Multihumped nondegenerate fundamental bright solitons in N-coupled nonlinear Schrödinger system
aa r X i v : . [ n li n . PS ] F e b Multihumped nondegenerate fundamental brightsolitons in N -coupled nonlinear Schr¨odinger system R. Ramakrishnan, S. Stalin † , M. Lakshmanan ‡ Department of Nonlinear Dynamics, Bharathidasan University, Tiruchirappalli - 620024, Tamilnadu, India.E-mail: ‡ [email protected] E-mail: † [email protected] (Corresponding Author) Abstract.
In this letter we report the existence of nondegenerate fundamentalbright soliton solution for coupled multi-component nonlinear Schr¨odinger equationsof Manakov type. To derive this class of nondegenerate vector soliton solutions, weadopt the Hirota bilinear method with appopriate general class of seed solutions. Veryinterestingly the obtained nondegenerate fundamental soliton solution of the N -couplednonlinear Schr¨odinger (CNLS) system admits multi-hump natured intensity profiles.We explicitly demonstrate this specific property by considering the nondegeneratesoliton solutions for 3 and 4-CNLS systems. We also point out the existence of a specialclass of partially nondegenerate soliton solutions by imposing appropriate restrictionson the wavenumbers in the already obtained completely nondegenerate soliton solution.Such class of soliton solutions can also exhibit multi-hump profile structures. Finally,we present the stability analysis of nondegenerate fundamental soliton of the 3-CNLSsystem as an example. The numerical results confirm the stability of triple-humpedprofile nature against perturbations of 5% and 10% white noise. The multi-humpnature of nondegenerate fundamental soliton solution will be usefull in multi-leveloptical communication applications with enhanced flow of data in multi-mode fibers. Multi-level optical communication with high bit-rate data transmission is a hotlydebated topic and is a challenging task in optical communication applications. Usingwavelength division multiplexing scheme, the conventional binary data transmissionapproaches its limit [1], where the maximum data-carrying rate of the fiber is restrictedby Shannon’s theorem [2] due to channel capacity crunch. In the conventional binarydata coding, the presence of light pulse is represented by logical “1” and logical “0”corresponds to its absence. However, the demand for fiber’s information carryingcapacity is increasing day by day. To improve the underlying technology it has beenproposed that soliton assisted fiber-optic telecommunication will play a crucial rolein determining the future communication systems. Several coding schemes have beenproposed in the past to develop this technology: For examble, solitons [3], whichare stable localized nonlinear wave solutions of nonlinear Schr¨odinger equation, arebeing proposed as constituting a model for optical pulse propagation in fibers asnatural bits for coding the information. Recently, the existence of soliton moleculesin dispersion-managed fiber [4] has been demonstrated and their possible usefulness inoptical telecommunications technology with enhanced data carrying capacity has beenpointed out [5]. Soliton molecule is a bound soliton state which can be formed whentwo antiphase solitons persist at a stable equilibrium separation distance, where theinteraction force is null among the individuals. Such stable equilibrium manifests thisbound state structure, reminiscent of a diatomic molecule in condensed matter physics.The binding force arises between the constituents of the soliton compound due to theKerr nonlinearity [6, 7] and the detailed mechanism can be found in Ref. [8]. Theexistence of two-pulse and three-pulse molecules complete the next level of alphabet ofsymbols. Such soliton molecules allow coding of two-bits of information simultanouslyin a single time slot. In this way, the soliton molecules increase the flow of data in fibers.It should be noted here that the initial shape (symmetric peaks with equal intensities) ofsolion molecules changes due to various losses in the fiber and its intrinsic nonlinearities.However, their fundamental