Nondegenerate Solitons and their Collisions in Manakov System
aa r X i v : . [ n li n . PS ] S e p Nondegenerate Solitons and their Collisions in Manakov System
R. Ramakrishnan, S. Stalin, and M. Lakshmanan ∗ Department of Nonlinear Dynamics, School of Physics,Bharathidasan University, Tiruchirapalli–620 024, India (Dated: December 2018)
Abstract
Recently, we have shown that the Manakov equation can admit a more general class of nonde-generate vector solitons, which can undergo collision without any intensity redistribution in generalamong the modes, associated with distinct wave numbers, besides the already known energy ex-changing solitons corresponding to identical wave numbers. In the present comprehensive paper,we discuss in detail the various special features of the reported nondegenerate vector solitons. Tobring out these details, we derive the exact forms of such vector one-, two- and three-soliton solu-tions through Hirota bilinear method and they are rewritten in more compact forms using Gramdeterminants. The presence of distinct wave numbers allows the nondegenerate fundamental soli-ton to admit various profiles such as double-hump, flat-top and single-hump structures. We explainthe formation of double-hump structure in the fundamental soliton when the relative velocity ofthe two modes tends to zero. More critical analysis shows that the nondegenerate fundamentalsolitons can undergo shape preserving as well as shape altering collisions under appropriate condi-tions. The shape changing collision occurs between the modes of nondegenerate solitons when theparameters are fixed suitably. Then we observe the coexistence of degenerate and nondegeneratesolitons when the wave numbers are restricted appropriately in the obtained two-soliton solution.In such a situation we find the degenerate soliton induces shape changing behavior of nondegener-ate soliton during the collision process. By performing suitable asymptotic analysis we analyze theconsequences that occur in each of the collision scenario. Finally we point out that the previouslyknown class of energy exchanging vector bright solitons, with identical wave numbers, turns outto be a special case of the newly derived nondegenerate solitons.
PACS numbers: ∗ Corresponding author E-mail: [email protected] . INTRODUCTION The propagation of light pulses in optical Kerr media is still one of the active areas ofresearch in nonlinear optics [1]. In particular the fascinating dynamics of light in multi-modefibers and fiber arrays has stimulated the investigation on temporal multi-component/vectorsolitons over different aspects, especially from the applications point of view [2]. In thenonlinear optics context, temporal vector solitons are formed due to the balance betweendispersion and Kerr nonlinearity. Mathematically these vector solitons are nothing but thesolutions of certain integrable coupled nonlinear Schr¨odinger family of equations. Thereexist many types of vector solitons which have been reported so far in the literaure and theirdynamics have also been investigated in various physical situations. For instance, bright-bright solitons [3–5], bright-dark solitons [6–9] and dark-dark solitons [6, 10] are some ofthe solitons which have been investigated in these systems. These vector solitons have alsoreceived considerable attention in other areas of science including Bose-Einstein condensates(BECs) [11, 12], bio-physics [13], plasma physics [14] and so on. Apart from the above,partially coherent solitons/soliton complexes have been reported in self-induced multi-modewaveguide system [15, 16], while polarization locked solitons and phase locked solitons infiber lasers [17] and dissipative vector solitons in certain dissipative systems [18–20] havealso been analyzed in the literature.From the above studies on vector solitons we have noted that the intensity pro-files of multi-component solitons reported, especially in the integrable coupled nonlinearSchr¨odinger systems, are defined by identical wave numbers in all the components. We callthese vector solitons as degenerate class of solitons. As a consequence of degeneracy inthe wave numbers, single-hump strcutured intensity profiles only emerge in these systemsin general [21]. In the coherently coupled system even degenerate fundamental soliton canalso admit double-hump profile when the four wave mixing process is taken into account[22, 23]. However, in this case one can not expect more than a double-hump profile. Veryinterestingly our theoretical [3, 4] and other experimental [24–26] studies confirm that thedegenerate vector solitons undergo in general energy redistribution among the modes duringthe collision, except for the special case of polarization parameters satisfying specific re-strictions, for example in the case of two component Manakov systems as α (1)1 α (1)2 = α (2)1 α (2)2 where α ( j ) i ’s, i, j = 1 ,
2, are complex numbers related to the polarization vectors. By exploitingthe fascinating shape changing collision scenario of degenerate Manakov solitons, it has been2heoretically suggested that the construction of optical logic gates is indeed possible, leadingto all optical computing [27]. We also note that logic gates have been implemented usingtwo stationary dissipative solitons of complex Ginzburg-Landau equation [29].Recently in Refs. [30–32] it has been reported that multi-hump structured dispersionmanaged solitons/double-hump intensity profile of soliton molecule may be useful for appli-cation in optical communications because they may provide alternative coding schemes fortransmitting information with enhanced data-carrying capacity. Multi-hump solitons havealso been identified in the literature in various physical situations [33–39]. They have beenobserved experimentally in a dispersive nonlinear medium [36]. Theoretically frozen double-hump states have been predicted in birefringent dispersive nonlinear media [33, 34]. Thesesolitons have been found in various nonlinear coupled field models also [37]. In the case ofsaturable nonlinear medium, stability of double and triple-hump optical solitons has alsobeen investigated [38]. Multi-humped partially coherent solitons have also been investigatedin photorefractive medium [15]. In addition to the above, the dynamics of double-humpsolitons have also been studied in mode-locked fiber lasers [17–20]. A double hump solitonhas been observed during the buildup process of soliton molecules in deployed fiber systemsand fiber laser cavities [30, 40].From the above studies, we observe that the various properties associated with the de-generate vector bright solitons of many integrable coupled field models have been well un-derstood. However, to our knowledge, studies on fundamental solitons with nonidenticalwave numbers in all the modes have not been considered so far and multi-hump structuresolitons have also not been explored in the integrable coupled nonlinear Schr¨odinger typesystems except in our recent work [45] and that of Qin et al [46] on the following Manakovsystem [48, 49], iq jz + q jtt + 2 X p =1 | q p | q j = 0 , j = 1 , , (1)where q j , j = 1 ,
2, describe orthogonally polarized complex waves in a birefringent medium.Here the subscripts z and t represent normalized distance and retarded time, respectively.Based on the above studies we are motivated to look for a new class of fundamental solitons,which possess nonidentical wave numbers as well as multi-hump profiles, which are usefulfor optical soliton based applications. We have successfully identified such a new classof solitons in [45]. We call the fundamental solitons with nonidentical wave numbers asnondegenerate vector solitons [21, 45]. Surprisingly this new class of vector bright solitons3xhibit multi-hump structure (double-hump soliton arises in the present Manakov systemand one can also observe N -hump soliton in the case of N -coupled Manakov type system)which may be useful for transmitting information in a highly packed manner. Therefore itis very important to investigate the role of additional wave number(s) on the new class offundamental soliton structures and collision scenario as well, which were briefly discussedin [45]. In the present comprehensive version we discuss the various properties associatedwith the nondegenerate solitons in a detailed manner by finding their exact analytical formsthrough Hirota bilinearization method. Then we discuss how the presence of additionaldistinct wave numbers and the cross phase modulation ( | q | + | q | ) q j , j = 1 ,
2, amongthe modes bring out double-hump profile in the structure of nondegenerate fundamentalsoliton. We find that the nondegenerate solitons undergo shape preserving collision generally,as reported by us in [45], and shape altering and shape changing collisions for specificparametric values. Further, we figured out the coexistence of degenerate and nondegeneratesolitons in the Manakov system. Such coexisting solitons undergo novel shape changingcollision scenario leading to useful soliton based signal amplification application. Finally, weshow that the degenerate class of vector solitons reported in [3, 4] can be deduced from theobtained nondegenerate two-soliton solution.The structure of the paper is organized as follows: In section II, we discuss the Hirotabilinear procedure in order to derive nondegenerate soliton solutions for Eq. (1). Using thisprocedure we obtained nondegenerate one- and two-soliton solutions in Gram determinantforms and also identified the coexistence of degenerate and nondegenerate solitons in SectionIII. In Section IV we discuss the various collision properties of nondegenerate solitons. Sec-tion V deals with the collision between degenerate and nondegenerate solitons. In Section VIwe recovered the degenerate one- and two-soliton solutions from the nondegenerate one- andtwo-soliton solutions by suitably restricting the wave numbers and in Section VII we pointout the possible experimental observations of nondegenerate solitons. In Section VIII wesummarize the results and discuss possible extension of this work. Finally in the AppendixA we present the three soliton solution in Gram determinant forms for completion whilein Appendix B we discuss about certain asymptotic forms of solitons. In Appendix C, weintroduce explicit forms of certain parameters appearing in the text. Finally in Appendix Dwe discuss the numerical stability analysis of nondegenerate solitons under different strengthof white noise as perturbation. 4
I. BILINEARIZATION
To derive the nondegenerate soliton solutions for the Manakov system we adopt the sameHirota bilinear procedure that has been already used to get degenerate vector bright solitonsolutions but with appropriate form of initial seed solutions. We point out later how such asimple form of new seed solutions will produce remarkably new physically important classof soliton solutions. In general, the exact soliton solutions of Eq. (1) can be obtained byintroducing the bilinearizing transformation, which can be identified from the singularitystructure analysis of Eq. (1) [50] as q j ( z, t ) = g ( j ) ( z, t ) f ( z, t ) , j = 1 , , (2)to Eq. (1). This results in the following set of bilinear forms of Eq. (1),( iD z + D t ) g ( j ) · f = 0 , j = 1 , , (3a) D t f · f = 2 X n =1 g ( n ) g ( n ) ∗ . (3b)Here g ( j ) ’s are complex functions whereas f is a real function and ∗ denotes complex con-jugation. The Hirota’s bilinear operators D z and D t are defined [51] by the expressions D mz D nt ( a · b ) = (cid:18) ∂∂z − ∂∂z ′ (cid:19) m (cid:18) ∂∂t − ∂∂t ′ (cid:19) n a ( z, t ) b ( z ′ , t ′ ) (cid:12)(cid:12) z = z ′ , t = t ′ . Substituting the standardexpansions for the unknown functions g ( j ) and f , g ( j ) = ǫg ( j )1 + ǫ g ( j )3 + ..., j = 1 , ,f = 1 + ǫ f + ǫ f + ..., (4)in the bilinear Eqs. (3a)-(3b) one can get a system of linear partial differential equations(PDEs). Here ǫ is a formal series expansion parameter. The set of linear PDEs arises aftercollecting the coefficients of same powers of ǫ . By solving these linear PDEs recursively (atan appropriate order of ǫ ), the resultant associated explicit forms of g ( j ) ’s and f constitutethe soliton solutions to the underlying system (1). We note that the truncation of seriesexpansions (4) for the nondegenerate soliton solutions is different from degenerate solitonsolutions. This is essentially due to the general form of seed solutions assigned to the lowestorder linear PDEs. 5 II. A NEW CLASS OF NONDEGENERATE SOLITON SOLUTIONS
To study the role of additional wave numbers on the structural, propagational and col-lisional properties of nondegenerate soliton it is very much important to find the exactanalytical form of it systematically. In this section by exploiting the procedure describedabove we intend to construct nondegenerate one- and two-soliton solutions which can begeneralized to arbitrary N -soliton case (For N = 3, see Appendix A below). In principlethis is possible because of the existence of nondegenerate N -soliton solution ensured by thecomplete integrability property of Manakov Eq. (1). Then we point out the possibility ofcoexistence of degenerate and nondegenerate solitons by imposing certain restriction on thewave numbers in the obtained nondegenerate two-soliton solution. Further we also point outthe possibility of deriving this partially nondegenerate two-soliton solution through Hirotabilinear method. We note that to avoid too many mathematical details we provide the finalform of solutions only since the NDS solution construction process is a lengthy one. A. Nondegenerate fundamental soliton solution
In order to deduce the exact form of nondegenerate one-soliton solution we consider twodifferent seed solutions for the two modes as g (1)1 = α (1)1 e η , g (2)1 = α (2)1 e ξ , (5)where η = k t + ik z and ξ = l t + il z , to the following linear PDEs ig ( j )1 z + g ( j )1 tt = 0 , j = 1 , . (6)In (5) the complex parameters α ( j )1 , j = 1 ,
2, are arbitrary. The above equations arise in thelowest order of ǫ . The presence of two distinct complex wave numbers k and l ( k = l ,in general) in the seed solutions (5) makes the final solution as nondegenerate one. Thisconstruction procedure is different from the standard one that has been followed in earlierworks on degenerate vector bright soliton solutions [3, 4] where identical seed solutions ofEq. (1) (solutions (5) with k = l and distinct α ( j )1 ’s, j = 1 ,
2) have been used as startingseed solutions for Eq. (6). We note that such degenerate seed solutions only yield degenerateclass of vector bright soliton solutions [3, 4, 45].With the starting solutions (5) we allow the series expansions (4) to terminate by them-selves while solving the system of linear PDEs. From this recursive process, we find that6 q | q ( a )-
30 0 4000.05 t | q , | q | q ( b ) -
30 0 3000.08 t | q , | q | q ( c ) -
30 0 3000.10.2 t | q , | q | q ( d ) -
40 0 4000.1 t | q , FIG. 1: Various symmetric intensity profiles of nondegenerate fundamental soliton: While (a)denotes double-hump solitons in both the modes (b) and (c) represent flat-top-double-hump solitonsand single-hump-double-hump solitons, respectively. Single-hump solitons in both the modes areillustrated in (d). The parameter values of each figures are: (a): k = 0 . . i , l = 0 . . i , α (1)1 = 0 .
