + γ sn( z ,
0) for α (2)1 = . i , while the other parameters as same as in Fig. 2. e ff ects in the intensity. Further, we have shown the nature of ICCSs (16) having double-hump andflat-top structures in A and A components, respectively, in Fig. 3 for ‘sine’ nonlinearity. A similardynamics can be observed for ‘cosine’ nonlinearity as well which are not shown here. The elliptic function nonlinearity becomes a pure hyperbolic one when m =
1, especially (17a) turnsout to be γ = γ + γ tanh( z ), and this influences a smooth transition of soliton identities similar tothat of a step-function. Soliton dynamics under such ‘tanh’ nonlinearity is shown in Figs. 4 and 5.Unlike in the previous case of periodic nonlinearities, here the amplitude, velocity, position and widthof the solitons are get modulated as a smooth step-like change. To be precise, the intensity of solitongets amplified with commensurate compression in its width (when γ >
0) so that the total energy10 igure 4: Controlled amplification and suppression of the intensity associated with pulse compression and widening ofsingle-hump IICSs soliton for γ = γ + γ tanh( z ) with γ = .
05 and γ = − .
05, respectively, while the other parametersare δ = (cid:15) = . (cid:15) = . γ = . k = + . i , α (1)1 = .
5, and α (2)1 = . i . is conserved during propagation. In contrast when γ <
0, the soliton under modulation becomesbroader with a suppression / decrease in its intensity. Further, the velocity of the soliton decreases(becomes slower) in the former while it travels faster (increases) in the latter, which consequentlychanges the actual position of the soliton at any given time. In one way, this clearly indicates theamplitude-independent velocity of solitons in the present system. For a better understanding, we havedemonstrated such compressed-amplification and widened-suppression of IICSs in Fig. 4, where theyadmit a standard single-hump profile. In the first case with γ >
0, the wider soliton having smallamplitude (at z = −
20) gets compressed and undergo significant enhancement in its amplitude (seeat z = γ <
0. Here the shorter / widersoliton travels faster while the taller / narrow soliton propagates slower. To be precise, the kink-likenonlinearity γ ( z ) influences a significant increase in the soliton intensity combined with compression.The nature of modulation in ICCSs under the kink nonlinearity is depicted in Fig. 5 which clearlyreveals the compressed-amplification of the solitons having double-hump and flat-top profiles withconsiderable change / reduction in its velocity that forces them to travel slower after the intensitygrowth. Such phenomenon can be utilized in soliton pulse-shaping dynamics. ff ects: Bell nonlinearity Next, we address an interesting concept of soliton tunneling. For this purpose, we consider a non-linearity of the form (17b) with m =
1, which leads to a ‘sech’ function of ‘ z ’ and in explicit formit can be written as γ = γ + γ sech( z ). This ‘sech’ type nonlinearity creates a localized structurewhich acts as a barrier. In the present ICCNLS system (10), it gives rise to tunneling of solitonsthrough the barrier as shown in Figs. 6 and 7. Here the parameter γ enables the formation of bar-rier in two categories, one with localized intensity peak (bell type for γ >
0) while the other withintensity dip (inverted bell type for γ < igure 5: Controlled amplification of the intensity associated with pulse compression of double-hump and flat-top ICCSswith γ = γ + γ tanh( z ) for the same choice of parameters given in Fig. 4 except for α (2)1 = . i . and velocity are preserved before and after tunneling. However, there occurs only a small phase-shiftdue to the tunneling dynamics. Further, when analyze the total energy of the traveling solitons, thereoccurs compression during the tunneling and widening during cross-over in the barrier regime, seeFig. 