Degenerate soliton solutions and their dynamics in the nonlocal Manakov system: II Interactions between solitons
aa r X i v : . [ n li n . S I] J un Noname manuscript No. (will be inserted by the editor)
Degenerate soliton solutions and their dynamics in thenonlocal Manakov system: II Interactions between solitons
S. Stalin · M. Senthilvelan · M. Lakshmanan
Received: date / Accepted: date
Abstract
In this paper, by considering the degenerate two bright soliton solutions ofthe nonlocal Manakov system, we bring out three different types of energy sharingcollisions for two different parametric conditions. Among the three, two of them arenew which do not exist in the local Manakov equation. By performing an asymp-totic analysis to the degenerate two-soliton solution, we explain the changes whichoccur in the quasi-intensity/quasi-power, phase shift and relative separation distanceduring the collision process. Remarkably, the intensity redistribution reveals that inthe new types of shape changing collisions, the energy difference of soliton in thetwo modes is not preserved during collision. In contrast to this, in the other shapechanging collision, the total energy of soliton in the two modes is conserved duringcollision. In addition to this, by tuning the imaginary parts of the wave numbers, weobserve localized resonant patterns in both the scenarios. We also demonstrate theexistence of bound states in the CNNLS equation during the collision process forcertain parametric values.
Keywords
Coupled nonlocal nonlinear Schr¨odinger equations · Hirota’s bilinearmethod · Soliton solutions
S. StalinCentre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli-620 024,Tamil Nadu, IndiaM. Senthilvelan (Corresponding Author)Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tamil Nadu, IndiaE-mail: [email protected]. LakshmananCentre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli-620 024,Tamil Nadu, IndiaE-mail: [email protected] S. Stalin et al.
Finding new localized wave solutions, studying their dynamics in the nonlocal in-tegrable equations and obtaining new integrable equations from the nonlocal reduc-tions are active areas of research in the study of integrable systems. Recently in [1],Ablowitz and Musslimani introduced a reverse space nonlocal nonlinear Schr¨odinger(NNLS) equation to explain the wave propagation in a nonlocal medium. A specialproperty associated with this equation is the existence of PT -symmetry when theself-induced potential obeys the PT -symmetry condition [2]. The presence of non-local field as well as PT -symmetric complex potential make the nonlocal equationsmore interesting subject. Various recent studies have shown that the analysis of NNLSequation and its variant have both physical and mathematical perspectives [3]-[35].Further only a few investigations on the dynamics of solitons in the coupled versionof NNLS equation have been reported in the literature [31]-[35]. In particular, breath-ing one soliton solution is constructed for the following nonlocal Manakov equationthrough inverse scattering transform [35], iq j,t ( x, t ) + q j,xx ( x, t ) + 2 X l =1 q l ( x, t ) q ∗ l ( − x, t ) q j ( x, t ) = 0 , j = 1 , . (1)However the soliton shows singularity in a finite time at x = 0 . The above equationis a vector generalization of reverse space NNLS equation. As we pointed out aboveEq. (1) also possess self-induced potential V ( x, t ) = 2 P l =1 q l ( x, t ) q ∗ l ( − x, t ) and PT -symmetry since the later obeys the PT -symmetric condition. In the first part ofaccompanying work [36], we have constructed general soliton solution for Eq. (1)and its augmented version iq ∗ j,t ( − x, t ) − q ∗ j,xx ( − x, t ) − X l =1 q ∗ l ( − x, t ) q l ( x, t ) q ∗ j ( − x, t ) = 0 , j = 1 , . (2)by bilinearizing themt in a non-standard way. The obtained soliton solutions are ingeneral non-singular [36]. In the present second part we study the dynamics of ob-tained two-soliton solution which has been reported in the previous part [36]. Beforeproceeding further, we first summarize the results presented in [36].To construct soliton solution of Eq. (1), we also augment Eq. (2) given above inthe bilinear process. Since, we have treated the fields q l ( x, t ) and q ∗ l ( − x, t ) , l = 1 , ,present in nonlocal nonlinearity as independent fields, to bilinearize them, we intro-duce two auxiliary functions, namely s (1) ( − x, t ) and s (2) ( − x, t ) in the non-standardbilinear process. By solving the obtained bilinear equations systematically, first wehave derived the non-degenerate one soliton solution. From this soliton solution, wehave deduced the degenerate one soliton solution and then the solutions which alreadyexist in the literature. We have also constructed degenerate two-soliton solution forEq. (1). As a continuation of the first part [36] in the present work we study theirdynamics.We show that there exists three types of shape changing collisions in (1) for twospecific parametric conditions, where the first shape changing collision is similar to itle Suppressed Due to Excessive Length 3 the one that arises in the case of local Manakov equation [37]. The second type ofshape changing collision is similar to the one that occurs in the mixed CNLS equation[38]. Besides these two collision scenario, we also observe a third type of collisionwhich is a variant of the second type of shape changing collision and it has not beenobserved in any local 2-CNLS equation. By carrying out the asymptotic analysis forthe fields q j ( x, t ) and q ∗ j ( − x, t ) in a novel way, we deduce the conservation equa-tion, expression for the phase shift and relative separation distances for all the threecollisions. More surprisingly, in the new types of shape changing collisions, the dif-ference in quasi-intensity of the two modes of a soliton before collision is not equal tothe difference in quasi-intensity of the same after collision. However, the total quasi-intensity of solitons before collision is equal to the total quasi-intensity of the solitonsafter collision in both the modes. In another type of collision the total quasi-intensityof individual solitons as well the total quasi-intensity of soliton before and after col-lision in both the modes are conserved. Finally, by tuning the imaginary part of thewave numbers we unearth a new type of localized resonant wave pattern that arisesduring first and second type of collision processes. We also demonstrate the existenceof bright soliton bound states in the nonlocal Manakov equation.The outline of the paper is as follows. In section 2, by performing an asymptoticanalysis, we investigate three types of shape changing collisions via intensity redistri-bution. Our aim here is to calculate the total energy of the solitons in both the modes,as well as phase shifts and relative separation distances. In Sec. 3, we explain theobservation of localized resonant pattern creation during the collision of degeneratetwo solitons. In this section, we also demonstrate the occurrence of bound states thatoccur between the interaction of degenerate two solitons. We present our conclusionsin Sec. 4. Differing from the local case, we perform the asymptotic analysis on the degeneratetwo-soliton solutions of Eq. (1) and Eq. (2). To perform the asymptotic analysis,we rewrite the two-soliton solution of (1) as a nonlinear superposition of two one-solitons. The resultant expressions are similar to the form given in