properties do not change during the evolution. Apart fromthe above, the concept of soliton molecules has been discussed earlier in detail in thecontext of non-dispersion managed fiber [9–11] and in fiber lasers [12–14]. In additionto the above, multi-soliton complexes in multimode fibers have also been discussed forincreasing the bit-rate in multi-level coding of information [15–17].Very recently we have identified a new class of nondegenerate vector bright solitons[18], with double-hump nature characterized by two distinct wavenumbers, for theManakov system [19]. Basically the Manakov system is a model for propagation oforthogonally polarized optical waves in birefringent fiber, where the solitons undergocollision without energy redistribution in general among the modes depending uponthe choice of soliton parameters [18, 20]. However, they encounter shape changingcollision for suitable choice of parameters whenever they interact with themselves orwhen they collide with degenerate vector brights solitons, that is solitons with single-peak intensity profile described by identical wavenumbers in both the modes [21]. Suchnondegenerate solitons (NDSs) exhibit multi-hump profiles, as we describe below inthe present letter, in the case of N -CNLS system which may be relevent for opticalcommunication applications. By exploiting the multi-peaks, with different peak powers,the nature of NDSs can be made useful to code the two bits of information as describedin [1] in the next level of binary coding. To the best of our knowledge study onnondegenerate solitons in multi-mode fibers or fiber arrays is missing in the literature andtheir existence in multi-component nonlinear Schr¨odinger system and their usefulnessin the context of higher bit-rate information transmission applications have not beenreported. In addition, the underlying interesting analytical forms of NDSs and theirgeometrical profiles have not been revealed so far in the literature and they need to beanalysed in detail.In this letter, we intend to investigate the multi-hump nature of nondegeneratefundamental solitons in the following system of multi-component nonlinear Schr¨odingerequations iq j,z + q j,tt + 2 N X p =1 | q p | q j = 0 , j = 1 , , ..., N, (1)by deriving their analytical forms through Hirota bilinear method. Equation (1)describes the optical pulse propagation in N -mode optical fibers [22] and it describesthe incoherent light beam propagation in photorefractive medium [16] and so on. In theabove, q j ’s are complex wave envelopes propagating in N -optical modes and z and t represent the normalized distance and retarded time, respectively. We note thatfor N = 2 in Eq. (1), we have studied the collision and stability properties of thenondegenerate solitons [21] and also we have identified their existence in other integrablenonlinear Schr¨odinger family of equations by revealing their analytical forms [23]. Toderive the exact form of the nondegenerate fundamental soliton solution for the N -CNLS sytem, we bilinearize Eq. (1) through the dependent variable transformation, q j ( z, t ) = g ( j ) ( z,t ) f ( z,t ) , j = 1 , , ..., N where g ( j ) ’s are in general complex functions and f isa real function. Substitution of this transformation in Eq. (1) brings out the followingbilinear forms: ( iD z + D t ) g ( j ) · f = 0 and D t f · f = 2( P Nn =1 g ( n ) · g ( n ) ∗ ). Here D z and D t are the usual Hirota bilinear operators [24]. Then we consider the standard Hirotaseries expansions g ( j ) = ǫg ( j )1 + ǫ g ( j )3 + ... , j = 1 , , ..., N and f = 1 + ǫ f + ǫ f + ... inthe solution construction process.To obtain the nondegenerate fundamental soliton solution of Eq. (1) we considerthe general forms of N -seed solutions, g ( j ) = α ( j )1 e η j , η j = k j t + ik j z , where α ( j )1 and k j , j = 1 , , ..., N are complex parameters and are nonidentical in general to the N -independent linear partial differential equations, ig ( j )1 ,z + g ( j )1 ,tt = 0, j = 1 , , ..., N , whicharise at the lowest order of ǫ . With such general choices of seed solutions, we proceedto solve