6. If we notice the figures clearly, there is no change in the amplitude and width of the soli-ton until it reaches the barrier and after a short-living compression / widening in the barrier regime,they reemerge with initial characteristics only with a phase-shift. For a complete understanding,we have also demonstrated the ICCSs exhibiting M-shape (double-hump) and flat-top structures fornonlinearity of the form γ = γ + γ sech( z ) in Fig. 7, there itself one can witness the identity preserv-ing tunneling propagation. Such type of tunneling mechanism is looking analogous to the quantumtunneling e ff ect with shape-preservation propagation nature beyond the barrier and they are also re-ferred to soliton spectral tunneling (SST) in the literature [60]. Such type of tunneling e ff ect appearin a wider context of science including matter wave tunneling in BECs, optical similariton tunnel-ing in photonic crystal fibers, tunneling of self similar optical rogue waves, etc. with an externalharmonic potential in both scalar and multicomponent nonlinear systems, see [36, 61–63] and refer-ences therein. These prescribe the possibility for observing phenomenon of tunneling experimentallywith localized compression. Another interesting observation is the appearance of a rogue-wave-likelocalized excitations with infinitely long tails when γ approaches zero, see Fig. 8. This phenomenoncan be understood in such a way that the solitons vanish along z and emerge / switch as long-lastingtails along x . In addition to the elliptic function nonlinearities, a classical exponential function also controls the dy-namics of soliton propagation in a straightforward rather significant way. Such type of nonlinearitiesenable the controllable ever-increasing energy of the associated waves. In our case, a simple exponen-tial nonlinearity given by (17c) increases the intensity of the solitons which is well substantiated with12 igure 6: Soliton tunneling and cross-over accompanied by phase-shift through the barrier given by the varying non-linearity γ = γ + γ sech( z ) for γ = .
05 and γ = − .
05, respectively, with δ = (cid:15) = . (cid:15) = . γ = . k = + . i , α (1)1 = .
5, and α (2)1 = . i .Figure 7: Tunneling of double-hump, and flat-top solitons for δ = (cid:15) = . (cid:15) = . γ = . + .
05 sech( z ), k = + . i , α (1)1 = .
5, and α (2)1 = . i . Solitons exhibit significant change in their phases after tunneling through thebarrier.Figure 8: Localized excitation of IICS (single-hump) and ICCSs (double-hump and flat-top solitons) at the barrier for thesame type of nonlinearity and arbitrary parameters given in Fig. 7, except for γ = . Figure 9: Exponential growth of (a) single-hump IICS and (b-c) ICCSs having double-hump and flat-top profilesassociated with compression for exponential (top panel) γ = γ + γ exp( γ z ) and combined (bottom panel) γ = γ sech z + γ exp( γ z ) nonlinearities. The choice of parameters are k = + . i , α (1)1 = .
5, with α (2)1 = . i forIICS and α (2)1 = . i for ICCS. The other parameters are δ = (cid:15) = . (cid:15) = . γ = . γ = .
05, and γ = . compression. For completeness, we have sown such modulation in Fig. 9 for γ = γ + γ exp( γ z ).Further, in the presence of superposed nonlinearities with exponential and sech functions, namely γ = γ sech z + γ exp( γ z ), one can understand the soliton growth as well as tunneling dynamics,which we have also depicted in Fig. 9. Additionally, we can observe a strange behaviour of solitongeneration from nowhere (for example before z < −