45 + 0 . i , α (2)1 = 0 .
49 + 0 . i . (b): k = 0 .
425 + 0 . i , l = 0 . . i , α (1)1 = 0 .
44 + 0 . i , α (2)1 = 0 .
43 + 0 . i . (c): k = 0 .
55 + 0 . i , l = 0 .
333 + 0 . i , α (1)1 = 0 . . i , α (2)1 = 0 . . i . (d): k = 0 .
333 + 0 . i , l = − .
316 + 0 . i , α (1)1 = 0 .
45 + 0 . i , α (2)1 = 0 . . i . the expansions (4) get terminated for the nondegenerate fundamental sliton solution as, g ( j ) = ǫg ( j )1 + ǫ g ( j )3 and f = 1 + ǫ f + ǫ f . The explicit expressions of g ( j )1 , g ( j )3 , f and f constitute a general form of new fundamental one-soliton solution to Eq. (1) as q = g (1)1 + g (1)3 f + f = ( α (1)1 e η + e η + ξ + ξ ∗ +∆ (1)1 ) /D q = g (2)1 + g (2)3 f + f = ( α (2)1 e ξ + e η + η ∗ + ξ +∆ (2)1 ) /D . (7)Here D = 1 + e η + η ∗ + δ + e ξ + ξ ∗ + δ + e η + η ∗ + ξ + ξ ∗ + δ , e ∆ (1)1 = ( k − l ) α (1)1 | α (2)1 | ( k + l ∗ )( l + l ∗ ) , e ∆ (2)1 = − ( k − l ) | α (1)1 | α (2)1 ( k + k ∗ ) ( k ∗ + l ) , e δ = | α (1)1 | ( k + k ∗ ) , e δ = | α (2)1 | ( l + l ∗ ) and e δ = | k − l | | α (1)1 | | α (2)1 | ( k + k ∗ ) ( k ∗ + l )( k + l ∗ )( l + l ∗ ) . In theabove one-soliton solution two distinct complex wave numbers, k and l , occur in both theexpressions of q and q simultanously. This confirms that the obtained solution is nonde-7enerate. We also note that the solution (7) can be rewritten in a more compact form usingGram determinants as g (1) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η e ξ η ∗ ( l + k ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ − | α (1)1 | ( k + k ∗ ) − | α (2)1 | ( l + l ∗ )
00 0 − α (1)1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , g (2) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η e ξ η ∗ ( l + k ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ − | α (1)1 | ( k + k ∗ ) − | α (2)1 | ( l + l ∗ )
00 0 0 − α (2)1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (8a) f = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ ξ ∗ ( l + l ∗ ) − | α (1)1 | ( k + k ∗ ) − | α (2)1 | ( l + l ∗ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (8b)The above Gram determinant forms satisfy the bilinear Eqs. (3a) and (3b) as well as Man-akov Eq. (1). To investigate the various properties associated with the above fundamentalsoliton solution, we rewrite Eq. (7) as q = e iη I e ∆(1)1 + ρ { cosh( ξ R + φ R φ I i sinh( ξ R + φ R φ I } /D , (9a) q = e iξ I e ∆(2)1 + ρ { cosh( η R + φ R φ I i sinh( η R + φ R φ I } /D , (9b)where D = e δ cosh( η R + ξ R + δ ) + e δ δ cosh( η R − ξ R + δ − δ ), η R = k R ( t − k I z ), η I = k I t + ( k R − k I ) z , ξ R = l R ( t − l I z ), ξ I = l I t + ( l R − l I ) z , ρ j = log α ( j )1 , j = 1 ,
2. Here, φ R , φ I , φ R and φ I are real and imaginary parts of φ = ∆ (1)1 − ρ and φ = ∆ (2)1 − ρ , respectively, and also k R , l R , k I and l I are the real and imaginaryparts of k and l , respectively. From the above, we can write φ R = log | k − l | | α (2)1 | | k + l ∗ | ( l + l ∗ ) , φ I = log ( k − l )( k ∗ + l )( k ∗ − l ∗ )( k + l ∗ ) , φ R = log | l − k | | α (1)1 | | k + l ∗ | ( k + k ∗ ) and φ I = log ( l − k )( k + l ∗ )( l ∗ − k ∗ )( k ∗ + l ) . The profilestructures of solution (9a)-(9b) are described by the four complex parameters k , l and α ( j )1 , j = 1 ,
2. For the nondegenerate fundamental soliton in the first mode, the amplitude,velocity and central position are found from Eq. (9a) as 2 k R , 2 l I and φ R l R , respectively.Similarly for the soliton in the second mode they are found from Eq. (9b) as 2 l R , 2 k I and φ R k R , respectively. Note that α ( j )1 , j = 1 ,
2, are related to the unit polarization vectors of thenondegenerate fundamental solitons in the two modes. They constitute different phases forthe nondegenerate soliton in the two modes as A = ( α (1)1 /α (1) ∗ ) / and A = ( α (2)1 /α (2) ∗ ) / .8o explain the various properties associated with solution (9a)-(9b) further we considertwo physically important special cases where the imaginary parts of the wave numbers k and l are either identical with each other ( k I = l I ) or nonidentical with each other( k I = l I ). Physically this implies that the former case corresponds to solitons in the twomodes travelling with identical velocities v = v = 2 k I but with k = l whereas the lattercase corresponds to solitons which propagate in the two modes with non-identical velocities v = v . In the identical velocity case, the quantity φ jI , j = 1 , k I = l I . This results in the following expression for the fundamental solitonpropagating with single velocity, v , = 2 k I , in the two modes, q = e iη I e ∆(1)1 + ρ cosh( ξ R + φ R /D ,q = e iξ I e ∆(2)1 + ρ cosh( η R + φ R /D , (10)where D = e δ cosh( η R + ξ R + δ )+ e δ δ cosh( η R − ξ R + δ − δ ) with η R = k R ( t − k I z ), η I = k I t + ( k R − k I ) z , ξ R = l R ( t − k I z ), ξ I = k I t + ( l R − k I ) z . Note that theconstants that appear in the above solution becomes equivalent to the one that appear in thesolution (9a)-(9b) after imposing the condition k I = l I in it. The solution (10) admits fourtypes of symmetric profiles (satisfying appropriate conditions on parameters, see below) andalso their corresponding asymmetric profiles. The symmetric profiles are: (i) double-humpsin both the modes (or a double-hump in q mode and a M-type double-hump in q mode),(ii) a flat-top in one mode and a double-hump in the other mode, (iii) a single-hump in thefirst mode and a double-hump in the second mode (or vice versa), (iv) single-humps in boththe modes. The corresponding four types of asymmetric wave profiles can be obtained bytuning the real parts of wave numbers k and l and the arbitrary complex parameters α ( j )1 ’s, j = 1 , k I = l I = 0 . k R < l R [47]. However to exhibit the generality of these structures, in the present paper,9 q | q ( a ) -
30 0 3000.06 t | q , | q | q ( b ) -
30 0 3000.08 t | q , | q | q ( c ) -
30 0 3000.10.2 t | q , | q | q ( d ) -
40 0 4000.1 t | q , FIG. 2: Various asymmetric intensity profiles of nondegenerate fundamental soliton: Figures (a),(b), (c) and (d) represent each of figures asymmetric intensity profiles as against the symmetricprofiles of Figs.1(a)-(d). The corresponding parameter values of each figures are: (a): k = 0 .
333 +0 . i , l = 0 .
315 + 0 . i , α (1)1 = 0 .
65 + 0 . i , α (2)1 = 0 .
49 + 0 . i . (b): k = 0 .
425 + 0 . i , l = 0 . . i , α (1)1 = 0 . . i , α (2)1 = 0 .
43 + 0 . i . (c): k = 0 .
55 + 0 . i , l = 0 .
333 + 0 . i , α (1)1 = 1 . . i , α (2)1 = 0 . . i . (d): k = 0 .
333 + 0 . i , l = − .
22 + 0 . i , α (1)1 = 0 .
45 + 3 i , α (2)1 = 0 . . i . we discuss these properties for k R > l R . It should be pointed out here that in Ref. [46] theauthors have derived this solution in the context of multi-component BEC using Darbouxtransformation and they have classified density profiles as we have reported in Ref. [45]for k R < l R in the context of nonlinear optics. They have also studied the stability ofdouble-hump soliton using Bogoliubov-de Gennes excitation spectrum.The symmetric nature of all the four cases can be confirmed by finding the extremumpoints of the nondegenerate one-soliton solution (10). For instance, to show that the double-hump soliton profile displayed in Fig. 1(a) is symmetric, we find the corresponding localmaximum and minium points by applying the first derivative test ( {| q j | } t = 0) and thesecond derivative test ( {| q j | } tt < >
0) to the expression of | q j | , j = 1 ,
2, at z = 0. Forthe first mode, the three three extremal points are identified, namely t = − . t = 5 . t = 11 .
9. We find another set of three extremal points for the second mode, namely10
10 0 10 - t z q -
10 0 10 - t z q (a) (b) FIG. 3: Node formation in the nonidentical velocity case. The parameter values are k = 1 +1 . i , l = 1 . . i , α (1)1 = 1 . . i , α (2)1 = 0 .
45 + 0 . i . t = − . t = 5 . t = 12 . {| q | } t = 0. The points t and t correspond tothe maxima (at which {| q | } tt <
0) of the double hump soliton whereas t corresponds to theminimum of the double hump soliton. Similarly the extremal points t and t represent themaxima and t corresponds to the minimum of the double hump soliton in the q mode. Inthe first component the two maxima t and t are symmetrically located about the minimumpoint t . This can be easily confirmed by finding the difference between t and t and t and t , that is t − t = 6 . t − t . This is true for the second component also, that is t − t = 6 . t − t . This implies that the two maxima t and t are located symmetricallyfrom the minimum point t . Then the magnitude ( | q | ) of each hump (of the double humpsoliton) corresponding to the maxima t is equal to 0 .
051 and t is equal to 0 . | q | ) corresponding to t is equal to 0 .
054 and t is equalto 0 . {| q j | } t very slowly tends to zero near the corresponding maximum for certain number of t values. This also confirms that the presence of almost flatness and symmetric nature of the11ne-soliton.We also derive the conditions analytically to corroborate the symmetric and asymmetricnature of soliton solution (10) in another way. For this purpose, we intend to calculate therelative separation distance ∆ t between the minima of the two components (modes)∆ t = ¯ t − ¯ t = ( t − t ) − ( t − t ) , = φ R l R − φ R k R . (11)If the above quantity ∆ t = 0 then the solution (10) exhibits symmetric profiles otherwiseit admits asymmetric profiles.The explicit form of relative separation distance turns out to be∆ t = 12 l R log ( k R − l R ) | α (2)1 | l R ( k R + l R ) − k R log ( l R − k R ) | α (1)1 | k R ( k R + l R ) . (12)We have explicitly calculated the relative separation distance values and confirmed thedisplayed profiles in Fig. 1 and 2 are symmetric and asymmetric, respectively. For instance,the ∆ t value corresponding to the symmetric double-hump soliton in both the modes (Fig.1(a)) is 0 .