the resulting inhomogeneous linear partial differential equations successively inorder to deduce the full series solution upto g ( j )2 N − in g ( j ) and f N in f . By combining theobtained forms of the unknown functions as per the series expansions we find a rathercomplicated form of the nondegenerate fundamental soliton solution for the N -CNLSequation. However, we have managed to rewrite it in a more compact form using thefollowing Gram determinants [25, 26], g ( N ) = (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) , f = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A I − I B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2)where the elements of the matrices A and B are A ij = e η i + η ∗ j ( k i + k ∗ j ) , B ij = κ ji = ψ † i σψ j ( k ∗ i + k j ) , C N = − (cid:16) α (1)1 , α (2)1 , . . . , α ( N )1 (cid:17) ,ψ j = (cid:16) α (1)1 , α (2)1 , . . . , α ( j )1 (cid:17) T , φ = (cid:16) e η , e η , . . . , e η n (cid:17) T , j, n = 1 , , .., N. In the above, g ( N ) and f are ((2 N ) + 1) and (2 N )th order determinants, T representsthe transpose of the matrices ψ j and φ , † denotes transpose complex conjugate, σ = I is an (n × n) identity matrix, φ denotes (n ×
1) column matrix, is a(1 × n) null matrix, C N is a (1 × n) row matrix and ψ represents a (n × κ ji ’s do not exist ( κ ji = 0) in the square matrix B when j = i . Also for a given set of N and j values the corresponding elements onlyexist and all the other elements are equal to zero in C N and ψ j matrices (we havedemonstrated the latter clearly for the 3-component case below). We have verifiedthe validity of the nondegenerate fundamental soliton solution (2) by substituting itin the bilinear equations of Eq. (1) along with the following derivative formula ofthe determinants, ∂M∂x = P ≤ i,j ≤ n ∂a i,j ∂x ∂M∂a i,j = P ≤ i,j ≤ n ∂a i,j ∂x ∆ i,j , where ∆ i,j ’s are thecofactors of the matrix M , the bordered determinant properties and the elementaryproperties of the determinants [24]. This action yields a pair of Jacobi identities andthus their occurrence confirms the validity of the obtained soliton solution. Multi-humpprofile nature is a special feature of the obtained nondegenerate fundamental solitonsolution (2). Such multi-hump structures and their propagation are characterized by2 N arbitrary complex wave parameters. The funamental nondegenerate soliton admitsa very interesting N -hump profile in the present N -CNLS system. In this system,in general, the nondegenerate solitons propagate with different velocities in differentmodes but one can make them to propagate with identical velocity by restricting theimaginary parts of all the wave numbers k j , j = 1 , , ..., N , to be equal. Interestingly, in1976, Nogami and Warke have obtained soliton solution for the multicomponent CNLSsystem [29]. We note that their soliton solution corresponds to the so called partiallycoherent soliton (PCS) which can be checked after replacing the function e j = exp( k j x )by e j = p k j a j exp( k j ¯ x j ), where ¯ x j = x − x j , a j = Π j = i c ij , c ij = k i + k j | k i − k j | and k j ’sare real constants, in their solution [30]. Since, the stationary N -PCS solution arisesfrom our solution (2) under the parametric restrictions α ( j )1 = e η j , j = 1 , , , ..., N and α (2)1 = − e η , ( η j : real), k j = k jR , k jI = 0, j = 1 , , ..., N , the solution of Nogami andWarke [29] and its time dependent version are essentially special cases of our generalsolution (2).It is interesting to note that if we set all the wavenumbers k j , j = 1 , , ..., N ,as identical, k j = k , j = 1 , , ..., N , which corresponds to the seed solutions gettingrestricted as g ( j ) = α ( j )1 e η , η = k t + ik z , for all j = 1 , , ..., N, in the fundamentalsoliton solution (2), the resultant form gets reduced to the following degenerate soliton(DS) solution for Eq. (1) [28] as( q , q , q , ..., q N ) T = ( A , A , A , ..., A N ) T k R e iη I sech (cid:18) η R + R (cid:19) , (3)where η R = k R ( t − k I z ), A j = α ( j )1 / ∆ and ∆ = (( P Nj =1 | α ( j )1 | )) / . Here α ( j )1 , k , j = 1 , , ..., N , are arbitrary complex parameters. Further, k R A j gives the amplitudeof the j th mode, R (= log ∆( k + k ∗ ) ) denotes the central position of the soliton and 2 k I is the soliton velocity [28]. It is evident that the degenerate soliton solution (3) alwaysadmits single-hump structure. Using this single peak intensity or power profile as signalin binary coding one cannot improve higher bit-rate in information transmission aspointed out in [4] whereas this class of degenerate solitons interestingly exhibit energyexhanging collision leading to the construction of all optical logic gates [31]. To enhancethe bit-rate multi-hump pulses with symmetric and asymmetric profiles, as we describebelow for 3 and 4-CNLS systems as examples, can be useful for optical communication.In order to show the multi-hump nature of the nondegenerate soliton, here wedemonstrate such special feature in the case of 3-CNLS and 4-CNLS systems. To startwith, we consider the three coupled nonlinear Schr¨odinger equation ( N = 3 in Eq. (1)).To get the nondegenerate fundamental soliton solution for this system, we considerthe solutions, g (1)1 = α (1)1 e η , g (2)1 = α (2)1 e η and g (3)1 = α (3)1 e η as seed solutions to thelowest order linear PDEs. These general form of seed solutions terminates the seriesexpansions as g ( j ) = ǫg ( j )1 + ǫ g ( j )3 + ǫ g ( j )5 , j = 1 , , f = 1 + ǫ f + ǫ f + ǫ f .By rewriting the explicit forms of the obtained unknown functions in terms of Gramdeterminants we get the resultant forms similar to the one (Eq. (2)) reported above forthe N -component case. We find that for the 3-CNLS system the matrices A and B areconstituted by the elements, A ij and B ij , i, j = 1 , , C N , ψ j and φ are deduced as C = (cid:16) α (1)1 (cid:17) , C = (cid:16) α (2)1 (cid:17) , C = (cid:16) α (3)1 (cid:17) , ψ = (cid:16) α (1)1 (cid:17) T , ψ = (cid:16) α (2)1 (cid:17) T , ψ = (cid:16) α (3)1 (cid:17) T and φ = (cid:16) e η e η e η (cid:17) T . From the resultant Gram-determinant forms, we deducethe following triple-humped nondegenerate fundamental soliton solution for the 3-CNLSsystem, q = 1 f e iη I (cid:16) e ∆51+ ρ cosh( η R + η R + φ e ∆11+∆212 cosh( η R − η R + φ (cid:17) ,q = 1 f e iη I (cid:16) e ∆52+ ρ cosh( η R + η R + ψ e ∆12+∆222 cosh( η R − η R + ψ (cid:17) ,q = 1 f e iη I (cid:16) e ∆53+ ρ cosh( η R + η R + χ e ∆13+∆232 cosh( η R − η R + χ (cid:17) ,f = e δ cosh( η R + η R + η R + δ e δ δ cosh( η R − η R − η R + δ − δ e δ δ cosh( η R − η R − η R + δ − δ e δ δ cosh( η R − η R − η R + δ − δ , (4)where η jR = k jR ( t − k jI z ), j = 1 , , φ = ∆ − ρ , φ = ∆ − ∆ , ψ = ∆ − ρ , ψ = ∆ − ∆ , χ = ∆ − ρ , χ = ∆ − ∆ , ρ j = α ( j )1 , j = 1 , ,
3, and the otherconstants given above are e δ = | α (1)1 | Λ , e δ = | α (2)1 | Λ , e δ = | α (3)1 | Λ , e ∆ = α (1)1 ̺ λ e δ , e ∆ = α (1)1 ̺ λ e δ , e ∆ = − α (2)1 ̺ λ ∗ e δ , e ∆ = α (2)1 ̺ λ e δ , e ∆ = − α (3)1 ̺ λ ∗ e δ , e ∆ = − α (3)1 ̺ λ ∗ e δ , e δ = | ̺ | | λ | e δ + δ , e δ = | ̺ | | λ | e δ + δ , e δ = | ̺ | | λ | e δ + δ , e δ = | ̺ | | ̺ | | ̺ | | λ | | λ | | λ | e δ + δ + δ , e ∆ = α (1)1 ̺ ̺ | ̺ | λ λ | λ | e δ + δ , e ∆ = − α (2)1 ̺ | ̺ | ̺ λ ∗ | λ | λ e δ + δ , e ∆ = α (3)1 | ̺ | ̺ ̺ | λ | λ ∗ λ ∗ e δ + δ ,Λ = ( k + k ∗ ) , Λ = ( k + k ∗ ) , Λ = ( k + k ∗ ) , ̺ = ( k − k ), ̺ = ( k − k ), ̺ = ( k − k ), λ = ( k + k ∗ ), λ = ( k + k ∗ ) and λ = ( k + k ∗ ). The abovenontrivial soliton solution is described by six arbitrary complex parameters, α ( j )1 , k j , j = 1 , ,
3. As a specific example, we can easily check that such multi-parameter | q | q | q ( a ) -
30 0 3000.15 t | q j | q | q | q ( b ) -
10 0 1000.5 t | q j | q | q | q | q ( c ) -
20 0 4000.14 t | q j Figure 1. (a) denotes triple-hump profiles of nondegenerate fundamental soliton in the3-CNLS system and (b) is its corresponding single-humped degenerate soliton profile.(c) represents a quadruple-humped nondegenerate soliton profiles in 4-CNLS system.The specific values of the soliton parameters are given in the text. solution admits a novel asymmetric triple-hump profile when we fix the velocity as k I = k I = k I = 0 .