15) and the intensity picking up when it ap-proaches the barrier and increase exponentially thereafter. Another striking feature is that the widthof the solitons are greatly a ff ected by this combined nonlinearity function. A wider (single-hump ordouble-hump or flat-top) soliton is getting localized when it reaches the barrier and cross-over it, thengrow with a very narrow width.Here, we wish to remark the integrability and stability of our considered general inhomogeneousmodel (1). From the obtained similarity transformation (9), equation (1) is turned to become condi-tionally integrable in the form of equation (10) via Riccati equation (12e). As these inhomogeneousequations are integrable, their solutions also stable. To ensure this, one can follow the stability anal-ysis as presented in Refs. [64,65] which demonstrated how one can identify whether a given solutionis stable or unstable on satisfying the conjectured criterion dP / dv > dP / dv >
0, respectively,where P is the normalized momentum and v is the normalized velocity. Proceeding along this direc-tion, we found that the obtained solution fulfils the condition for stability dP / dv > . Inhomogeneous Two-soliton Solution and their Collisions Being motivated by the e ff ects of inhomogeneous nonlinearities in the one-soliton propagation,here we proceed further to explore their significance in all possible bright soliton collisions of generalsystem (1). For this purpose, first we wish to construct explicit two-soliton solution by following thestandard algorithm of Hirota method [58] and by using the bilinear form (13) as well as the similaritytransformation (9). The bright inhomogeneous two-soliton solution of the general CCNLS system(1) is obtained as A j ( x , z ) = (cid:15) √ γ G ( j ) F e i ζ ( x , z ) ⇒ (cid:15) √ γ G ( j )1 + G ( j )3 + G ( j )5 + G ( j )7 + F + F + F + F e i ζ ( x , z ) , j = , , (18a)where the explicit expression of dependent functions G (1) = G , G (2) = H , and F takes the followingform: G ( j )1 = α ( j )1 e η + α ( j )2 e η , (18b) G ( j )3 = e η + η ∗ + δ ( j )11 + e η + η ∗ + δ ( j )12 + e η + η ∗ + δ ( j )21 + e η + η ∗ + δ ( j )22 + e η + η ∗ + η + δ ( j )1 + e η + η ∗ + η + δ ( j )2 , (18c) G ( j )5 = e η + η ∗ + η + µ ( j )11 + e η + η ∗ + η + µ ( j )12 + e η + η ∗ + η + µ ( j )21 + e η + η ∗ + η + µ ( j )22 + e η + η ∗ + η + η ∗ + µ ( j )1 + e η + η ∗ + η + η ∗ + µ ( j )2 , (18d) G ( j )7 = e η + η ∗ + η + η ∗ + φ ( j )1 + e η + η + η ∗ + η ∗ + φ ( j )2 , (18e) F = e η + η ∗ + R + e η + η ∗ + δ + e η + η ∗ + δ ∗ + e η + η ∗ + R , (18f) F = e η + η ∗ + (cid:15) + e η + η ∗ + (cid:15) + e η + η ∗ + (cid:15) + e η + η ∗ + (cid:15) + e η + η ∗ + η ∗ + τ + e η ∗ + η + η + τ ∗ + e η + η ∗ + η ∗ + τ + e η ∗ + η + η + τ ∗ + e η + η ∗ + η + η ∗ + R , (18g) F = e η + η ∗ + η + η ∗ + θ + e η + η ∗ + η + η ∗ + θ + e η + η ∗ + η + η ∗ + θ + e η + η ∗ + η + η ∗ + θ , (18h) F = e η + η ∗ + η + η ∗ ) + R . (18i)Here η j = k j ( X + i δ k j Z ), X = (cid:15) (cid:16) γ x − δ(cid:15) (cid:15) (cid:82) γ dz (cid:17) , and Z = (cid:15) (cid:82) γ dz , while the auxiliary function S utilized in the mathematical process and other quantities are described in the appendix. Note thathere k j represents the wave vectors, while α ( (cid:96) ) j denotes the polarization parameters ( j , (cid:96) = ,
2) whichare going to play significant role in soliton collisions we discuss in this section.The propagation nature of IICSs and ICCSs under inhomogeneous nonlinearities discussed in theprevious section, excites one to explore their collision behaviour driven by the four-wave mixing ef-fect, which is also of considerable attraction. In this connection, the collision of bright solitons in thegeneral CCNLS system (2) can be broadly divided into three categories: (i) IICS × IICS, (ii) ICCS × ICCS, and (iii) IICS × ICCS. Here one should note that the subsystem resulting for the choice (6)supports only the second type of collision between two ICCSs due to the non-availability of IICSsas the choice leads to singular solutions. The remaining two versions of the system exhibit all the15bove three types of bright soliton collisions. Through a systematic asymptotic analysis and withthe aid of already available knowledge on soliton collisions in homogeneous systems, we investi-gate a categorical analysis on the collision scenario for these three cases, which show elastic andinelastic / shape-changing type collisions of bright solitons with di ff erent profile structures. Mathe-matically, we identify the form of solitons well before ( z → −∞ ) and well after ( z → + ∞ ) collision,as one could not exactly analyze at the collision point around z →