Eqs. (15a)-(15b)in [36] but differ in amplitudes and phases.As far as Eqs. (1) and (2) are concerned, one can identify different types of shapechanging collisions. In particular, here we point out three interesting cases, which wedesignate as Type-I, Type-II and a variant of Type-II collisions.2.1 Type-I shape changing collisionWe visualize Type-I collision for the following choice, namely ¯ k R < , ¯ k R > , ¯ k R < ¯ k R , ¯ k I , ¯ k I > , ¯ k I < ¯ k I ,k R > , k R < , k R > k R , k I , k I > , k I < k I , (3) S. Stalin et al. where k j = k jR + ik jI , ¯ k j = ¯ k jR + i ¯ k jI , j = 1 , , For this choice, the nonlocalsolitons exhibit a shape changing collision similar to the one that occurs in the localManakov equation [37]. We call this type of collision as local Manakov type collisionor Type-I collision. For the parametric restrictions (3) the solitons S and S are wellseparated initially. The variables ξ jR and ¯ ξ jR ’s in the two nonlocal solitons behaveasymptotically as (i) ξ R , ¯ ξ R ∼ , ξ R , ¯ ξ R → ±∞ as t ± ∞ (soliton 1 ( S )) and (ii) ξ R , ¯ ξ R ∼ , ξ R , ¯ ξ R → ∓∞ as t ± ∞ (soliton 2 ( S )). Here the variables, ξ jR and ¯ ξ jR are the real parts of the wave variables ξ j and ¯ ξ j and are equal to − k jI ( x +2 k jR t ) and − ¯ k jI ( x − k jR t ) , respectively.2.2 Type-II shape changing collision and its variantThe nonlocal solitons exhibit another interesting collision scenario for the followingparametric condition: ¯ k R > , ¯ k R < , ¯ k R > ¯ k R , ¯ k I , ¯ k I > , ¯ k I < ¯ k I ,k R < , k R > , k R < k R , k I , k I > , k I < k I . (4)For this choice, the nonlocal Manakov equation admits a collision similar to the onethat occurs in the local mixed CNLS equation [38]. To differentiate this second typeof collision from the earlier one which is pointed out in the previous paragraph wecall this collision as nonlocal mixed CNLS like collision or Type-II collision. For theparametric restriction given in (4) the wave variables ξ jR and ¯ ξ jR ’s behave asymp-totically as (i) ξ R , ¯ ξ R ∼ , ξ R , ¯ ξ R → ∓∞ as t ± ∞ (soliton 1 ( S )) and (ii) ξ R , ¯ ξ R ∼ , ξ R , ¯ ξ R → ±∞ as t ± ∞ (soliton 2 ( S )).For the same parametric restriction chosen in (4) we also come across anothertype of new shape changing collision which we call as variant of Type-II collisionwhich has not been observed in any local -CNLS equation. Thus in the nonlocalManakov equation, we observe three types of shape changing collisions whereas inthe local Manakov equation we come across only one type of shape changing colli-sion [37]. We perform the asymptotic analysis for all the three types of shape chang-ing collisions. Our results show that the asymptotic analysis carried out on Type-IIand its variant collisions match with each other.2.3 Asymptotic forms in Type-I shape changing collisionNow we can check that the parametric choice given in (3) for the Type-I collisionleads to the following asymptotic forms.(i) Before Collision: ( t → −∞ )In the limit t → −∞ , the two-soliton solution reduces to the following two indepen-dent one soliton solutions:(a) Soliton 1: ( ξ R , ¯ ξ R ) ∼ , ξ R , ¯ ξ R → + ∞ q j ( x, t ) = A − j ( k + ¯ k ) e (¯ ξ R − ξ R )2 + i (¯ ξ I − ξ I )2 i [cosh( χ − jR ) cos( χ − jI ) + i sinh( χ − jR ) sin( χ − jI )] , (5a) itle Suppressed Due to Excessive Length 5 where A − j = i ( k +¯ k ) e ρ j − θ j − , ρ j = ln α ( j )1 , θ − j = ∆ ( j )1 − ρ j , χ − jR = ξ R +¯ ξ jR + θ − jR , χ − jI = ξ I +¯ ξ I + θ − jI , j = 1 , . Here, subscript j ( = 1 , ) represents the modes q and q , respectively, and the superscript denotes the soliton 1 at t → −∞ . The parameters A − j and θ − jR denote the amplitude and phase of the soliton 1 in both the componentsbefore collision. The corresponding asymptotic analysis on the fields q ∗ j ( − x, t ) yieldsthe following expression, q ∗ j ( − x, t ) = ˆ A − j ( k + ¯ k ) e − (¯ ξ R − ξ R )2 − i (¯ ξ I − ξ I )2 i [cosh( ˆ χ − jR ) cos( ˆ χ − jI ) + i sinh( ˆ χ − jR ) sin( ˆ χ − jI )] , (5b)where ˆ A − j = i ( k +¯ k ) e ˆ ρ j − ˆ θ j − , ˆ ρ j = ln β ( j )1 , ˆ θ − j = γ ( j )1 − ˆ ρ j , ˆ χ − jR = ξ R +¯ ξ jR +ˆ θ − jR , ˆ χ − jI = ξ I +¯ ξ I +ˆ θ − jI , j = 1 , . In Eq. (5b), ‘ hat’ corresponds to field q ∗ j ( − x, t ) .