002 (to get the perfect zero value one has to fine tune the parameters suitably)and for asymmetric double-hump solitons the value is equal to 0 . k I = l I ), v = v ,the distinct wave numbers k and l influence drastically the propagation of nondegeneratesolitons in the two modes. If the relative velocity (∆ v = v − v ) of the solitons betweenthe two modes is large, then there is a node created in the structure of the fundamentalsolitons of both the modes [46]. This is due to the cross phase modulation between themodes. In this situation the intensity of the fast moving soliton ( v = 2 l I >
0) in the firstmode starts to decrease and it gets completely suppressed after z = 0. At the same valueof z the fast moving soliton reappears in the second mode after a finite time. Similarlythis fact is true in the case of slow moving soliton ( v = 2 k I <
0) as well. Consequentlythe intensity of solitons is unequally distributed among the two modes. This is clearlydemonstrated in Fig. 3 and Figs. 4(a)-4(b). On the otherhand, if the relative velocity tendsto zero (∆ v → I total = | q | + | q | , of nondegenerate solitonsstarts to get distributed equally among the two components. As a consequence of this, a12 IG. 4: Double-hump formation in the profile structure of nondegenerate fundamental soliton:(a) and (b) represent the node formation in soliton profiles. (c) and (d) denote the emergenceof double-hump in both the modes. The corresponding parameter values for (a) and (b) are: k = 0 . − . i , l = 0 . − . i , α (1)1 = 1 and α (2)1 = 0 .
5; For figures (c) and (d) the values arechosen as k = 0 . − . i , l = 0 . − . i , α (1)1 = 1 and α (2)1 = 0 . double-hump profile starts to emerge in each of the modes as displayed in Fig. 4(c)-4(d).At perfect zero relative velocity (∆ v = 0), the double-hump fundamental soliton emergescompletely in both the modes. As we have already pointed out in [45] the nondegeneratesoliton solution exhibits symmetric and asymmetric profiles in the nonidentical velocity casealso but the relative velocity of the solitons should be minimum. We have not displayedtheir plots here for brevity.Recently we found that the occurence of multi-humps depends on the number of distinctwave numbers and modes [53] apart from the nonlinearities. In the present two compo-nent case, the resultant nondegenerate fundamental soliton solution (9a)-(9b) yields only adouble-hump soliton. However a triple-hump soliton and a quadruple hump soliton are alsoobserved in the cases of 3 and 4 component Manakov system cases, respectively. For the N -component case one may expect a more complicated profile, as mentioned in the case oftheory of incoherent solitons [54, 55], involving N -number of humps which are character-ized by 2 N -complex parameters. These results will be published elsewhere. Very recently13e have also reported the existence of nondegenerate fundamental solitons and their vari-ous novel profile structures in other integrable coupled NLS type systems [21] as well. Itshould be pointed out that the multi-hump nature of nondegenerate fundamental solitonis somewhat analogous to partially coherent solitons/soliton complexes [15, 16] where suchpartially coherent solitons can be obtained when the number of modes is equal to the numberof degenerate vector soliton solution [3, 56]. We also note here that the 2-partially coherentsoliton can be deduced from the double-humped nondegenerate fundamental soliton (9a)-(9b) in the Manakov system by imposing the restrictions α (1)1 = e η , α (2)1 = − e η , k = k R , l = k R , k I = l I = 0, where η and η are real constants, in solution (7) [56]. The solitoncomplex reported in [57] is a special case of nondegenerate fundamental soliton solution (7)when the parameters k and l are chosen as real constants and α (1)1 = α (2)1 = 1. B. Nondegenerate two-soliton solution
In order to investigate the collision dynamics of nondegenerate soliton of the form (7), itis essential to derive the expression for the corresponding two soliton solution. To constructit, we consider the seed solutions as g (1)1 = α (1)1 e η + α (1)2 e η and g (2)1 = α (2)1 e ξ + α (2)2 e ξ , η j = k j t + ik j z and ξ j = l j t + il j z , j = 1 ,
2, for Eqs. (6). By proceeding with the proceduregiven in the previous subsection along with these seed solutions we find that the seriesexpansions for g ( j ) , j = 1 , f get terminated as g ( j ) = ǫg ( j )1 + ǫ g ( j )3 + ǫ g ( j )5 + ǫ g ( j )7 and f = 1 + ǫ f + ǫ f + ǫ f + ǫ f . The other unknown functions, g ( j )9 , g ( j )11 , f , f andetc., are found to be identically zero. We further note here that the termination of theseperturbation series occurs at the order of ǫ in g ( j ) ’s and at the level of ǫ in f for deriving thedegenerate two-soliton solution. The resulting explicit forms of the unknown functions inthe truncated series expansions constitute the following nondegenerate two-soliton solution,14n Gram determinant form, to Eq. (1), g (1) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η ξ ∗ ( k + l ∗ ) e η e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η ξ ∗ ( k + l ∗ ) e η e ξ η ∗ ( l + k ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ e ξ η ∗ ( l + k ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ − | α (1)1 | ( k ∗ + k ) α (1) ∗ α (1)2 ( k ∗ + k ) − α (1)1 α (1) ∗ ( k ∗ + k ) | α (1)2 | ( k + k ∗ ) − | α (2)1 | ( l ∗ + l ) α (2) ∗ α (2)2 ( l ∗ + l )
00 0 0 − α (2)1 α (2) ∗ ( l ∗ + l ) | α (2)2 | ( l ∗ + l )
00 0 0 0 − α (1)1 − α (1)2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (13a) g (2) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η ξ ∗ ( k + l ∗ ) e η e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η ξ ∗ ( k + l ∗ ) e η e ξ η ∗ ( l + k ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ e ξ η ∗ ( l + k ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ − | α (1)1 | ( k ∗ + k ) α (1) ∗ α (1)2 ( k ∗ + k ) − α (1)1 α (1) ∗ ( k ∗ + k ) | α (1)2 | ( k + k ∗ ) − | α (2)1 | ( l ∗ + l ) α (2) ∗ α (2)2 ( l ∗ + l )
00 0 0 − α (2)1 α (2) ∗ ( l ∗ + l ) | α (2)2 | ( l ∗ + l )
00 0 0 0 0 0 − α (2)1 − α (2)2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (13b) f = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η ξ ∗ ( k + l ∗ ) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η ξ ∗ ( k + l ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ ξ ∗ ( l + l ∗ ) − | α (1)1 | ( k ∗ + k ) α (1) ∗ α (1)2 ( k ∗ + k ) − α (1)1 α (1) ∗ ( k ∗ + k ) | α (1)2 | ( k + k ∗ ) − | α (2)1 | ( l ∗ + l ) α (2) ∗ α (2)2 ( l ∗ + l ) − α (2)1 α (2) ∗ ( l ∗ + l ) | α (2)2 | ( l ∗ + l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (13c)In the above, the eight arbitrary complex parameters k j , l j , α ( j )1 and α ( j )2 , j = 1 , N -soliton solutionof the Manakov system can be obtained. To derive the N -nondegenerate soliton solution,the power series expansion should be as in the following form g ( j ) = P N − n =1 ǫ n − g ( j )2 n − and f = 1 + P Nn =1 ǫ n f n . The 4 N complex parameters, which are present in the N -solitonsolution, determine the shape of the N -solitons. In Appendix A, we have given the three-soliton solution form explicitly using the Gram determinants. C. Partially nondegenerate two-soliton solution
To show the possibility of occurrence of degenerate and nondegenerate solitons simul-tanously in the Manakov system (1), we restrict the wave numbers k and l (or k and l ) as k = l (or k = l ) but k = l (or k = l ) in the obtained completely nondegeneratetwo-soliton solution (13a)-(13c). As a consequence of this restriction, the wave variables η and ξ automatically get restricted as ξ = η . By imposing such a restriction in the fullynondegenerate two-soliton solution (13a)-(13c) we deduce the following form of partiallynondegenerate two-soliton solution as g (1) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η e ξ η ∗ ( l + k ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ − | α (1)1 | ( k ∗ + k ) α (1) ∗ α (1)2 ( k ∗ + k ) − α (1)1 α (1) ∗ ( k ∗ + k ) | α (1)2 | ( k + k ∗ ) − | α (2)1 | ( k ∗ + k ) α (2) ∗ α (2)2 ( k ∗ + l )
00 0 0 − α (2)1 α (2) ∗ ( l ∗ + k ) | α (2)2 | ( l ∗ + l )
00 0 0 0 − α (1)1 − α (1)2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (14a)16 (2) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η e ξ η ∗ ( l + k ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ ξ ∗ ( l + l ∗ ) e ξ − | α (1)1 | ( k ∗ + k ) α (1) ∗ α (1)2 ( k ∗ + k ) − α (1)1 α (1) ∗ ( k ∗ + k ) | α (1)2 | ( k + k ∗ ) − | α (2)1 | ( k ∗ + k ) α (2) ∗ α (2)2 ( k ∗ + l )
00 0 0 − α (2)1 α (2) ∗ ( l ∗ + k ) | α (2)2 | ( l ∗ + l )
00 0 0 0 0 0 − α (2)1 − α (2)2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (14b) f = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η η ∗ ( k + k ∗ ) e η ξ ∗ ( k + l ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ η ∗ ( l + k ∗ ) e ξ ξ ∗ ( l + l ∗ ) − | α (1)1 | ( k ∗ + k ) α (1) ∗ α (1)2 ( k ∗ + k ) − α (1)1 α (1) ∗ ( k ∗ + k ) | α (1)2 | ( k + k ∗ ) − | α (2)1 | ( k ∗ + k ) α (2) ∗ α (2)2 ( k ∗ + l ) − α (2)1 α (2) ∗ ( l ∗ + k ) | α (2)2 | ( l ∗ + l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (14c)The above new class of solution (14a)-(14c) can be derived through Hirota bilinear methodwith the following seed solutions, g (1)1 = α (1)1 e η + α (1)2 e η and g (2)1 = α (2)1 e η + α (2)2 e ξ , η j = k j t + ik j z and ξ = l t + il z , j = 1 ,
2, for Eqs. (6). Such coexistence of degenerate andnondegenerate solitons and their dynamics are characterized by seven complex parameters k j , l , α ( j )1 and α ( j )2 , j = 1 ,