0, where the dynamics is quiteunpredictable because of its shorter span and nonlinear superposition due to interaction. In a standardway for two-soliton collision, we consider the following asymptotic relations:Soliton-1: η R ≈ ⇒ η R = δ k R (cid:15) ( k I − k I ) (cid:90) γ dz ≈ ±∞ as z → ±∞ , (19a)Soliton-2: η R ≈ ⇒ η R = δ k R (cid:15) ( k I − k I ) (cid:90) γ dz ≈ ∓∞ as z → ±∞ , (19b)along with the required conditions on k jR and k jI parameters, which we have chosen here as k R , k R > k I > k I with opposite signs showing head-on collision (one can also choose same sign to k jI for overtaking collision). The mathematical form for asymptotic analyses seems similar to thatof homogeneous models reported already [47–49] and we do not present their detailed expressionshere. Considering the length of the article, we devote this section only for discussion on the abovecollisions and how they can be controlled / altered by the inhomogeneous nonlinearities. As mentioned in the previous section, the inhomogeneous incoherently coupled solitons possessa standard single hump profile mathematically represented by a hyperbolic secant function. As thefour wave mixing nonlinearity is vanishing for this condition, they behave much similar to the Man-akov or two-coupled NLS type solitons. Analysis on their asymptotic dynamics reveals a simpleelastic collision between two IICSs by retaining their amplitude, width and velocity after collision,except with a phase-shift. To be explicit, the amplitude of solitons after collision to that of beforecollision takes the form B + j = ( k − k )( k + k ∗ )( k ∗ − k ∗ )( k ∗ + k ) B − j for right-moving Soliton-1 and B + j = ( k ∗ − k ∗ )( k + k ∗ )( k − k )( k ∗ + k ) B − j forleft-moving Soliton-2, which ultimately results into | B + j | = | B − j | representing the unaltered intensi-ties. Here j = , A , A and − / + denotes before / after collision. However,these identities are greatly manipulated by the inhomogeneity appearing in the medium which bydefault a ff ects their collision outcomes as well. For elucidation, we have depicted such a variationimposed by periodic, kink-like, bell-type, and exponentially growing nonlinearities in Figs. 10-11.Under constant nonlinearity, both colliding solitons reappears with same amplitude, width and ve-locity Fig. 10(a). On the other hand, wth periodic type nonlinearities ‘sn(z,0)’ and ‘cn(z,0)’, theseidentities exhibit periodic variation along temporal direction z both before and after collision. How-ever, the maximum amplitude of both solitons remain same amidst their periodic oscillation whichcontinuously alters the width as velocity as shown in Fig. 10(b-c).Further, the e ff ect of kink-like nonlinearity is to enhance the amplitude of solitons accompaniedby a compression after collision. This results in an escalated intensity for both the interacting solitonson the same background. Note that the velocity of both solitons are significantly a ff ected (moving too16a) (b) (c) Figure 10: Elastic collisions of two single-hump shaped IICSs under (a) constant nonlinearity and periodic nonlinearitieswith (b) γ ( z ) = γ + γ sn( z ,
0) and (c) γ ( z ) = γ + γ cn( z , k = + . i , k = . − . i , α (1)1 = . α (2)1 = . i , α (1)2 = . α (2)2 = . i , δ = (cid:15) = . (cid:15) = . γ = .
0, and γ = . (a) (b) (c) Figure 11: Elastic collisions of two single-hump shaped IICSs under (a) kink-like nonlinearity γ ( z ) = γ + γ sn( z ,
1) for γ = .
0, and γ = .
0, (b) bell-type nonlinearity γ ( z ) = γ + γ cn( z ,
1) for γ = .
0, and γ = .
5, and (c) exponentialnonlinearity γ ( z ) = γ + γ exp( γ z ) for γ = . γ = .