(b) Soliton 2: ( ξ R , ¯ ξ R ) ∼ , ξ R , ¯ ξ R → −∞ q j ( x, t ) = A − j ( k + ¯ k ) e (¯ ξ R − ξ R )2 + i (¯ ξ I − ξ I )2 i [cosh( χ − jR ) cos( χ − jI ) + i sinh( χ − jR ) sin( χ − jI )] , (5c)where A − j = i ( k +¯ k ) e ∆ ( j )7 − δ − θ − j , θ − j = µ ( j )4 − ∆ ( j )7 , χ − jR = ξ R +¯ ξ R + θ − jR , χ − jI = ξ I +¯ ξ I + θ − jI , j = 1 , . In the above, amplitude and phase of the soliton 2 before col-lision is represented by A − j and θ − jR , respectively. Here, the superscript denotes thesoliton 2 before collision. In the same limit, the asymptotic expression of q ∗ j ( − x, t ) turns out to be q ∗ j ( − x, t ) = ˆ A − j ( k + ¯ k ) e − (¯ ξ R − ξ R )2 − i (¯ ξ I − ξ I )2 i [cosh( ˆ χ − jR ) cos( ˆ χ − jI ) + i sinh( ˆ χ − jR ) sin( ˆ χ − jI )] , (5d)where ˆ A − j = i ( k +¯ k ) e γ ( j )7 − δ − ˆ θ − j , ˆ θ − j = ϕ ( j )4 − γ ( j )7 , ˆ χ − jR = ξ R +¯ ξ R +ˆ θ − jR , ˆ χ − jI = ξ I +¯ ξ I +ˆ θ − jI , j = 1 , .(ii) After Collision: ( t → + ∞ )In this limit, t → + ∞ , the two-soliton solution reduces to the following two onesoliton solutions:(a) Soliton 1: ( ξ R , ¯ ξ R ) ∼ , ξ R , ¯ ξ R → −∞ q j ( x, t ) = A j ( k + ¯ k ) e (¯ ξ R − ξ R )2 + i (¯ ξ I − ξ I )2 i [cosh( χ jR ) cos( χ jI ) + i sinh( χ jR ) sin( χ jI )] , (6a)where A j = i ( k +¯ k ) e µ ( j )1 − δ − θ j , θ j = µ ( j )5 − µ ( j )1 , χ jR = ξ R +¯ ξ R + θ jR , χ jI = ξ I +¯ ξ I + θ jI , j = 1 , . Here, the quantities A j and θ jR define the amplitude and S. Stalin et al. phase of the soliton 1 after collision. In the superscript of the above expressions denotes the soliton 1 at t → + ∞ . q ∗ j ( − x, t ) = ˆ A j ( k + ¯ k ) e − (¯ ξ R − ξ R )2 − i (¯ ξ I − ξ I )2 i [cosh( ˆ χ jR ) cos( ˆ χ jI ) + i sinh( ˆ χ jR ) sin( ˆ χ jI )] , (6b)where ˆ A j = i ( k +¯ k ) e ϕ ( j )1 − δ − ˆ θ j , ˆ θ j = ϕ ( j )5 − ϕ ( j )1 , ˆ χ jR = ξ R +¯ ξ R +ˆ θ jR , ˆ χ jI = ξ I +¯ ξ I +ˆ θ jI , j = 1 , .(b) soliton 2: ( ξ R , ¯ ξ R ) ∼ , ξ R , ¯ ξ R → −∞ q j ( x, t ) = A j ( k + ¯ k ) e (¯ ξ R − ξ R )2 + i (¯ ξ I − ξ I )2 i [cosh( χ jR ) cos( χ jI ) + i sinh( χ jR ) sin( χ jI )] , (6c)where A j = i ( k +¯ k ) e ρ j − θ j , ρ j = ln α ( j )2 , θ j = ∆ ( j )4 − ρ j , χ jR = ξ R +¯ ξ R + θ jR , χ jI = ξ I +¯ ξ I + θ jI , j = 1 , . The amplitude and phase of the soliton 2 in the non-linear Schr¨odinger field after collision is represented by A j and θ j , respectively. q ∗ j ( − x, t ) = ˆ A j ( k + ¯ k ) e − (¯ ξ R − ξ R )2 − i (¯ ξ I − ξ I )2 i [cosh( ˆ χ jR ) cos( ˆ χ jI ) + i sinh( ˆ χ jR ) sin( ˆ χ jI )] , (6d)where ˆ A j = i ( k +¯ k ) e ˆ ρ j − ˆ θ j , ˆ ρ j = ln β ( j )2 , ˆ θ j = γ ( j )4 − ˆ ρ j , ˆ χ jR = ξ R +¯ ξ R +ˆ θ jR , ˆ χ jI = ξ I +¯ ξ I +ˆ θ jI , j = 1 , . One can find the explicit forms of the various constantswhich appear in the asymptotic forms given in Appendix A.Similarly we can calculate the asymptotic forms of Type-II and its variant colli-sions. However, to avoid too many details, we do not present their explicit forms, butonly demonstrate numerically the typical cases.From the above asymptotic forms of solitons S and S , we conclude that a def-inite intensity redistribution has occurred among the modes of the nonlocal solitonswhich can be identified from the amplitude changes in the solitons S and S . Duringthe collision process, the phases of the solitons have also changed. The conservationof total energy (or intensity) of the solitons is yet another quantity which character-izes these three shape changing collisions. The conservation of energy which occursin the Type-II and its variant collisions is entirely different from the collision in thelocal mixed CNLS equation. In order to show the intensity redistribution among themodes of the nonlocal solitons from the asymptotic forms, we calculate the explicitexpressions of the amplitudes and phases of the solitons. The obtained expressions ofall the quantities which appear from asymptotic forms are given in the Appendix A.2.4 Intensity redistributionIn this subsection, first we demonstrate how the intensity redistribution and conser-vation of energy occur between the solitons in the Type-I, Type-II and variant of itle Suppressed Due to Excessive Length 7 Type-II collisions. To demonstrate this, we begin our analysis with the asymptoticforms obtained in the previous sub-section.
In Type-I collision, the analysis reveals that the amplitudes of the solitons S and S are changing from ( k +¯ k ) A − j i and ( k +¯ k ) A − j i to ( k +¯ k ) A j i and ( k +¯ k ) A j i , j =1 , , respectively, due to collision. Similarly the amplitudes of the fields q ∗ j ( − x, t ) , j = 1 , are also changing, during the evolution process, from ( k +¯ k ) ˆ A − j i and ( k +¯ k ) ˆ A − j i to ( k +¯ k ) ˆ A j i and ( k +¯ k ) ˆ A j i , j = 1 , . Here, A ± ij ’s are polarizationvectors of the i th soliton. This is because of the energy sharing interaction that oc-curs between them.In Type-I collision, the quasi-intensity (quasi-power) of the soliton S in the firstmode q ( x, t ) shares with the soliton S in q ( x, t ) mode and the same kind of inten-sity sharing occurs between the modes of the soliton S also. This in turn confirmsthat the intensity redistribution occurs in between the modes. Even though the inten-sity redistribution occurs among the solitons that are present in the modes q ( x, t ) and q ( x, t ) the total energy of the individual solitons is conserved which can beconfirmed from A − · ˆ A − + A − · ˆ A − = A · ˆ A + A · ˆ A = 1 , (7a) A − · ˆ A − + A − · ˆ A − = A · ˆ A + A · ˆ A = 1 , (7b)where the explicit forms of A ± ij , i, j = 1 , are given in Appendix A. In the above,subscripts denote the modes while superscripts represent the soliton number. Theabove conservation form, reveals the fact that the total quasi-intensity of the indi-vidual solitons is conserved. In the local case, the total energy of each soliton iscalculated by adding the absolute squares of the amplitudes of the individual modesof the solitons [37]. Even though the amplitudes of both the co-propagating solitonsare altered after the interaction, the total energy does not vary and is conserved.In addition to the above, the total energy of the solitons is also conserved. Thiscan be verified by the following conservation form A − · ˆ A − + A − · ˆ A − + A − · ˆ A − + A − · ˆ A − = A · ˆ A + A · ˆ A + A · ˆ A + A · ˆ A = 2 . (8)Eq. (8) confirms that the total energy of the solitons S and S before collision isequal to the total energy of the solitons after collision.The change in amplitude of each one of the solitons in both the components canbe evaluated by introducing the transition amplitude T lj by T lj = A l + j A l − j , where A l + j isthe amplitude of the l -th soliton in the j -th component after collision and A l − j is theamplitude of the soliton in the corresponding mode before collision. To calculate the S. Stalin et al. intensity exchange among the modes of the solitons we multiply the transition ampli-tude T lj = A l + j A l − j by the transition amplitude ˆ T lj = ˆ A l + j ˆ A l − j of field q ∗ j ( − x, t ) , where ˆ A l + j and ˆ A l − j are the amplitudes of the solitons of the field q ∗ j ( − x, t ) of each mode afterand before collision respectively. This definition also differs from the local Manakovcase in which we multiply T lj by its own complex conjugate transition element T l ∗ j ,to get | T lj | [ ? ]. The intensity exchange between the solitons S and S due to Type-I Fig. 1