2. The interesting collision behaviour of the coexisting degenerateand nondegenerate solitons is discussed in section V.
IV. VARIOUS SHAPE PRESERVING AND SHAPE CHANGING COLLISIONSOF NONDEGENERATE SOLITONS
The several interesting collision properties associated with the nondegenerate solitonscan be explored by analyzing the asymptotic forms of the two-soliton solution (13a)-(13c)of Eq. (1). By doing so, we observe that the nondegenerate solitons undergo three typesof collision scenarios. For either of the two cases (i) Equal velocities: k I = l I , k I = l I k I = l I , k I = l I , the nondegenerate two solitons undergoshape preserving, shape altering and shape changing collision behaviours. Here we presentthe asymptotic analysis for the case of shape preserving collision only and it can be carriedout for other cases also in a similar manner.. A. Asymptotic analysis
In order to study the interaction dynamics of nondegenerate solitons completely, we per-form a careful asymptotic analysis for the nondegenerate two soliton solution (13a)-(13c)and we deduce the explicit forms of individual solitons at the limits z → ±∞ . To explorethis, we consider k jR , l jR > j = 1 , k I > k I , l I > l I , k I = l I and k I = l I , whichcorresponds to the case of a head-on collision between the two symmetric nondegeneratesolitons. In this situation the two symmetric fundamental solitons S and S are well sep-arated and subsequently the asymptotic forms of the individual solitons can be deducedfrom the solution (13a)-(13c) by incorporating the asymptotic nature of the wave variables η jR = k jR ( t − k jI z ) and ξ jR = l jR ( t − l jI z ), j = 1 ,
2, in it. The wave variables η jR and ξ jR behave asymptotically as (i) Soliton 1 ( S ): η R , ξ R ≃ η R , ξ R → ∓∞ as z ∓ ∞ and (ii)Soliton 2 ( S ): η R , ξ R ≃ η R , ξ R → ∓∞ as z ± ∞ . Correspondingly these results leadto the following asymptotic forms of nondegenerate individual solitons.(a) Before collision: z → −∞ Soliton 1: In this limit, the asymptotic forms of q and q are deduced from the two solitonsolution (13a)-(13c) for soliton 1 as below: q ≃ A − k R e iη I cosh( ξ R + φ − ) (cid:2) ( k ∗ − l ∗ ) ( k ∗ + l ) cosh( η R + ξ R + φ − ) + ( k + l ∗ ) ( k − l ) cosh( η R − ξ R + φ − ) (cid:3) , (15a) q ≃ A − l R e iξ I cosh( η R + φ − ) (cid:2) ( k ∗ − l ∗ ) ( k + l ∗ ) cosh( η R + ξ R + φ − ) + ( k ∗ + l ) / ( k − l ) / cosh( η R − ξ R + φ − ) (cid:3) . (15b)Here, φ − = log ( k − l ) | α (2)1 | ( k + l ∗ )( l + l ∗ ) , φ − = log ( l − k ) | α (1)1 | ( k ∗ + l )( k + k ∗ ) , φ − = log | k − l | | α (1)1 | | α (2)1 | | k + l ∗ | ( k + k ∗ ) ( l + l ∗ ) , φ − = log | α (1)1 | ( l + l ∗ ) | α (2)1 | ( k + k ∗ ) , A − = [ α (1)1 /α (1) ∗ ] / and A − = i [ α (2)1 /α (2) ∗ ] / . In the latter,superscript (1 − ) represents soliton S before collision and subscript (1 ,
2) denotes the twomodes q and q respectively.Soliton 2: The asymptotic expressions for soliton 2 in the two modes before collision turn18ut to be q ≃ k R A − e i ( η I + θ − ) cosh( ξ R + ϕ − ) (cid:2) ( k ∗ − l ∗ ) ( k ∗ + l ) cosh( η R + ξ R + ϕ − ) + ( k + l ∗ ) ( k − l ) cosh( η R − ξ R + ϕ − ) (cid:3) , (16a) q ≃ l R A − e i ( ξ I + θ − ) cosh( η R + ϕ − ) (cid:2) ( k ∗ − l ∗ ) ( k + l ∗ ) cosh( η R + ξ R + ϕ − ) + ( k ∗ + l ) ( k − l ) cosh( η R − ξ R + ϕ − ) (cid:3) . (16b)In the above, ϕ − = 12 log ( k − l ) | α (2)2 | ( k + l ∗ )( l + l ∗ ) + 12 log | k − l | | l − l | | k + l ∗ | | l + l ∗ | ,ϕ − = 12 log ( l − k ) | α (1)2 | ( k ∗ + l )( k + k ∗ ) + 12 log | k − l | | k − k | | k + l ∗ | | k + k ∗ | ,ϕ − = 12 log | k − l | | α (1)2 | | α (2)2 | | k + l ∗ | ( k + k ∗ ) ( l + l ∗ ) + 12 log | k − k | | l − l | | k − l | | k − l | | k + k ∗ | | k + l ∗ | | k + l ∗ | | l + l ∗ | ,ϕ − = 12 log | α (1)2 | ( l + l ∗ ) | α (2)2 | ( k + k ∗ ) + 12 log | k − k | | l + l ∗ | | k − l | | k + l ∗ | | k + k ∗ | | k + l ∗ | | k − l | | l − l | ,e iθ − = ( k − k )( l − l )( l ∗ + l )( k − l ) ( k + k ∗ )( k ∗ + l ) ( k ∗ − k ∗ )( l + l ∗ )( l ∗ − l ∗ )( k ∗ − l ∗ ) ( k ∗ + k )( k + l ∗ ) , A − = [ α (1)2 /α (1) ∗ ] / ,e iθ − = ( l − l )( k − l ) ( k + l ∗ ) ( l + l ∗ )( k ∗ − l ∗ ) ( l ∗ − l ∗ )( k ∗ + l ) ( l ∗ + l ) , A − = [ α (2)2 /α (2) ∗ ] / . Here, superscript (2 − ) refers to soliton S before collision.(b) After collision: z → + ∞ Soliton 1: The asymptotic forms for soliton 1 after collision deduced as, q ≃ k R A e i ( η I + θ +1 ) cosh( ξ R + φ +1 ) (cid:2) ( k ∗ − l ∗ ) ( k ∗ + l ) cosh( η R + ξ R + δ − ς ) + ( k + l ∗ ) ( k − l ) cosh( η R − ξ R + φ − δ ) (cid:3) , (17a) q ≃ l R A e i ( ξ I + θ +2 ) cosh( η R + φ +2 ) (cid:2) ( k ∗ − l ∗ ) ( k + l ∗ ) cosh( η R + ξ R + δ − ς ) + ( k ∗ + l ) ( k − l ) cosh( η R − ξ R + φ − δ ) (cid:3) . (17b)Here, φ +1 = φ − + 12 log | k − l | | l − l | | k + l ∗ | | l + l ∗ | , φ +3 = φ − + 12 log | k − k | | k − l | | k − l | | l − l | | k + k ∗ | | k + l ∗ | | k + l ∗ | | l + l ∗ | ,φ +2 = φ − + 12 log | k − l | | k − k | | k + l ∗ | | k + k ∗ | , φ +4 = φ − + 12 log | k − k | | k + l ∗ | | k − l | | l + l ∗ | | k + k ∗ | | k − l | | k + l ∗ | | l − l | ,e iθ +1 = ( k − k )( k − l ) ( k ∗ + k )( k ∗ + l ) ( k ∗ − k ∗ )( k ∗ − l ∗ ) ( k + k ∗ )( k + l ∗ ) , e iθ +2 = ( l − l )( k − l ) ( k + l ∗ ) ( l ∗ + l )( k ∗ − l ∗ ) ( l ∗ − l ∗ )( k ∗ + l ) ( l + l ∗ ) ,A = [ α (1)1 /α (1) ∗ ] / and A = [ α (2)1 /α (2) ∗ ] / , in which superscript (1+) denotes soliton S after collision. 19oliton 2: The expression for soliton 2 after collision deduced from the two soliton solutionis q ≃ A k R e iη I cosh( ξ R + ϕ +1 ) (cid:2) ( k ∗ − l ∗ ) ( k ∗ + l ) cosh( η R + ξ R + ϕ +3 ) + ( k + l ∗ ) ( k − l ) cosh( η R − ξ R + ϕ +4 ) (cid:3) , (18a) q ≃ A l R e iξ I cosh( η R + ϕ +2 ) (cid:2) i ( k ∗ − l ∗ ) ( k + l ∗ ) cosh( η R + ξ R + ϕ +3 ) + ( k ∗ + l ) ( l − k ) cosh( η R − ξ R + ϕ +4 ) (cid:3) , (18b)where ϕ +1 = log ( k − l ) | α (2)2 | ( k + l ∗ )( l + l ∗ ) , ϕ +2 = log ( l − k ) | α (1)2 | ( k ∗ + l )( k + k ∗ ) , ϕ +3 = log | k − l | | α (1)2 | | α (2)2 | | k + l ∗ | ( k + k ∗ ) ( l + l ∗ ) , ϕ +4 = log | α (1)2 | ( l + l ∗ ) | α (2)2 | ( k + k ∗ ) , A = [ α (1)2 /α (1) ∗ ] / and A = i [ α (2)2 /α (2) ∗ ] / . In the latter,superscript (2+) represents soliton S after collision.In the above, η jR = k jR ( t − k jI z ), η jI = k jI t + ( k jR − k jI ) z , ξ jR = l jR ( t − l jI z ), ξ jI = l jI t + ( l jR − l jI ) z , j = 1 , , and that the phase terms ϕ − j , j = 1 , , , ϕ − = ϕ +1 + log | k − l | | l − l | | k + l ∗ | | l + l ∗ | , ϕ − = ϕ +4 + log | k − k | | l + l ∗ | | k − l | | k + l ∗ | | k + k ∗ | | k + l ∗ | | k − l | | l − l | , ϕ − = ϕ +2 + log | k − l | | k − k | | k + l ∗ | | k + k ∗ | , ϕ − = ϕ +3 + log | k − k | | l − l | | k − l | | k − l | | k + k ∗ | | k + l ∗ | | k + l ∗ | | l + l ∗ | . The above asymptotic analysisclearly shows that the shape preserving collision always occur among the nondegeneratesolitons whenever the phase terms obey the conditions, φ − j = φ + j , ϕ − j = ϕ + j , j = 1 , , , . (19) B. Shape preserving and altering collisions: Elastic collision
From the above analysis, we observe that the intensities of nondegenerate solitons S and S in the two modes are the same before and after collision whenever the phase conditions(19) are satisfied. This implies that the initial amplitudes do not get altered after collision j = 1 ,
2. It is also evident from the transition amplitude calculations, T lj = A l + j A l − j , j, l = 1 , j represents the modes and the superscript l ± denotes the nondegen-erate soliton numbers 1 and 2 in the asymptotic regimes z → ±∞ . Again to confirm thatthe intensities of the nondegenerate solitons are preserved during the collision process, wecalculate the transition intensities as well, | T lj | , l, j = 1 ,
2, which can be obtained by takingthe absolute squares of the transition amplitudes T lj ’s. The transition intensities turn out tobe unimodular, that is | T lj | = 1, l, j = 1 ,
2. Physically this implies that the nondegeneratesolitons, for k I = l I , k I = l I , k = l , corresponding to two distinct wave numbers undergoelastic collision without any intensity redistribution between the modes q and q except for20 IG. 5: Shape preserving collision of symmetric nondegenerate solitons - The energy does notget exchanged among the nondegenerate solitons during the shape preserving collision process:(a) and (b) represent collision between two symmetric double-hump solitons. (c) and (d) denoteinteraction among flattop and symmetric double-hump soliton. The parameter values: (a) and(b): k = 0 .
333 + 0 . i , l = 0 .
315 + 0 . i , k = 0 . − . i , l = 0 . − . i , α (1)1 = 0 .
45 + 0 . i , α (1)2 = 0 .
49 + 0 . i , α (2)1 = 0 .
49 + 0 . i and α (2)2 = 0 .
45 + 0 . i . (c) and (d): k = 0 .
43 + 0 . i , l = 0 . . i , k = 0 . − . i , l = 0 . − . i , α (1)1 = 0 . . i , α (1)2 = 0 . . i , α (2)1 = 0 . . i and α (2)2 = 0 .