75 and γ = . γ ( z ) = γ + γ cn( z , ff ect around the collision point z → γ > γ <
0, respectively. In Fig. 11(b), we have shown the tunneling e ff ect in two-soliton collision which have an excess intensity in the tunneling region only. The final form of interestis the familiar one resulting into an exponentially growing intensity of the solitons starting well beforethe collision and last indefinitely along with commensurate compression of the beam as given in Fig.11(c). We have observed a prominent feature of kink-like and exponential nonlinearities, that hasthe ability to change the nature of soliton collisions. For example, from elastic collision to inelasticcollision. In the present case, these nonlinearities changes the elastic collision of two IICSs into aninelastic collision and the intensity of both solitons get increased after interaction with a considerablecompression in their width, see Figs. 11(a) and (c). When the four-wave mixing nonlinearity comes into picture, we obtain optical soliton profilestructures ranging from asymmetric single-hump to symmetric double-hump as well as flat-top.Here, we discuss the collision scenario of such solitons in the presence of inhomogeneous non-linearities. Our asymptotic analysis shows that the collision between ICCSs is purely elastic as C + j = ( k − k )( k + k ∗ )( k ∗ − k ∗ )( k ∗ + k ) C − j for right-moving Soliton-1 and C + j = ( k ∗ − k ∗ )( k + k ∗ )( k − k )( k ∗ + k ) C − j for left-moving soliton whichlead to | C + j | = | C − j | . Note that such ICCSs having di ff erent profiles reappear unaltered after collisionby retaining their shapes along with other identities such as amplitude, width, and velocity. As anexample, we have demonstrated such elastic collision of two double-hump ICCSs in the A compo-nent and collision between a single-hump and flat-top ICCSs in the A component in Fig. 12(a). Asmentioned in the previous section for collision of two IICSs, here these ICCSs undergo modulationby the inhomogeneous nonlinearity without a ff ecting their elastic nature of collision. The sn and cnnonlinearities introduce periodic variation of their amplitudes and velocities with smaller magnitudeof temporal oscillations near z → / suppression at the barrier / well occurs in thelatter. This is quite simple in the case of bell-type nonlinearity which by default preserves the in-tensities. Finally, the exponential nonlinearity increases the intensities throughout their propagationas well as under collision by retaining its elastic nature. Also, here we should note their continu-ous beam compression which is quite opposite to that of amplification. Similar to the previous case,here also the kink-like and exponential nonlinearities alters the elastic collision of two ICCSs intoan inelastic collision. To be precise, both ICCSs collide and exhibit a step / continuous amplificationin both components for kink-like / exponential nonlinearity by retaining their profile identities evenafter collisions, such as both double-hump ICCSs in A and a single-hump–double-hump ICCSa in A . For illustrative purpose and completeness, we have shown such periodically oscillating, step-like18ompressed amplification, cross-over of the well and amplification with uniform compression of twoICCSs collisions in Figs. 12-13 for the appropriate choices of arbitrary parameters. Compared to the previous two collision scenario, the collision between a IICS and ICCS is turnedout to be more interesting. For this purpose, we consider a IICS resulting for the choice S = ( α (1)1 ) + δ ( α (2)1 ) = S = ( α (1)2 ) + δ ( α (2)2 ) (cid:44)