5. The other parameter values are chosen as k R = 0 . k R = 0 . k R = 0 . α (1)1 = 0 .
65 + 0 . i , α (2)1 = 0 . − . i and α (3)1 = 0 .
35 + 0 . i . InFig. 1(a), we display the asymmetric triple-hump profiles in all the components for theabove choice of parameter values. It is important to note that for the specific choice ofparameter values, the solution (4) also exhibits symmetric triple-hump soliton profile.The symmetric and asymmetric nature of solution (4) can be identified by calculatingthe following relative separation distances between the solitons of the modes,∆ t = t − t = 12 log | α (1)1 | ( k R − k R )( k R + k R ) k R | α (2)1 | ( k R − k R )( k R + k R ) k R , (5 a )∆ t = t − t = 12 log | α (1)1 | ( k R − k R )( k R + k R ) k R | α (3)1 | ( k R − k R )( k R + k R ) k R , (5 b )∆ t = t − t = 12 log | α (2)1 | ( k R − k R )( k R + k R ) k R | α (3)1 | ( k R − k R )( k R + k R ) k R . (5 c )It is evident from Eqs. (5 a )-(5 c ) the solution (4), with k I = k I = k I , always admitsasymmetric triple-hump profiles when ∆ t = ∆ t = ∆ t = 0. In contrast to this,almost symmetric (not perfect symmetric) triple-hump profile arises in all the modeswhen the soliton parameters obey the condition, ∆ t = ∆ t = ∆ t →
0. The doublenode (or multi-node) formation occurs when the relative velocities among the solitonsof the modes, q j ’s j = 1 , ,
3, do not tend to zero. Such node formation is demonstratedin Fig. 2 for the unequal velocity case (of the modes) in the present 3-CNLS system.We wish to point out here that the triple peak power profiles obeying the above relativeseparation distance condition, both symmetric and asymmetric, could be useful in thelaunching of the initial signal in binary coding scheme. In the practical situation theinitial profiles can vary their shape due to fiber’s loss and nonlinear higher order effects.This situation cannot be avoided in a fiber. However, the solution (4) retains thefundamental property, namely the triple-hump soliton profile, of the nondegeneratesoliton during the evolution along the fiber. It is interesting to note that when weimpose the condition k = k = k in the solution (4), it turns out to be a single- Figure 2.
Double-node formation in the unequal velocities case in the profile ofnondegenerate fundamental soliton in 3-CNLS system. The parameter values are k = 0 . . i , k = 0 . . i , k = 0 . . i α (1)1 = 0 . . i , α (2)1 = 0 . − . i and α (3)1 = 0 .
35 + 0 . i . humped degenerate fundamental soliton for the 3-CNLS system. This can be seen fromFig. 1(b) for the values k = k = k = 1 + i , α (1)1 = 0 .
65 + 0 . i , α (2)1 = 0 . − . i and α (3)1 = 0 .