45 + 0 . i . a finite phase shift. The latter confirms that the polarization vectors associated with thenondegenerate fundamental solitons do not contribute to the energy redistribution amongthe modes. Consequently the nondegenerate solitons in each mode exhibit elastic collision.The total intensity of each soliton is conserved which can be verfied from | A l − j | = | A l + j | , j, l = 1 ,
2. In addition to this, the total intensity in each of the modes is also conserved | A − j | + | A − j | = | A j | + | A j | = constant.During the collision process, the initial phase of each of the soliton is also changed. The21hase shift of soliton S in the two modes gets modified after collision asΦ = φ +1 − φ − = log | k − l || l − l | | k + l ∗ || l + l ∗ | , Φ = φ +2 − φ − = log | k − l || k − k | | k + l ∗ || k + k ∗ | . (20)Similarly the phase shift suffered by soliton S in the two modes are given byΦ = ϕ +1 − ϕ − = log | k + l ∗ || l + l ∗ | | k − l || l − l | , Φ = ϕ +2 − ϕ − = log | k + l ∗ || k + k ∗ | | k − l || k − k | . (21)From the above expressions we conclude that the phases of all the solitons are mainlyinfluenced by the wave numbers k j and l j , j = 1 ,
2, and not by the complex parameters α ( j )1 ’s and α ( j )2 ’s, j = 1 ,
2. This peculiar property of nondegenerate solitons is different in thecase of degenerate vector bright solitons (see Sec. V below) where the complex parameters α ( j )1 ’s and α ( j )2 ’s, associated with polarization constants, play a crucial role in shifting theposition of solitons after collision.Further, to confirm that the profile shapes of the nondegenerate solitons S and S are in-variant under the above elastic collision, we explicitly deduce the relative separation distancebetween the modes of the solitons. This is similar to the analysis which we have alreadydiscussed for the one-soliton solution to confirm the symmetric and asymmetric profile na-tures of the fundamental soliton. As a consequence of this analysis, one would expect thatthe relative separation distance values corresponding to solitons S and S before collisionshould be equal to the values after collision in order to ensure the shape preserving natureof the collision. For this purpose first we deduce the following expressions for relative sep-aration distance for the solitons S and S before and after collisions from the asymptoticforms as ∆ t − = 1 l R log | α (2)1 | ( k − l ) / l R ( k + l ∗ ) / − k R log ( l − k ) / | α (1)1 | k R ( k ∗ + l ) / , (22a)∆ t − = 1 l R log | α (2)2 || k − l | ( k − l ) / | l − l | l R | k + l ∗ | ( k + l ∗ ) / | l + l ∗ | − k R log | α (1)2 || k − k | | k − l | ( l − k ) / k R | k + k ∗ | | k + l ∗ | ( k ∗ + l ) / , (22b)22 t = 1 l R log | α (2)1 || k − l | ( k − l ) / | l − l | l R | k + l ∗ | ( k + l ∗ ) / | l + l ∗ | − k R log | α (1)1 || k − k | | k − l | ( l − k ) / k R | k + k ∗ | | k + l ∗ | ( k ∗ + l ) / , (23a)∆ t = 1 l R log | α (2)2 | ( k − l ) / l R ( k + l ∗ ) / − k R log ( l − k ) / | α (1)2 | k R ( k ∗ + l ) / . (23b)To identify the profile change of a given soliton S (or S ) during the collision, we analyticallyfind the total change in relative separation distance by subtracting the quantity ∆ t n − from∆ t n +12 , n = 1 ,
2. This results in the following expressions for soliton S ,∆ t = ∆ t − ∆ t − = 1 l R log | k − l || l − l | | k + l ∗ || l + l ∗ | − k R log | k − l || k − k | | k + l ∗ || k + k ∗ | , (24)and for soliton S ,∆ t = ∆ t − ∆ t − = 1 l R log | k − l || l − l | | k + l ∗ || l + l ∗ | − k R log | k − l || k − k | | k + l ∗ || k + k ∗ | . (25)To demonstrate the shape preserving collision property of nondegenerate solitons, for thecase k I = l I , k I = l I , we start with various symmetric profiles as initial conditions. InFigs. 5(a) and 5(b) we set two well separated symmetric double-hump soliton profiles asinitial profiles in both the modes. From these figures, we observe that the symmetric natureof double-hump soliton S is preserved in both the modes after collision while interacting withanother symmetric double-hump soliton S except for a finite phase shift, which is alreadydeduced in Eqs. (20) and (21). This can be easily verified from the asymptotic analysis itself.Further, in order to ensure the shape preserving collision scenario of symmetric double-humpsolitons we explicitly compute the numerical value of relative separation distance betweenthe modes of each double-hump solitons by substituting all the parameter values in Eqs.(24) and (25). This action yields the final values as ∆ t = − . t = − . φ − j = φ + j and ϕ − j = ϕ + j , j = 1 , , ,
4, given by Eq. (19). We also show theshape preserving collision between flattop soliton and double-hump soliton occurs in Figs.5(c) and 5(d). The same type of collision behaviour is also observed while the symmetricsingle-hump soliton collides with the symmetric double-hump soliton, which is illustrated23
IG. 6: Shape preserving collision of symmetric nondegenerate solitons: (a) and (b) deonte collisionbetween single-hump and double-hump solitons: The values corresponding to this collision scenarioare k = 0 .
55 + 0 . i , l = 0 .
333 + 0 . i , k = 0 . − . i , l = 0 . − . i , α (1)1 = 0 .
45 + 0 . i , α (1)2 = 0 .
43 + 0 . i , α (2)1 = 0 .
43 + 0 . i and α (2)2 = 0 .
45 + 0 . i . (c) and (d) denote two single-hump solitons interaction: The corresponding parameter values are chosen as k = 0 .
333 + 0 . i , l = − .
316 + 0 . i , k = − . − . i , l = 0 . − . i , α (1)1 = 0 .
45 + 0 . i , α (1)2 = 0 . . i , α (2)1 = 0 . . i and α (2)2 = 0 .
45 + 0 . i . in Figs. 6(a) and 6(b). In Figs. 6(c) and 6(d) we depict the elastic collision between twosymmetric single-hump solitons. From Figs. 6, we find that each soliton retains its structureduring the collision scenario.Next, we illustrate the shape preserving collision among the asymmetric solitons. As wepointed out earlier, the nondegenerate fundamental soliton also admits asymmetric profilesfor k I = l I . To bring out one more asymmetric soliton we set k I = l I in the two-soliton solution (13a)-(13c). In order to study the shape preserving collision of such twoasymmetric solitons, first we locate asymmetric double-hump soliton S along the line η R = k R ( t − k I z ) ≃ ξ R = l R ( t − k I z ) ≃ S alongthe line η R = k R ( t − k I z ) ≃ ξ R = l R ( t − k I z ) ≃
0. These asymmetric structured24
IG. 7: Shape preserving collision of asymmetric nondegenerate solitons: (a) and (b) represent twoasymmetric soliton collision: k = 0 . − . i , l = 0 . − . i , k = 0 . . i , l = 0 . . i , α (1)1 = 0 .
65 + 0 . i , α (1)2 = 0 .
49 + 0 . i , α (2)1 = 0 .
49 + 0 . i and α (2)2 = 0 .
65 + 0 . i (c) and (d)denote asymmetric flattop-double-hump soliton: The corresponding parameter values are chosenas (a): k = 0 . − . i , l = 0 . − . i , k = 0 . . i , l = 0 .
425 + 1 . i , α (1)1 = 0 . . i , α (1)2 = 0 .
43 + 0 . i , α (2)1 = 0 .
43 + 0 . i and α (2)2 = 0 . . i . double-hump solitons also preserve their structure after collision. This is clearly depicted inFigs. 7(a) and 7(b). To ensure the shape preserving nature of asymmetric solitons, we againexplicitly calculate the relative separation distance values for both the asymmetric solitons S and S as ∆ t = ∆ t = − . k I = l I and k I = l I . We illustratesuch collision scenario in Fig. 10. We explain the profile alteration in the head-on colli-sion between slowly moving symmetric double-hump soliton and fastly moving asymmetricdouble-hump soliton as displayed in Figs. 10(a)-(b). To draw this figure we fix the para-metric choice as k = 0 .
41 + 0 . i , l = 0 .
305 + 0 . i , k = 0 . − . i , l = 0 . − . i , α (1)1 = α (2)2 = 0 .
44 + 0 . i and α (1)2 = α (2)1 = 0 .
44 + 0 . i in solution (13a)-(13c). Fromthis figure, we find that while symmetric double-hump soliton S − in the first mode slightlychanges into an asymmetric structure, the asymmetric double-hump soliton S − becomessymmetric. For this kind of shape altering collision the parameter values corresponding toFigs. 10(a)-(b) are inconsistent with the condition (19), eventhough the unimodular con-dition of transition amplitudes is still preserved. Similar kind of profile alteration occursin the second mode also. This is due to the incoherent interaction between the modes q and q . Again similar type of collision property has been observed when a symmetric(or asymmetric) flattop soliton collides with an asymmetric (or symmetric) double-humpsoliton in the q (or q ) component, which is demonstrated in Figs. 10(c) and 10(d) for k = 0 .
425 + 0 . i , l = 0 . . i , k = 0 . − . i , l = 0 . − . i , α (1)1 = α (2)2 = 0 . . i and α (1)2 = α (2)1 = 0 . . i . In Figs. 10(e) and 10(f), we illustrate shape alteration collisionbetween symmetric single-hump and double-hump solitons in both the components by fixingthe parameter values as k = 0 . − . i , l = 0 . − . i , k = 0 .
333 + 1 . i , l = 0 .
55 + 1 . i , α (1)1 = α (2)2 = 0 . . i and α (1)2 = α (2)1 = 0 .
45 + 0 . i . In each of the modes, the collisiontransforms the symmetric double-hump soliton into a slightly asymmetric double-hump soli-ton leaving the symmetric single-hump soliton unaltered. However, in all the above casesthe energy does not get redistributed among the modes eventhough the shape of the solitonsgets altered during the collision. One can prove the unimodular nature of the transitionamplitudes in these cases by following the procedure mentioned earlier in this section. Aswe pointed out earlier, the similar kind of shape preserving and shape altering collisions arealso observed in the case of k I = l I and k I = l I . Here, we have not displayed their plotsand their corresponding asymptotic analysis for brevity.Additionally, in Fig. 11, we display another type of collision scenario for the velocitycondition k I = l I , k I = l I . In this collision scenario the asymmetric double-hump solitonsthat are present in the two modes change dramatically. However, the single-hump solitons26 IG. 8: Shape preserving collision of asymmetric nondegenerate solitons: (a) and (b) representasymmetric single-hump and double-hump soliton collision: k = 0 . − . i , l = 0 . − . i , k = 0 .
333 + 1 . i , l = 0 .
55 + 1 . i , α (1)1 = 1 . . i , α (1)2 = 0 . . i , α (2)1 = 0 . . i and α (2)2 = 1 . . i . (c) and (d) denote collision of two asymmetric single-hump solitons: Theparameter values of each figure are chosen as : k = 0 . − . i , l = − . − . i , k = − . . i , l = 0 .
333 + 1 . i , α (1)1 = 0 .
45 + 3 . i , α (1)2 = 0 . . i , α (2)1 = 0 . . i and α (2)2 = 0 .
45 + 3 . i . undergo collision without any change in their intensity profiles. Due to the incoherentcoupling between the modes, the change occured only in the profile of the double-humpsoliton. One can carry out an appropriate asymptotic analysis for this kind of collisionprocess also. We also note here that this kind of shape changing collision is not observed inthe degenerate case. We remark that elastic collision is also noticed in the case of dissipativesolitons where a new soliton pair (doublet) is formed when single soliton state (singlet)destroys initial doublet state. During this interaction, energy or momentum is not conservedin the fiber laser cavity [66, 68]. But the elastic collision observed in the present conservativesystem is entirely different from the above collision which has been observed in the dissipativesystem. The vector solitons in dissipative systems exhibit several interesting dynamicalfeatures, especially in fiber lasers. Fiber lasers are very useful nonlinear systems to study27 =- = S - S - S + S + -
80 0 8000.1 t | q S - S - S + S + -
80 0 8000.1 t | q FIG. 9: Shape preserving collision between symmetric double-hump soliton and asymmetric double-hump soliton: The parameter values are k = 0 .
333 + 0 . i , l = 0 .
315 + 0 . i , k = 0 . − . i , l = 0 . − . i , α (1)1 = 0 .
45 + 0 . i , α (1)2 = 2 .
49 + 2 . i , α (2)1 = 0 .
49 + 0 . i and α (2)2 = 0 .
45 + 0 . i . the formation and dynamics of temporal optical solitons experimentally. In fact several typesof solitons were observed experimentally in fiber lasers. For instance, vector multi-solitonoperation and vector soliton interaction in an erbium doped fiber laser [41], and a novel typeof vector dark domain wall soliton have been observed in a fiber ring laser [42]. Also vectordissipative soliton operation of erbium-doped fiber lasers mode locked with atomic layergraphene was experimentally investigated [43] and the coexistence of polarization-lockedand polarization rotating vector solitons in a fiber laser with a semiconductor saturableabsorber mirror have been observed experimentally [44]. C. Shape changing collision
Further, here we demonstrate the shape changing collision scenario of nondegeneratesolitons for unequal velocities, that is k I = l I and k I = l I (We also note here that forappropriate choices of parameters for this unequal velocity case as pointed out above bothshape preserving and shape altering cases do occur). During this interaction, we observe thatan intensity redistribution occurs among the modes of nondegenerate fundamental solitonsalong with profile change. We display such a collision dynamics in Figs. 12 and 13. Atypical intensity redistribution phenomenon is demonstrated in Fig. 12 when two asymmetricdouble-hump solitons collide with each other. To bring out this nonlinear phenomenon wechoose the parameter values as k = 1 . − . i , l = 0 . . i , k = 1 . . i , l = 1 . − . i , α (1)1 = α (2)2 = 0 . . i and α (1)2 = α (2)1 = 0 .