0, wherethe first is assumed to be right-moving while the second is left-moving one. The detailed asymp-totic analysis reveals the amplitude variation due to collision as B + j = ( k − k )( k + k ∗ )( k ∗ − k ∗ )( k ∗ + k ) B − j for right-movingIICS-1 and C + j = (cid:18) ( k ∗ − k ∗ )( k + k ∗ ) | ( α ( j )1 κ − α ( j )2 κ ) + α ( j ) ∗ ( α (1)1 α (1)2 + α (2)1 α (2)2 ) / ( k − k ) | ( k ∗ − k ∗ )( k ∗ + k ) κ | α ( j )2 | (cid:19) / C − j for left-moving ICCS-2. Thisamplitude alteration results into the case | B + j | = | B − j | representing the unaltered intensities repre-senting elastic collision of IICS. However, | C + j | (cid:44) | C − j | , which accounts the important reason forinelastic collision of ICCS. Thus in both components, IICS reappears without any change in theirintensity, while the ICSS undergoes a change in its intensities in opposite sense in both components.For example, an increase in A is accompanied by a commensurate decrease in A which conservesits intensity as well as the total energy of the system. Thus, it can be inferred that the IICS inducesthe switching of intensity from one component to another component through ICCS, which leads tothe name energy-switching collision. Note that these intensity variations are purely dependent on thewave vectors and polarization parameters, not on the nonlinearity of the system. Such an energy-switching collision between IICS and ICCS with constant nonlinearity is shown in Fig. 14(a), wherea single-hump right-moving IICS is undergoing elastic collision in both components and the inten-sity of right-moving ICCS with flat-top profile decreases to a single-hump in A , while its intensity isincreasing in A along with a profile change from double-hump to single-hump. This shows that theextra energy of ICCS after collision in A is taken / switched from A , see Fig. 14(a).Next, we consider the situation where the nonlinearity is temporally modulated and look for thechange in the collision scenario. As discussed earlier, here also, these varying nonlinearities modulatethe amplitudes, velocity and width of the participating solitons substantially. The energy-switchingnature of ICCS does not change for any choice of considered inhomogeneous nonlinearity, whilethe IICS is manifesting itself from elastic into an inelastic-switch due to kink-like and exponentialnonlinearities in addition to the appropriate modulation in its width through a cascaded compression.For a clear understanding, we have demonstrated energy-switching collision of IICS × ICCS withconstant, periodic, kink-like, bell-type, and exponentially varying nonlinearities in Fig. 14(b-c) andFig. 15(a-c). As their implications are well discussed in the previous cases, we refrain from givinghere again. All these nonlinearities induce change in their identities of the colliding solitons IICSand ICCS, but without a ff ecting the switching nature. Thus, we are getting IICS-ICCS collision withperiodical variations, step-like amplitude enhancement along with compression, tunneling throughan high amplitude barrier, and continuous amplification with compression.19a) (b) (c) Figure 12: Elastic collisions of two double-hump ICCSs in A and collision of single-hump–flat-top ICCSs in A components under (a) constant nonlinearity and periodic nonlinearities with (b) γ ( z ) = γ + γ sn( z ,
0) and (c) γ ( z ) = γ + γ cn( z , k = + . i , k = . − . i , α (1)1 = . α (2)1 = . α (1)2 = . α (2)2 = . i , δ = (cid:15) = . (cid:15) = . γ = .
0, and γ = . (a) (b) (c) Figure 13: Elastic collisions of two ICCSs with double-hump, flat-top and single-hump structures under (a) kink-likenonlinearity γ ( z ) = γ + γ sn( z ,
1) for γ = .
0, and γ = .
0, (b) bell-type nonlinearity γ ( z ) = γ + γ cn( z ,
1) for γ = . γ = − .
5, and (c) exponential nonlinearity γ ( z ) = γ + γ exp( γ z ) for γ = . γ = . γ = . Figure 14: Energy-switching collision of left-moving ICCS due to elastically reappearing right-moving IICS under (a)constant nonlinearity and periodic nonlinearities with (b) γ ( z ) = γ + γ sn( z ,
0) and (c) γ ( z ) = γ + γ cn( z , k = + . i , k = . − . i , α (1)1 = . i , α (2)1 = . α (1)2 = . α (2)2 = . δ = (cid:15) = . (cid:15) = . γ = .
0, and γ = . (a) (b) (c) Figure 15: Energy-switching collision of left-moving ICCS due to elastically reappearing right-moving IICS under (a)kink-like nonlinearity γ ( z ) = γ + γ sn( z ,
1) for γ = .
0, and γ = .