35 + 0 . i . We note that the 3-partially coherent soliton or multi-solitoncomplexes arise from the nondegenerate fundamental soliton solution (4) of the 3-CNLSsystem when the soliton parameters are fixed as α (1)1 = e η , α (2)1 = − e η , α (3)1 = e η , k = k R , k = k R , k = k R and k jI = 0, j = 1 , ,
3, where η j , j = 1 , ,
3, areconsidered as real constants [15, 28].Next we illustrate the multi-hump nature of nondegenerate soliton in the 4-CNLSsystem. To obtain such solution one has to proceed with the analysis for the N = 4 case,as we have described in the above 3-component case. For brevity, we do not give thedetails of the final solution due to its complex nature. However, one can easily deducethe form of the solution from the soliton solution of the N -component case, Eq. (2), asgiven above. The final solution contains eight arbitrary complex parameters, namely α ( j )1 and k j , j = 1 , , ,
4. These parameters play a significant role in determining the profilenature of the underlying soliton in the 4-component case. In general, the nondegenerateone-soliton solution in the 4-CNLS system exhibits asymmetric quadruple-hump profilein all the modes. Such novel quadruple-hump profile is displayed in Fig. 1(c) for theparameter values k = 0 .
48 + 0 . i , k = 0 . . i , k = 0 .
53 + 0 . i , k = 0 .
55 + 0 . i , α (1)1 = 0 .
65 + 0 . i , α (2)1 = 0 . − . i , α (3)1 = 0 .
45 + 0 . i and α (4)1 = 0 . − . i . Wehave verified the asymmetric quadruple-hump profile nature by calculating the relativeseparation distance, ∆ t = ∆ t = ∆ t = 0. However we do not present their explicitforms due to size limitation of the letter article. It is evident from Figs. 1(a) and 1(c)that the nondegenerate soliton (in 3, 4 and also in the arbitrary N ( >
4) CNLS systems)exhibits multi-hump nature. This multi-peak nature can increase the bit-rate in codingthe information. Consequently it can help to uplift the flow of data in fiber. In thepresent 4-CNLS system case also multi-node forms when the relative velocities of thesolitons among the modes do not tend to zero. One can also recover the already knowndegenerate soliton solution by fixing the condition k = k = k = k in the final form | q | q | q ( a ) -
30 0 3000.15 t | q j | q | q | q | q ( b ) -
20 0 3000.12 t | q j | q | q | q | q ( c ) -
30 0 3000.15 t | q j Figure 3. (a) denotes double-humped profile of the partially nondegenerate one solitonsolution of 3-CNLS system. (b) and (c) represent triple and double-humped profiles ofpartially nondegenerate soliton solution of 4-CNLS system when the conditions k = k and k = k = k on wavenumbers are imposed, respectively. of nondegenerate soliton solution of the 4-CNLS system.In the following, we further report the fact that the N -CNLS system can also admitvery interesting partially nondegenerate soliton solution when the wavenumbers arerestricted suitably. Such partial nondegenerate soliton solutions also exhibit multi-humpprofiles (but less than N in number). For instance, here we demonstrate their existencefor the 3 and 4-CNLS systems and this procedure can be generalized to the N -componentcase in principle. For the 3-component case, the partially nondegenerate soliton solutioncan be obtained by imposing the condition, k = k (or k = k or k = k ), on thewave numbers in the solution (4). This restriction reduces the asymmetric triple-humpprofile, as depicted in Fig. 1(a), into the asymmetric double-hump intensity profile asdisplayed in Fig. 3(a) for the choice of parameters k = k = 0 . . i , k = 0 .
45 + 0 . i , α (1)1 = 0 .
65 + 0 . i , α (2)1 = 0 . − . i and α (3)1 = 0 .