45 + 0 . i . From Fig. 12, one can easily observe28hat the profiles of asymmetric double-hump solitons S and S change dramatically aftercollision, where the initial asymmetric solitons S and S lose their identities and reemergewith another set of asymmetric profiles. In addition to the profile changes, there is alsoa finite intensity redistribution which takes place between the two modes of the solitons.However, the total energy of the individual solitons as well as modes is conserved in order tohold the energy conservation of system (1). Similar kind of collision is also depicted in Fig.13, where a drastic change only occurs in the profile of asymmetric double-hump soliton butwithout any change in the asymmetric single-hump soliton. This can be witnessed in Fig.13 by setting the values of the parameters as k = 0 .
36 + 0 . i , l = 0 . − . i , k = 0 . − . i , l = 0 . − . i , α (1)1 = α (2)2 = 0 . . i and α (1)2 = 1 . . i , α (2)1 = 0 . . i in the solution(13a)-(13c). From this figure one can confirm that the intensity redistribution only occursamong the modes of the asymmetric double-hump soliton. A detailed asymptotic analysishas been carried out in order to ensure this peculiar intensity redistribution, which we havegiven in Appendix B. We remark that the nondegenerate solitons also exhibit shape changingcollision for the equal velocity case as well with k I = l I and k I = l I for appropriate choiceof parameters, which are inconsistent with Eq. (19). V. COLLISION BETWEEN NONDEGENERATE AND DEGENERATE SOLI-TONS
In this section, we discuss the collision among degenerate and nondegenerate solitonsadmitted by the two-soliton solution (13a)-(13c) of Manakov system (1) in the partial non-degenerate limit k = l and k = l . The following asymptotic analysis assures that thereis a definite energy redistribution occurs among the modes q and q . A. Asymptotic analysis
To elucidate this new kind of collision behaviour, we analyze the partial nondegeneratetwo-soliton solution (14a)-(14c) in the asymptotic limits z → ±∞ . The resultant actionyields the asymptotic forms corresponding to degenerate and nondegenerate solitons. As wepointed out in the shape preserving collision case, to obtain the asymptotic forms for thepresent case we incorporate the asymptotic nature of the wave variables η jR = k jR ( t − k Ij z )and ξ R = l R ( t − l I z ), j = 1 ,
2, in the solution (14a)-(14c). Here the wave variable29 =- = ( a ) S - S + S - S + -
60 0 6000.040.08 t | q S - S + S - S + ( b ) -
60 0 6000.040.08 t | q z =- = ( c ) S - S + S - S + -
70 0 7000.09 t | q ( d ) S - S + S - S + -
70 0 7000.09 t | q z =- = ( e ) S - S + S - S + -
60 0 600 t | q ( f ) S - S + S - S + -
60 0 600 t | q FIG. 10: Shape altering collision: (a) and (b) denote shape altering collision between symmetricdouble-hump soliton and asymmetric double-hump soliton. (c) and (d) refer to collision betweensymmetric flattop and asymmetric double-hump soliton. (e) and (f) represent interaction betweensingle-hump and asymmetric double-hump soliton. z =- = S - S + S - S + -
70 0 7000.1 t | q S - S + S - S + -
70 0 7000.19 t | q FIG. 11: Shape changing collision between asymmetric double-hump soliton and single-hump soli-ton: k = 0 .
333 + 0 . i , l = 0 .
315 + 0 . i , k = 0 .
315 + 2 . i , l = 0 . − . i , α (1)1 = α (2)2 = 0 . . i , α (1)2 = α (2)1 = 0 .
45 + 0 . i . =- = S - S - S + S + ( c )-
25 0 2501.3 t | q S - S - S + S + ( d )-
25 0 2501.7 t | q FIG. 12: Shape changing collision between two asymmetric double-hump solitons: k = 1 . − . i , l = 0 . . i k = 1 . . i , l = 1 . − . i , α (1)1 = α (2)2 = 0 . . i , α (1)2 = α (2)1 = 0 .
45 + 0 . i . η R corresponds to the degenerate soliton and η R , ξ R correspond to the nondegeneratesoliton. In order to find the asymptotic behaviour of these wave variables we considerthe parametric choice as k R , k R , l R > k I > k I , l I < k I > k I , k I > l I .For this choice, the wave variables behave asymptotically as follws: (i) degenerate soliton S : η R ≃ η R , ξ R → ∓∞ as z → ∓∞ (ii) nondegenerate soliton S : η R , ξ R ≃ η R → ±∞ as z → ±∞ . By incorporating these asymptotic behaviours of wave variablesin the solution (14a)-(14c), we deduce the following asymptotic expressions for degenerateand nondegenerate solitons.(a) Before collision: z → −∞ Soliton 1: In this limit, the asymptotic form for the degenerate soliton deduced from thepartially nondegenerate two soliton solution (14a)-(14c) is q j ≃ A − A − k R e iη I sech( η R + R , j = 1 , , (26)where A − j = α ( j )1 / ( | α (1)1 | + | α (2)1 | ) / , j = 1 , R = ln ( | α (1)1 | + | α (2)1 | )( k + k ∗ ) . Here, in A − j thesuperscript 1 − denote soliton S before collision and subscript j refers to the mode number.Soliton 2: The asymptotic expressions for the nondegenerate soliton S which is present31 =- = ( c ) S - S - S + S + -
70 0 700 t | q S - S - S + S + ( d ) -
70 0 700 t | q FIG. 13: Shape changing collision between asymmetric single-hump and double-hump solitons: k = 0 .
36 + 0 . i , l = 0 . − . i k = 0 . − . i , l = 0 . − . i , α (1)1 = α (2)2 = 0 . − . i , α (1)2 = 1 . . i , α (2)1 = 0 .
45 + 0 . i .FIG. 14: Energy sharing collision between degenerate and nondegenerate soliton: k = l = 1 + i , k = 1 − i , l = 1 . − . i , α (1)1 = 0 . . i , α (2)2 = 0 . . i , α (1)2 = 0 .
25 + 0 . i , α (2)1 = 1 + i . in the two modes before collision are obtained as q ≃ k R A − D (cid:18) e iξ I +Λ cosh( η R + Φ − ∆ e iη I +Λ cosh( ξ R + λ − λ (cid:19) , (27a) q ≃ l R A − D (cid:18) e iη I +Λ cosh( ξ R + Γ − γ e iξ I +Λ cosh( η R + λ − λ (cid:19) , (27b) D = e Λ cosh( η R − ξ R + λ − λ e Λ cosh( i ( η I − ξ I ) + ϑ − ϕ e Λ cosh( η R + η R + λ − R . A − = [ α (1)2 /α (1) ∗ ] / , A − = [ α (2)2 /α (2) ∗ ] / . In the latter the superscript 2 − denotenondegenerate soliton S before collision.(b) After collision: z → + ∞ Soliton 1: The asymptotic forms for degenerate soliton S after collision deduced from thesolution (14a)-(14c) as, q j ≃ A A e i ( η I + θ + j ) k R sech( η R + R ′ − ς , j = 1 , , (28)where A = α (1)1 / ( | α (1)1 | + χ | α (2)1 | ) / , A = α (1)1 / ( | α (1)1 | χ − + | α (2)1 | ) / , χ =( | k − l | | k + k ∗ | ) / ( | k − k | | k + l ∗ | ), e iθ +1 = ( k − k )( k ∗ + k )( k − l ) ( k ∗ + l ) ( k ∗ − k ∗ )( k + k ∗ )( k ∗ − l ∗ ) ( k + l ∗ ) , e iθ +2 = ( k − k ) ( k ∗ + k ) ( k − l )( k ∗ + l )( k ∗ − k ∗ ) ( k + k ∗ ) ( k ∗ − l ∗ )( k + l ∗ ) . Here 1+ in A refers to degenerate soliton S after collision.Soliton 2: Similarly the expression for the nondegenerate soliton, S , after collision deducedfrom the two soliton solution (14a)-(14c) is q ≃ k R A e iη I cosh( ξ R + Λ − ρ ) (cid:2) ( k ∗ − l ∗ ) ( k ∗ + l ) cosh( η R + ξ R + ς ) + ( k + l ∗ ) ( k − l ) cosh( η R − ξ R + R − R ) (cid:3) , (29) q ≃ l R A e iξ I cosh( η R + µ − ρ ) (cid:2) ( k ∗ − l ∗ ) ( k + l ∗ ) cosh( η R + ξ R + ς ) + ( k ∗ + l ) ( k − l ) cosh( η R − ξ R + R − R ) (cid:3) . (30)where ρ j = log α ( j )2 , j = 1 , A = [ α (1)2 /α (1) ∗ ] / , A = i [ α (2)2 /α (2) ∗ ] / . The explicitexpressions of all the constants are given in Appendix C. B. Degenerate soliton collision induced shape changing scenario of nondegeneratesoliton
The coexistence of nondegenerate and degenerate solitons can be brought out from thepartially nondegenerate soliton solution (14a)-(14c). Such coexisting solitons undergo anovel collision property, which has been illustrated in Fig. 14. From this figure, one canobserve that the intensity of the degenerate soliton S is enhanced after collision in thefirst mode and it gets suppressed in the second mode. As we expected the degenerate soli-ton undergoes energy redistribution among the modes q and q . In the degenerate solitoncase, the polarization vectors, A lj = α ( j ) l / ( | α (1)1 | + | α (2)1 | ) / , l, j = 1 ,
2, play crucial rolein changing the shape of the degenerate solitons under collision, where the intensity/energy33edistribution happens between the modes q and q . As we have pointed out in the nextsection, the shape preserving collision arises in the pure degenerate case when the polariza-tion parameters obey the condition, α (1)1 α (1)2 = α (2)1 α (2)2 where α ( j ) i ’s, i, j = 1 ,
2, are complex numbersrelated to the polarization vectors as given above. The above collision is similar to the onewhich occurs in the completely degenerate case [3, 4]. However, this is not true in the caseof nondegenerate solitons. The nondegenerate asymmetric double-hump soliton S exhibitsa novel collision property as depicted in Fig. 14. In both the modes, the nondegenerate soli-ton S experiences strong effect when it interacts with a degenerate soliton. As a result thenondegenerate soliton swtiches its asymmetric double-hump profile into single-hump profilewith an enhancement of intensity along with a phase shift. In addition to the latter case, wealso noticed that the nondegenerate soliton loses its asymmetric double-hump profile intoanother form of asymmetric double-hump profile when it interacts with a degenerate soli-ton. In the nondegenerate case, the relative separation distances (or phases) are in generalnot preserved during the collision. Therefore the mechanism behind the occurence of shapepreserving and changing collisions in the nondegenerate solitons is quite new. These novelcollision properties can be understood from the corresponding asymptotic analysis given inthe previous subsection. The asymptotic analysis reveals that energy redistribution occursbetween modes q and q . In order to confirm the shape changing nature of this interestingcollision process we obtain the following expression for the transition amplitudes, T = ( | α (1)1 | + | α (2)1 | ) / ( | α (1)1 | + χ | α (2)1 | ) / , T = ( | α (1)1 | + | α (2)1 | ) / ( | α (1)1 | χ − + | α (2)1 | ) / . (31)In general, the transition amplitudes are not equal to unity. If the quantity T lj is notunimodular (for this case the constant χ = 1) then the degenerate and nondegenerate solitonsalways exhibit shape changing collision. The standard elastic collision can be recovered when χ = 1. One can calculate the shift in the positions of both degenerate and nondegeneratesolitons after collision from the asymptotic analysis. This new kind of collision property hasnot been observed in the degenerate vector bright solitons of Manakov system [3, 4]. Theproperty of enhancement of intensity in both the components of nondegenerate soliton issimilar to the one observed earlier in the mixed coupled nonlinear Schr¨odinger system [58].The amplification process of a single-humped nondegenerate soliton in both the modes canbe viewed as an application for signal amplification where the degenerate soliton acts as apumping wave. 34 q | q -
30 0 3000.08 t | q , FIG. 15: Degenerate one-soliton: The values are k = 0 . . i , α (1)1 = 1 . . i , α (2)1 = 0 . . i . VI. DEGENERATE VECTOR BRIGHT SOLITON SOLUTIONS AND THEIRCOLLISION DYNAMICS
The already reported degenerate vector one-bright soliton solution of Manakov system(1) can be deduced from the one-soliton solution (7) by imposing k = l in it. The formsof q j given in Eq. (7) degenerates into the standard bright soliton form [3, 48] q j = α ( j )1 e η e η + η ∗ + R , j = 1 , , (32)which can be rewritten as q j = k R ˆ A j e iη I sech( η R + R , (33)where η R = k R ( t − k I z ), η I = k I t +( k R − k I ) z , ˆ A j = α ( j )1 q ( | α (1)1 | + | α (2)1 | ) , e R = ( | α (1)1 | + | α (2)1 | )( k + k ∗ ) , j = 1 ,