0, (b) bell-type nonlinearity γ ( z ) = γ + γ cn( z ,
1) for γ = .
0, and γ = .
5, and (c) exponential nonlinearity γ ( z ) = γ + γ exp( γ z ) for γ = . γ = .
75 and γ = . . Inhomogeneous Soliton Bound States Further from the soliton collisions, one shall also explore the dynamics of soliton bound statesresulting for the choice of equal velocity solitons. In recent years, this attracted much attentionin the aspects of soliton molecule formation in optical and atomic systems and referred as velocityresonant solitons. If we consider the case of homogeneous medium with constant nonlinearity and thevelocity of two solitons are depend on the wave vectors, in particular their imaginary part k jI for j -thsoliton. Under such velocity resonance k I = k I , there occurs a periodic attraction and repulsion ofcontributing solitons, which are usually called as breathing of solitons. Unlike the standard breathersappearing on constant non-zero background, these breathing solitons can exist on both zero as wellas non-zero background. In our system, we can form soliton bound states for the above discussedthree cases of collisions. Further, the bound states among di ff erent profile solitons are also possiblewith appropriate choice of polarization parameters determining the contribution of four-wave mixingnonlinearity. Without providing much mathematical forms, we have demonstrated such soliton boundstates arising between two ICCSs exhibiting double-hump–single-hump (in A ) and flat-top–single-hump (in A ) structures are shown in Fig. 16. A similar breathing solitons can be observed for theother two cases as well as with non-zero initial velocities which we have not given here consideringthe length of the article. Apart from the two-soliton-bound-states, one can investigate the formationof multi-soliton bound structures as well as the interaction between solitons and bound states whichis of further interest and we do not discuss the details here.
6. Conclusion
In conclusion, we have investigated the propagation and collision dynamics of inhomogeneoussolitons in a system of non-autonomous coherently coupled nonlinear Schr ¨odinger (CCNLS) mod-els. By identifying an appropriate similarity transformation that reduces the considered CCNLS intothe canonical integrable CCNLS systems and with the aid of the Hirotas bilinear method, we haveconstructed general soliton solutions. In particular, we classify the solutions into two categories andrefer them as inhomogeneous coherently coupled solitons and inhomogeneous incoherently coupledsolitons that respectively appear in the presence and absence of four wave mixing e ff ects. After-wards, the dynamics of these solitons featuring non-trivial profiles are explored by considering var-ious temporal modulation of nonlinearities namely, step-like switching nonlinearity, optical latticesand exponential nonlinarity. Our analysis revealed that depending upon the nature of nonlineari-ties the modulated CCNLS system can admit various special localized coherent structures displayingdistinct behaviours, like periodically varying solitons, soliton compression with an unusual hike inits intensity, tunneling / cross-over e ff ects due to a localized barrier / well, monotonous amplificationwith compression of width and a sudden appearance of a solitonic excitation that grows in amplitudeduring propagation. Then, influence of such inhomogeneous nonlinearities on soliton collisions arealso investigated briefly along with soliton bound states. Interestingly, we point out that the collisionnature can be altered substantially for certain type of nonlinearity management, namely kink-likenonlinearities. The results presented in this work will be applicable to the studies on engineering22a) (b) (c)(d) (e) (f) Figure 16: Soliton bound states between two ICCSs having double-hump–single-hump in A and flat-top–single-hump in A components under (a) constant, (b) periodic-sine, (c) periodic-cosine, (d) kink-like, (e) bell-type, and (f) exponential-growth type nonlinearities. Here the parameters are chosen as (a) γ = . γ = γ = . (cid:15) = .
025 and (cid:15) = .
0, (b) γ = . γ = . γ = − . (cid:15) = . (cid:15) = .
25, (c) γ = . γ = . γ = − . (cid:15) = . (cid:15) = .
25, (d) γ = . γ = . (cid:15) = .
75 and (cid:15) = .
25, (e) γ = . γ = . (cid:15) = .
75 and (cid:15) = .
75, (f) γ = . γ = . (cid:15) = . (cid:15) = .