Type-I shape changing collision in CNNLS equation: (a) and (b) are the local Manakov type energysharing collision plotted for the parametric values k = 0 . i . , ¯ k = − . i . , k = − i , ¯ k = 2+ i , α (1)1 = 1+ i , α (1)2 = 1 . i , α (2)1 = 0 . i , α (2)2 = 2+ i , β (1)1 = 1 − i , β (1)2 = − . − i , β (2)1 = − . − i and β (2)2 = 2 − i . collision is defined by T lj · ˆ T lj = A l + j A l − j · ˆ A l + j ˆ A l − j , l, j = 1 , , (9)where all the quantities in the expression (9) are given in the Appendix A. By suitablyfixing the parameters, we can make the right hand side of the above expression to beequal to one. For this special parametric choice, we can come across a pure elasticcollision (or shape preserving collision). For all other parametric values there occursa change in the amplitude in solitons and it leads to the shape changing collision.As in the local CNLS case, one can make one of the transition matrices vanish bysuitably fixing the values of the parameters. For this case, the intensity of any one ofthe solitons in one of the modes becomes zero. In Type-II collision process also the amplitude of the solitons changes in both thefields q j ( x, t ) and q ∗ j ( − x, t ) . In this collision scenario, the quasi-intensity of soliton S is enhanced in both the modes while the quasi-intensity of soliton S is sup-pressed. This collision scenario is entirely different from the one that occurs in theType-I collision. A remarkable feature of the Type-II collision is that the total energyof individual solitons is not conserved, so that A l − · ˆ A l − − A l − · ˆ A l − = A l +1 · ˆ A l +1 − A l +2 · ˆ A l +2 , l = 1 , . (10)From Eq. (10), we infer that the difference in quasi-intensity of soliton S in both themodes before collision is not equal to the same after collision. This is also true for itle Suppressed Due to Excessive Length 9 the soliton S as well. In the local mixed CNLS equation the energy difference turnsout to be the same before and after collision [38]. The free parameters that appear inthe degenerate nonlocal two soliton solution (28a)-(28c) given in [36] do allow thesimilar kind of shape changing collision as the one happens in the case of local mixedCNLS equation.In Type-II shape changing collision, the total intensity of the solitons S and S inboth the components before collision is equal to the the total intensity of the solitons S and S after collision, that is A − · ˆ A − + A − · ˆ A − + A − · ˆ A − + A − · ˆ A − = A · ˆ A + A · ˆ A + A · ˆ A + A · ˆ A = 2 . (11)The intensity exchange between the solitons in the Type-II collision can also be cal-culated by defining the transition matrices. In this case the transition matrices aredefined by T lj · ˆ T lj = A l + j A l − j · ˆ A l + j ˆ A l − j , l, j = 1 , . A special case in which the right handside becomes one produces shape preserving elastic collision.In addition to the above Type-II collision, we also observe a variant of it. In thevariant of Type-II collision, the intensity of soliton S is suppressed in both the modeswhereas the intensity of soliton S is suppressed in q mode and is enhanced in q mode. This collision scenario is entirely different from the previous collision pro-cesses and has not been encountered in any local -CNLS equation. The variant ofType-II collision also obeys the non-conservation and conservation relations (10) and(11), respectively. Fig. 2