2. Note that the above fundamental bright soliton always propagates in both themodes q and q with the same velocity 2 k I . The polarization vectors ( ˆ A , ˆ A ) † have differentamplitudes and phases, unlike the case of nondegenerate solitons where they have onlydifferent phases. The presence of single wave number k in the solution (33) restricts thedegenerate soliton to have a single-hump form only. A typical profile of the degeneratesoliton is shown in Fig. 15. As already pointed out in [3, 4] the amplitude and centralposition of the degenerate vector bright soliton are obtained as 2 k R ˆ A j , j = 1 , R k R ,respectively.Further, the degenerate two-soliton solution can be deduced from the nondegeneratetwo-soliton solution (13a)-(13c) by applying the degenerate limits k = l and k = l . Thisresults in the following standard degenerate two-soliton solution [3], that is35 j ( t, z ) = α ( j )1 e η + α ( j )2 e η + e η + η ∗ + η + δ j + e η + η + η ∗ + δ j e η + η ∗ + R + e η + η ∗ + δ + e η ∗ + η + δ ∗ + e η + η ∗ + R + e η + η ∗ + η + η ∗ + R , (34)where j = 1 , η j = k j ( t + ik j z ), e δ = k k + k ∗ , e R = k k + k ∗ , e R = k k + k ∗ , e δ j = ( k − k )( α ( j )1 k − α ( j )2 k )( k + k ∗ )( k ∗ + k ) , e δ j = ( k − k )( α ( j )2 k − α ( j )1 k )( k + k ∗ )( k + k ∗ ) , e R = | k − k | ( k + k ∗ )( k + k ∗ ) | k + k ∗ | ( k k − k k )and k il = µ P n =1 α ( n ) i α ( n ) ∗ i ( k i + k ∗ l ) , i, l = 1 , µ = +1. The N degenerate vector bright soliton solu-tion can be recovered from the nondegenerate N -soliton solutions by fixing the wave numbersas k i = l i , i = 1 , , ..., N . In passing we also note that the nondegenerate fundamental solitonsolution (7) can arise when we fix the parameters α (1)2 = α (2)1 = 0 in Eq. (34) and renamethe constants k as l and α (2)2 as α (2)1 in the resultant solution. We also note that the abovedegenerate two-soliton solution (34) can also be rewritten using Gram determinants fromthe Gram determinant forms of nondegenerate two-soliton solution (13a)-(13c). Such Gramdeterminant forms of degenerate two-soliton solution are new to the literature.As reported in [3, 4], the degenerate fundamental solitons ( k i = l i , i = 1 ,
2) in theManakov system undergo shape changing collision due to intensity redistribution among themodes. The energy redistribution occurs in the degenerate case because of the polarizationvectors of the two modes combine with each other. This shape changing collision illustratedin Fig. 3 where the intensity redistribution occurs because of the enhancement of soliton S in the first mode and the corresponding intensity of the same soliton is suppressed in thesecond mode. To hold the conservation of energy between the modes the intensity of thesolitons S gets suppressed in the first mode and it is enhanced in the second mode. Thestandard elastic collision has already been brought out in the degenerate case for the veryspecial case α (1)1 α (1)2 = α (2)1 α (2)2 [4, 56]. VII. POSSIBLE EXPERIMENTAL OBSERVATIONS OF NONDEGENERATESOLITONS
To experimentally observe the nondegenerate vector solitons (single hump/double humpsolitons) one may adopt the mutual-incoherence method which has been used to observethe multi-hump multi-mode solitons experimentally (please see Ref. [36]). The Manakovsolitons (degenerate solitons) can also be observed by the same experimental procedure withappropriate modifications (please see Ref. [24]). In the following, we briefly envisage howthe procedure given in Ref. [36] can be modified to generate the single hump/double hump36
IG. 16: Shape changing collision of degenerate two-solitons: k = l = 1+ i , k = l = 1 . − . i , α (1)1 = 0 . . i , α (1)2 = α (2)1 = α (2)2 = 1. soliton (nondegenerate soliton) discussed in our work.To generate the nondegenerate vector solitons it is essential to consider two laser sourcesof different characters, so that the wavelength of the first laser beam is different from thesecond one. Using polarizing beam splitters, each one of the laser beams can be split intoordinary and extraordinary beams. The extraordinary beam coming out from the firstsource can be further split into two individual fields F and F by allowing it to fall on abeam splitter. These two fields are nothing but the reflected and transmitted extraordinarybeams coming out from the beam splitter. The intensities of these two fields are different.Similarly the second beam which is coming out from the second source can also be split intotwo fields F and F by passing through another beam splitter. The intensities of thesetwo fields are also different. As a result one can generate four fields that are incoherentto each other. To set the incoherence in phase among these four fields one should allowthem to travel sufficient distance before coupling is performed. The fields F and F now become nondegenerate two individual solitons in the first mode whereas F and F form another set of two nondegenerate solitons in the second mode. The coupling betweenthe fields F and F can be performed by combining them using another beam splitter.Similarly, by suitably locating another beam splitter, one can combine the fields F and F ,respectively. After appropriate coupling is performed the resultant optical field beams cannow be focused through two individual cylindrical lenses and the output may be recorded inan imaging system, which consists of a crystal and CCD camera. The collision between thenondegenerate two-solitons in both the modes can now be seen from the recorded images.To observe the elastic collision between nondegenerate solitons (single hump/double hump37olitons), one must make arrangements to vanish the mutual coherence property betweenthe solitons F and F in the first mode q and F and F in the second mode q (pleasesee Ref. [24]). The four optical beams are now completely independent and incoherentwith one another. The collision angle at which the nondegenerate solitons interact shouldbe sufficiently large enough. Under this situation, no energy exchange is expected to occurbetween the nondegenerate solitons of the two modes. VIII. CONCLUSION
From the present study, we point out a few applications of our above reported solitonsolutions. The shape preserving collision property of the nondegenerate solitons can be usedfor optical communication applications. The nondegenerate solitons of Manakov system canbe seen as a soliton molecule when k I ≈ k I and l I ≈ l I . Therefore as explained in thecontext of soliton molecule, the double hump (or multi-hump) structure of the nondegeneratesolitons can be useful for sending information of densely packed data [30]. Degenerate solitoncollision induced enhancement of intensity property of nondegenerate soliton is consideredas signal amplification application. Recently the various properties associated with solitonmolecule have been explored in the literature [30, 31, 40, 63, 64]. Also breather wavemolecule has been identified in [65]. The interesting collision property of degenerate solitonhas already been shown that it is useful for optical computing [28, 56]. Our results providea new possibility to investigate nondegenerate type solitons in both integrable and non-integrable systems. The present study can also be extended to fiber arrays and multi-modefibers where Manakov type equations describe the pulse propagation. Recently we haveinvestigated the novel dynamics of nondegenerate solitons in N -coupled system and theresults will be published elsewhere.We have derived a general form of nondegenerate one-, two- and three-soliton solutionsfor the Manakov model through Hirota bilinear method. Such new class of solitons admitvarious interesting profile structures. The double-hump formation is elucidated by analysingthe relative velocities of the modes of the solitons. Then we have pointed out the coexistenceof degenerate and nondegenerate solitons in the Manakov system by imposing a wave numberrestriction on the obtained two-soliton solution. We have found that nondegenerate solitonsundergo shape preserving, shape altering and shape changing collision scenarios for bothequal velocities and unequal velocities cases. However, for partially equal velocity case, we38ave demonstrated shape changing collision. By performing appropriate asymptotic analysis,the novel shape changing collision has been explained while the degenerate soliton interactswith the nondegenerate soliton. Finally we recovered the well known energy exchangingcollision exhibiting degenerate soliton solutions from the newly identified nondegenerate oneand two-soliton solutions. We have also verified the stability nature of double hump solitonseven during collision using Crank-Nicolson method as explained in Appendix D. It is alsovery interesting to investigate many possibilities of collision dynamics using three-solitonsolution as deduced in Appendix A. Now we are investigating what will happen when (i)two degenerate solitons interact with a nondegenerate soliton and (ii) two nondegeneratesolitons collide with a degenerate soliton and so on. The results will be published elsewhere. Acknowledgement
The authors are thankful to Prof. P. Muruganandam, Department of Physics, Bharathi-dasan University, Tiruchirapalli - 620 024 for his personal help in verifying the shape pre-serving collision nature of symmetric double-hump solitons numerically with white noiseand Gaussian noise as perturbations. The work of R.R., S.S. and M.L. are supported byDST-SERB Distinguished Fellowship program (SB/DF/04/2017) of M.L.
Appendix A: Three-soliton solution
The explicit form of nondegenerate three-soliton solution of Eq. (1) can be deduced byproceeding with the Eqs. (4) using the series representation upto orders ǫ for g ( N ) and ǫ for f . Then the solution can be expressed using Gram determinant in the following way: g ( N ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A I φ − I B T C N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , f = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A I − I B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , N = 1 , . (A1a)Here the matrices A and B are of the order (6 ×
6) defined as A = A mm ′ A mn A nm A nn ′ , B = κ mm ′ κ mn κ nm κ nn ′ , m, m ′ , n, n ′ = 1 , , . (A1b)The various elements of matrix A are obtained from the following, A mm ′ = e η m + η ∗ m ′ ( k m + k ∗ m ′ ) , A mn = e η m + ξ ∗ n ( k m + l ∗ n ) , (A1c)39 nn ′ = e ξ n + ξ ∗ n ′ ( l n + l ∗ n ′ ) , A nm = e η ∗ n + ξ m ( k ∗ n + l m ) , m, m ′ , n, n ′ = 1 , , . (A1d)The elements of matrix B is defined as κ mm ′ = ψ † m σψ m ′ ( k ∗ m + k m ′ ) , κ mn = ψ † m σψ ′ n ( k ∗ m + l n ) , κ nm = ψ ′ † n σψ m ( l ∗ n + k m ) , κ nn ′ = ψ ′ † n σψ ′ n ′ ( l ∗ n + l n ′ ) . (A1e)In (A1e) the column matrices are ψ j = α (1) j , ψ ′ j = α (2) j , j = m, m ′ , n, n ′ = 1 , , η j = k j t + ik j z and ξ j = l j t + il j z , j = 1 , ,
3. The other matrices in Eq. (A1a) are definedbelow: φ = (cid:16) e η e η e η e ξ e ξ e ξ (cid:17) T , C = − (cid:16) α (1)1 α (1)2 α (1)3 (cid:17) , C = − (cid:16) α (2)1 α (2)2 α (2)3 (cid:17) , = (cid:16) (cid:17) and σ = I is a (6 ×
6) identity ma-trix.