25, with other parameters as k = . k = . α (1)1 = . i , α (2)1 = . α (1)2 = + i , and α (2)2 = − i . ff ects. Now, it is quite natural to look for the influence of such modulations in higher-dimensionalnonlinear optical systems featuring vortex solitons, soliton bullets, resonant solitons, lump solitonsand dromians. An advantage of our present study it can be straightforwardly extended to multicom-ponent systems with more than two fields, to name a few multimode propagation in GRIN mediawith anisotropy, spinor condensates with hyperfine spins F = Acknowledgment
The work KS was supported by Department of Science and Technology - Science and EngineeringResearch Board (DST-SERB), Government of India, sponsored National Post-Doctoral Fellowship(File No. PDF / / / / Appendix
The auxiliary function S and other quantities appearing in the two-soliton solution take the fol-lowing form: the auxiliary function s is given by S = S e η + S e η + S e η + η + e η + η ∗ + η + λ + e η + η ∗ + η + λ + e η + η ∗ + η + λ + e η + η ∗ + η + λ + e η + η ∗ + η + λ + e η + η + η ∗ + λ + e η + η ∗ + η + η ∗ + λ . e R u = κ uu ( k u + k ∗ u ) , e δ = κ ( k + k ∗ ) , e δ ( j ) uv = δδ α ( j ) ∗ v S u k u + k ∗ v ) , e (cid:15) uv = S u S ∗ v k u + k ∗ v ) , e δ ( j ) u = δδ α ( j ) ∗ u S + ( k − k )( α ( j )1 κ u − α ( j )2 κ u )( k + k ∗ u )( k + k ∗ u ) , e τ u = S u S ∗ k u + k ∗ ) ( k u + k ∗ ) , e λ uv = ( k − k ) κ uv S − u ( k u + k ∗ v )( k − u + k ∗ v ) , e µ ( j ) uv = ( k − k ) α ( j )3 − u S u S ∗ v k u + k ∗ v ) ( k − u + k ∗ v ) , e θ uv = | k − k | S u S ∗ v D ( k u + k ∗ v ) ( α (1)3 − u α (1) ∗ − v + δα (2)3 − u α (2) ∗ − v ) , e λ u = ( k − k ) S S S ∗ u k + k ∗ u ) ( k + k ∗ u ) , e λ = ( k − k ) S S S ∗ D , e R = | k − k | | S | | S |
16 ˜ D , e φ ( j ) u = δδ ( k − k ) ( k ∗ − k ∗ ) S S α ( j ) ∗ − u S ∗ u D ( k + k ∗ u ) ( k + k ∗ u ) , e R = | k − k | ( κ κ − κ κ ) + | S | ( k + k ∗ ) | k + k ∗ | ( k + k ∗ ) , e µ ( j ) u = ( k − k ) S u D (cid:16)(cid:104) ( k − u + k ∗ ) + ( k ∗ − k ∗ )( k − u + k ∗ ) (cid:105) α ( j )3 − u α ( j ) ∗ α ( j ) ∗ + ( k − u + k ∗ )( k ∗ − k ∗ ) α ( j ) ∗ ( α (3 − u )3 − u α (3 − u ) ∗ ) − ( k ∗ − k ∗ )( k − u + k ∗ ) α ( j ) ∗ ( α (3 − u )3 − u α (3 − u ) ∗ ) + ( k − u + k ∗ )( k − u + k ∗ ) α ( j )3 − u ( α (3 − u ) ∗ α (3 − u ) ∗ ) (cid:17) , S u = ( α (1) u ) + δ ( α (2) u ) , S = α (1)1 α (1)2 + δα (2)1 α (2)2 , ˜ D = ( k + k ∗ ) ( k ∗ + k ) ( k + k ∗ ) ( k + k ∗ ) ,κ uv = (cid:16) α (1) u α (1) ∗ v + δ α (2) u α (2) ∗ v (cid:17) / ( k u + k ∗ v ) . Here u , v , j , l = ,
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