Type-II shape changing collision: (a) and (b) are the mixed CNLS like shape changing collisiondrawn for the parametric values, k = − . i . , ¯ k = 0 . i . , k = 2 + i , ¯ k = − i , α (1)1 = 1 + i , α (1)2 = 1 . i , α (2)1 = 0 . i , α (2)2 = 2 + i , β (1)1 = 1 − i , β (1)2 = − . − i , β (2)1 = − . − i , β (2)2 = 2 − i . We recall here that Eq. (1) corresponds to three different equations, namely non-local version of (i) Manakov equation, (ii) defocusing CNLS equation and (iii) mixedCNLS equation, depending upon the sign of σ l . It is noted that the shape changingcollision that occurs in the local Manakov system differs from the one that occurs inthe local mixed CNLS system [38]. For example, the shape changing collision thatoccurs in the mixed coupled NLS equation can be viewed as an amplification processin which the amplification of signal (say soliton 1) using pump wave (say soliton 2)without any external amplification medium and without any creation of noise thatdoes not exist in the local Manakov case [37]. Very surprisingly, the nonlocal Man-akov equation simultaneously admits both the types of shape changing collisions mentioned above, that is the one occurs in the 2-CNLS equation and the other thatoccurs in the mixed coupled NLS equation. This type of collision has not been ob-served in any other (1 + 1) -dimensional nonlocal integrable system. In Figs. 2, 3 and Fig. 3
A variant of Type-II shape changing collision: (a) and (b) represent the intensity switching collisionplotted for the parametric values k = − . i . , ¯ k = 1+ i . , k = 2+ i , ¯ k = − i , α (1)1 = 1+ i , α (1)2 = 1 . i , α (2)1 = 0 . i , α (2)2 = 2 + i , β (1)1 = 1 − i , β (2)1 = − . − i , β (1)2 = − . − i and β (2)2 = 2 − i .
4, we have demonstrated the shape changing collisions that occur in (1) for σ l = +1 .The local Manakov type shape changing collision that occurs in the system (1) is il-lustrated in Figs. 2a-2b whereas in Figs. 3a-3b the shape changing collision that occurin (2) as in the case of mixed CNLS equation is shown. A variant of Type-II inten-sity switching collision is illustrated in Figs. 4a-4b. These three figures also revealthat besides the change in amplitudes, changes also occur in phase shift and relativeseparation distances. In the following, we calculate these changes.2.5 Phase shiftsDuring the collision process, another important quantity, namely the phase is alsobeing altered. The phase identifies essentially the position of the solitons. The changein phase can be calculated from the expressions already obtained. The initial phase ofthe soliton S (= θ − jR k I +¯ k I ) ) changes to θ jR k I +¯ k I ) . Similarly, the initial phase of thesoliton S (= θ − jR k I +¯ k I ) ) changes to θ jR k I +¯ k I ) . Therefore, the phase shift suffered bythe soliton S in both the modes during collision is Φ = 12( k I + ¯ k I ) ln | ρ ¯ ρ ( ̺ Γ Γ − ̺ ν ν − ̺ Γ Γ ) || Γ Γ κ κ | . (12a)Similarly the phase shift suffered by the soliton S is Φ = 12( k I + ¯ k I ) ln | Γ Γ κ κ || ρ ¯ ρ ( ̺ Γ Γ − ̺ ν ν − ̺ Γ Γ ) | . (12b)From the above two phase shift expressions of solitons S and S , we find that Φ = − ( k I + ¯ k I )( k I + ¯ k I ) Φ . (12c) itle Suppressed Due to Excessive Length 11 In the Type-II and its variant shape changing collisions, the phase shift suffered bythe solitons S and S is equal to the phase shift suffered by the soliton S and S inthe Type-I collision, respectively. In the Type-II and its variant collisions, the phaseshift of the solitons one and two is related by the same relation given in (12c). Fromthis relation, we infer that in both the collision processes