Appendix B: Asymptotic analysis of shape changing collision of nondegeneratesolitons in the unequal velocity case: k I = l I , k I = l I To carry out the asymptotic analysis for the shape changing collision we fix the parametersas k I < k I , l I > l I , k jR , l jR > j = 1 , k I = l I , k I = l I . For this choice thenondegenerate two-soliton solution (13a)-(13c) reduces to the following asymptotic forms:(a) Before collision: z → −∞ Soliton 1: ( η R , ξ R ≃ , η R → + ∞ , ξ R → −∞ ) q ≃ A − k R e i ( η I + θ − ) cosh( ξ R + ψ − ) (cid:2) ( k ∗ − l ∗ ) ( k ∗ + l ) cosh( η R + ξ R + ψ − ) + ( k + l ∗ ) ( k − l ) cosh( η R − ξ R + ψ − ) (cid:3) , (B1a) q ≃ A − l R e i ( ξ I + θ − ) cosh( η R + ψ − ) (cid:2) ( k ∗ − l ∗ ) ( k + l ∗ ) cosh( η R + ξ R + ψ − ) + ( k ∗ + l ) / ( k − l ) / cosh( η R − ξ R + ψ − ) (cid:3) . (B1b)Here, ψ − = log ( k − l ) | k − l | | α (2)1 | ( k + l ∗ ) | k + l ∗ | ( l + l ∗ ) , ψ − = log ( l − k ) | k − k | | α (1)1 | ( k ∗ + l ) | k + k ∗ | ( k + k ∗ ) , e iθ − = ( k − k )( k ∗ + k )( k ∗ − k ∗ )( k + k ∗ ) , ψ − = log | k − k | | k + l ∗ | | α (1)1 | ( l + l ∗ ) | α (2)1 | | k + k ∗ | | k − l | ( k + k ∗ ) , ψ − = log | k − k | | k − l | | k − l | | α (2)1 | | α (1)1 | | k + k ∗ | | k + l ∗ | | k + l ∗ | ( k + k ∗ ) ( l + l ∗ ) , e iθ − = ( k − l ) ( k ∗ + l ) ( k ∗ − l ∗ ) ( k + l ∗ ) , A − = [ α (1)1 /α (1) ∗ ] / and A − = i [ α (2)1 /α (2) ∗ ] / .40oliton 2: ( η R , ξ R ≃ , η R → −∞ , ξ R → + ∞ ) q ≃ k R A − e i ( η I + θ − ) cosh( ξ R + χ − ) (cid:2) ( k ∗ − l ∗ ) ( k ∗ + l ) cosh( η R + ξ R + χ − ) + ( k + l ∗ ) ( k − l ) cosh( η R − ξ R + χ − ) (cid:3) , (B2a) q ≃ l R A − e i ( ξ I + θ − ) cosh( η R + χ − ) (cid:2) ( k ∗ − l ∗ ) ( k + l ∗ ) cosh( η R + ξ R + χ − ) + ( k ∗ + l ) ( k − l ) cosh( η R − ξ R + χ − ) (cid:3) . (B2b)In the above, χ − = 12 log | l − l | ( k − l ) | α (2)2 | | l + l ∗ | ( k + l ∗ )( l + l ∗ ) , χ − = 12 log | k − l | ( l − k )( l + l ∗ ) | α (1)2 | | k + l ∗ | ( k ∗ + l )( k + k ∗ ) ( k + k ∗ ) ,e iθ − = ( k − l ) ( k ∗ + l ) ( k ∗ − l ∗ ) ( k + l ∗ ) , e iθ − = ( l − l )( l + l ∗ )( l ∗ − l ∗ )( l ∗ + l ) , A − = [ α (1)2 /α (1) ∗ ] / ,χ − = 12 log | l − l | | k − l | | k − l | | α (1)2 | | α (2)2 | | l + l ∗ | | k + l ∗ | | k + l ∗ | ( k + k ∗ ) ( l + l ∗ ) , A − = [ α (2)2 /α (2) ∗ ] / ,χ − = 12 log | k − l | | l + l ∗ | | α (1)2 | ( l + l ∗ ) | α (2)2 | | k + l ∗ | | l − l | ( k + k ∗ ) . (b) After collision: z → + ∞ Soliton 1: ( η R , ξ R ≃ , η R → −∞ , ξ R → + ∞ ) q ≃ k R A e i ( η I + θ ) cosh( ξ R + ψ +1 ) (cid:2) ( k ∗ − l ∗ ) ( k ∗ + l ) cosh( η R + ξ R + ψ +3 ) + ( k + l ∗ ) ( k − l ) cosh( η R − ξ R + ψ +4 ) (cid:3) , (B3a) q ≃ l R A e i ( ξ I + θ ) cosh( η R + ψ +2 ) (cid:2) ( k ∗ − l ∗ ) ( k + l ∗ ) cosh( η R + ξ R + ψ +3 ) + ( k ∗ + l ) ( k − l ) cosh( η R − ξ R + ψ +4 ) (cid:3) . (B3b)Here, ψ +1 = 12 log | l − l | ( k − l ) | α (2)1 | | l + l ∗ | ( k + l ∗ )( l + l ∗ ) , ψ +2 = 12 log | k − l | ( l − k ) | α (1)1 | | k + l ∗ | ( k ∗ + l )( k + k ∗ ) ,e iθ = ( k − l ) ( k ∗ + l ) ( k ∗ − l ∗ ) ( k + l ∗ ) , e iθ = ( l − l )( l ∗ + l )( l ∗ − l ∗ )( l + l ∗ ) , A = [ α (1)1 /α (1) ∗ ] / ψ +3 = 12 log | k − l | | k − l | | l − l | | α (1)1 | | α (2)1 | | k + l ∗ | | k + l ∗ | | l + l ∗ | ( k + k ∗ ) ( l + l ∗ ) , A = [ α (2)1 /α (2) ∗ ] / ψ +4 = 12 log | k − l | | l + l ∗ | | α (1)1 | ( l + l ∗ ) | α (2)1 | | k + l ∗ | | l − l | ( k + k ∗ ) . η R , ξ R ≃ , η R → + ∞ , ξ R → −∞ ) q ≃ A k R e i ( η I + θ ) cosh( ξ R + χ +1 ) (cid:2) ( k ∗ − l ∗ ) ( k ∗ + l ) cosh( η R + ξ R + χ +3 ) + ( k + l ∗ ) ( k − l ) cosh( η R − ξ R + χ +4 ) (cid:3) , (B4a) q ≃ A l R e i ( ξ I + θ ) cosh( η R + χ +2 ) (cid:2) i ( k ∗ − l ∗ ) ( k + l ∗ ) cosh( η R + ξ R + χ +3 ) + ( k ∗ + l ) ( l − k ) cosh( η R − ξ R + χ +4 ) (cid:3) , (B4b)where χ +1 = log ( k − l ) | k − l | | α (2)2 | ( k + l ∗ ) | k + l ∗ | ( l + l ∗ ) , χ +2 = log α (2)1 | k − k | ( k − l )( k − l )( k ∗ + l ) | α (1)2 | α (2)2 | k + k ∗ | ( k ∗ + l )( k ∗ + l )( l − k )( k + k ∗ ) , e iθ = ( k − k )( k + k ∗ )( k ∗ − k ∗ )( k ∗ + k ) , e iθ = ( k − l ) ( k + l ∗ ) ( k ∗ − l ∗ ) ( k ∗ + l ) , χ +3 = log | k − k | | k − l | | k − l | | α (1)2 | | α (2)2 | | k + k ∗ | | k + l ∗ | | k + l ∗ | ( k + k ∗ ) ( l + l ∗ ) , A =[ α (1)2 /α (1) ∗ ] / , χ +4 = log | k − k | | k + l ∗ | | α (1)2 | ( l + l ∗ ) | α (2)2 | | k + k ∗ | | k − l | ( k + k ∗ ) and A = i [ α (2)2 /α (2) ∗ ] / .From the above analysis, we find that the structures of individual solitons are invariantbefore and after collisions except for the terms corresponding to the various phases ψ − j , χ − j , ψ + j , χ + j , j = 1 , , ,
4. For instance, from Eqs. (B1a) and (B3a), the phase terms ψ − j , j = 1 , , , q mode change into ψ + j , j = 1 , , , ψ − j = ψ + j , χ − j = χ + j , j = 1 , , , . (B5)Using the complicated shape changing collision property of nondegenerate solitons wecould not identify a linear fractional transformation (as in the case of the degenerate case)in order to construct optical logic gates. 42 ppendix C: Constants which appear in the asymptotic expressions in Section V The various constants which arise in the asymptotic analysis of collision between degen-erate and nondegenerate solitons in Sec. V are given below. e Λ = iα (1)1 ( k − k ) ( k − l ) ( k ∗ + k ) ( k + k ∗ )( k + l ∗ ) | k + l ∗ | α (1)2 ( k ∗ − l ∗ ) ( k ∗ − l ∗ ) e R ∗ + R − R ,e Λ = ( k − k ) ( k ∗ + l ) ( k + k ∗ ) ˆΛ ˆΛ ( k ∗ − k ∗ ) ( k ∗ − l ∗ ) ( k ∗ + k ) , e Λ = | α (1)1 || α (2)1 | ( k + k ∗ )( k + k ∗ )( l + l ∗ ) | k − l | ,e Λ = ( | α (1)1 | + | α (2)1 | ) / ( | α (1)1 | | k − k | | k + l ∗ | + | α (2)1 | | k − l | | k + k ∗ | ) / ,e Λ = | k + l ∗ || k − l | ( | α (1)1 | | k + l ∗ | + | α (2)1 | | k − l | ) / ( | α (1)1 | | k − k | + | α (2)1 | | k + k ∗ | ) / ,e Λ = ( k − l ) ( k + l ∗ ) ( k + l ∗ ) ˆΛ ˆΛ ( k ∗ − l ∗ ) ( k ∗ − l ∗ ) ( k ∗ + l ) , ˆΛ = ( | α (1)1 | ( k − k ) − | α (2)1 | ( k ∗ + k )) / ,e Λ = α (2)1 ( k − k ) ( k − l ) ( k ∗ + l ) ( k + k ∗ )( k ∗ + l ) | k + k ∗ | α (2)2 ( k ∗ − k ∗ ) ( k ∗ − l ∗ ) e R ∗ + R − R , ˆΛ = ( | α (1)1 | ( k − k ) | k + l ∗ | − | α (2)1 | | k − l | ( k ∗ + k )) / , ˆΛ = ( | α (1)1 | | k − k | ( k ∗ + l ) − | α (2)1 | ( k − l ) | k + k ∗ | ) / , ˆΛ = ( | α (2)1 | ( k − l ) − | α (1)1 | ( k ∗ + l )) / ,e Φ21 − ∆212 = | α (1)2 | ( k − k )( k ∗ − k ∗ ) ( k − l ) ( k + k ∗ )( k + k ∗ )( k + k ∗ ) ( k ∗ + l ) , e λ − λ = | α (2)2 || k − l | ( k − l ) ˆΛ ( k + l ∗ ) | k + l ∗ | ( l + l ∗ ) ˆΛ ,e λ − R = | k − k || k − l || k − l | ˆΛ | k + k ∗ | | k + l ∗ | | k + l ∗ | ( | α (1)1 | + | α (2)1 | ) / e R R ,e ϑ − ϕ = ( k − k ) ( k ∗ − l ∗ ) ( k ∗ + l ) ( k + l ∗ ) ( k ∗ − k ∗ ) ( k − l ) e R ∗ R − ( R R ∗ , e λ − λ = | k − k | ˆΛ | k + l ∗ | e R − R | k + k ∗ | | k − l | ˆΛ ,e Γ21 − γ = ( k − l ) ( k − l )( k ∗ − l ∗ ) ( k + l ∗ ) ( k + l ∗ )( k ∗ + l ) e R , e λ − λ = ( k − k )( k − l ) ˆΛ | k + k ∗ | ( k ∗ + l ) ˆΛ e R , ˆΛ = ( | α (1)1 | | k − k | | k + l ∗ | + | α (2)1 | | k − l | | k + k ∗ | ) / ,e R ′− ς = | k − k || k − l | ˆΛ | k + k ∗ | | k + l ∗ | ( k + k ∗ ) , e ς = | k − l || k + l ∗ | e R R , e R − R = | α (1)2 | ( l + l ∗ ) | α (2)2 | ( k + k ∗ ) , ˆΛ = ( | α (1)1 | | k − k | + | α (2)1 | | k + k ∗ | ) / , ˆΛ = ( | α (1)1 | | k + l ∗ | + | α (2)1 | | k − l | ) / ,e Λ22 − ρ = ( k − l ) ( k + l ∗ ) e R , e µ − ρ = ( l − k ) ( k ∗ + l ) e R , e R = | α (1)1 | ( k + k ∗ ) , e R = α (1)1 α (1) ∗ ( k + k ∗ ) ,e R = | α (1)2 | ( k + k ∗ ) , e R = | α (2)1 | ( k + k ∗ ) , e R = α (2)1 α (2) ∗ ( k + l ∗ ) , e R = | α (2)2 | ( l + l ∗ ) . ppendix D: Numerical stability analysis corresponding to Figs. 5(a) and 5(b)under perturbation In this appendix, we wish to point out the stability nature of the obtained nondegeneratesoliton solutions numrerically using Crank-Nicolson procedure even under the addition ofsuitable white noise or Gaussian noise to the initial conditions. Specifically we consider theshape preserving collision of symmetric double hump solitons discussed in Figs. 5. For thispurpose, we have considered the Manakov system (1) with the initial conditions, q j (0 , t ) = [1 + Aζ ( t )] q j, ( t ) , j = 1 , . (D1)In the above, q j, ’s, j = 1 ,
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