35 + 0 . i . The partially NDSdouble-hump profile is described by the following explicit form of solution, deducedfrom solution (4), q = 1 f e iη I e ∆21+ ρ cosh( η R + ∆ − ρ , q = 1 f e iη I e ∆+ ρ cosh( η R + ∆ − ρ ,q = 1 f e iη I (cid:16)
12 [cosh(2 η R − η R + ∆ ) + sinh(2 η R − η R + ∆ )]+ e ∆22+ ρ cosh( η R + ∆ − ρ (cid:17) ,f = e ¯ δ cosh( η R + η R + ¯ δ e ¯ δ δ cosh( η R − η R + ¯ δ − δ . (6)In the above e ¯ δ = e δ + e δ , e ¯ δ = e δ + e δ , e ∆ = e ∆ + e ∆ , η = η = k t + ik z , η = k t + ik z and the other constants are deduced from the constants of the solution(4) by imposing the condition k = k in them. We point out that one can get thedegenerate soliton solution by imposing the restriction further on the wavenumbers,that is as we mentioned above k = k = k leads to completely degenerate solitonsolution. It is important to note that partially nondegenerate soliton solution of the3-CNLSE can exhibit only upto double hump profile in all the three modes due tothe degeneracy among the modes and the nature of this solution is controlled by fivearbitrary complex parameters.Similarly, for the 4-CNLS equation, partially nondegenerate soliton solution canbe deduced from the solution (2) of N -component case. However, due to the complexnature of the resultant solution we do not present the expression here. Very interestinglysuch solution provides the following three possibilities: (i). k = k , (ii). k = k = k and (iii) k = k = k = k . The quadruple-hump soliton profile of the 4-CNLSsystem becomes a triple-hump profile when we consider the first possibility, k = k .This triple-humped partially nondegenerate soliton solution is diplayed in Fig. 3(b)for k = k = 0 .
55 + 0 . i , k = 0 . . i , k = 0 .
45 + 0 . i , α (1)1 = 0 .
65 + 0 . i , α (2)1 = 0 . − . i , α (3)1 = 0 .
45 + 0 . i and α (4)1 = 0 . − . i . In contrast to thelatter, we observe that the double-hump soliton profile emerges while considering thesecond possibility, k = k = k , in the full nondegenerate form of solution of the 4-CNLS system. Such double-humped partially NDS solution profile is depicted in Fig.3(c) for the values k = k = k = 0 .
55 + 0 . i , k = 0 .
45 + 0 . i , α (1)1 = 0 .
35 + 0 . i , α (2)1 = 0 .
45 + 0 . i , α (3)1 = 0 .
55 + 0 . i and α (4)1 = 0 . − . i . The final possibilty, k = k = k = k , corresponds to complete degeneracy. This choice brings out thecompletely degenerate soliton solution for the 4-CNLS system. In general, for the N -component case, one would expect N − k = k = ... = k n , whereas the partial nondegeneracy appearsfrom out of the remaining N − q j ( − , t ) = [1 + Aζ ( t )] q j, − ( t ), j = 1 , ,
3, where q j, − ( t ), j = 1 , ,
3, are the initial profiles obtained from the solution (4) at z = − A is the amplitude of the white noise which is generated from the random numbers in theinterval [ − ,
1] and ζ ( t ) is the noise function. The space and time step sizes are fixedin the numerical calculation, respectively, as dz = 0 . dt = 0 .
2. We also fix thedomain range values for both t and z as [ − , N -coupled nonlinear Schr¨odinger equations (1). This new class ofsolitons exhibit multi-hump nature among all the modes. The existence of such specialmulti-humped profiles is demonstrated explicitly by considering the nondegenerate0 −
100 0 100 t − z | q | −
100 0 100 t − z | q | −
100 0 100 t − z | q | −
100 0 100 t − − z | q | . . . −
100 0 100 t − − | q | . . . −
100 0 100 t − − | q | . . . Figure 4.
Numerical plots for the asymmetric nondegenerate triple hump solitonprofile with 5% of white noise as perturbation. Top panel denotes the triple-humpprofile of 3-dimensional surface plot and the bottom panel represents the correspondingdensity plots. The soliton parameters correspond to Fig. 1(a). −
100 0 100 t − z | q | −
100 0 100 t − z | q | −
100 0 100 t − z | q | −
100 0 100 t − − z | q | . . . . −
100 0 100 t − − | q | . . . −
100 0 100 t − − | q | . . . Figure 5.
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