the soliton S gets phaseshifted opposite to the soliton S . We also find that the phase shifts not only dependon the amplitude parameters α ( j ) i and β ( j ) i , i, j = 1 , but also on the wave numbers k j , ¯ k j . This is similar to the local Manakov equation and mixed CNLS equation.2.6 Relative separation distancesThe changes which occur in phases of both the solitons in turn cause a change in theirrelative separation distances during both the collision process. The relative separationdistance is nothing but the distance between the positions of the solitons after andbefore collision [ ? ]. We denote them by x ± , where x +12 is equal to the position ofsoliton S minus the position of soliton S after collision (at t → + ∞ ) and x − isequal to the position of soliton S minus the position of soliton S before collision(at t → −∞ ), that is x +12 = x +2 − x +1 , x − = x − − x − , where x − and x − denote the positions of S and S at t → −∞ , respectively,whereas x +1 and x +2 are the positions of S and S at t → + ∞ , respectively. Theirexplicit forms can be obtained from the phase shifts of the solitons which turns out tobe x − = 12( k I + ¯ k I ) ln | ρ ¯ ρ ( ̺ Γ Γ − ̺ ν ν − ̺ Γ Γ ) || Γ κ κ κ |− k I + ¯ k I ) ln | Γ || κ | , (13a) x +12 = 12( k I + ¯ k I ) ln | Γ || κ |− k I + ¯ k I ) ln | ρ ¯ ρ ( ̺ Γ Γ − ̺ ν ν − ̺ Γ Γ ) || Γ κ κ κ | . (13b)The total change in relative separation distance is given by ∆x = x +12 − x − = ( k I + ¯ k I + k I + ¯ k I )2( k I + ¯ k I )( k I + ¯ k I ) ln | Γ Γ κ κ || ρ ¯ ρ ( ̺ Γ Γ − ̺ ν ν − ̺ Γ Γ ) | . The above expression can also be rewritten as ∆x = − (cid:18) k I + ¯ k I k I + ¯ k I (cid:19) Φ . (13c) We observe that the amplitude dependent relative separation distance found aboveturns out to be the same as in the case of local Manakov equation and mixed CNLSequation. Similarly, we obtain the same expression for the relative separation in allthree types of shape changing collisions. It is clear from the expression (13c), that therelative separation distance non-trivially depends on all the complex parameters, k j , ¯ k j , α ( j ) i ’s and β ( j ) i ’s, i, j = 1 , . Thus the amplitudes, phases and the relative sepa-ration distances are all get changed during the interaction between the two nonlocalbright solitons.2.7 Role of complex parameters in the collision processFrom the above results it is clear that all the complex parameters, k j , ¯ k j , α ( j ) i ’s and β ( j ) i ’s, i, j = 1 , , play important roles in the soliton collision process. In the localManakov case, the parameters α ( j ) i ’s play a crucial role in the shape changing col-lision process but not the wave numbers [37]. Hereafter we focus only on the threetypes of shape changing collisions which occur in the nonlocal Manakov system.In our investigations, we have identified three kinds of collisions. In Type-I col-lision, the quasi-intensity of soliton S is enhanced and the quasi-intensity of soliton S is suppressed in the first mode q ( x, t ) . In order to obey the conservation law, theswitching of quasi-intensity reversed in the second mode q ( x, t ) , that is the quasi-intensity of solitons S is suppressed in the first mode whereas the quasi-intensity ofsoliton S is enhanced in the second mode. In this case, the quasi-intensities of soli-tons are either partially enhanced or partially suppressed. It is demonstrated in Fig.2a and 2b for the parametric values k = 0 . i . , ¯ k = − . i . , k = − i , ¯ k = 2 + i , α (1)1 = 1 + i , α (1)2 = 1 . i , α (2)1 = 0 . i , α (2)2 = 2 + i , β (1)1 = 1 − i , β (1)2 = − . − i , β (2)1 = − . − i and β (2)2 = 2 − i . The second type of shapechanging collision is illustrated in Figs. 3a-3b for the parameters k = − . i . , ¯ k = 0 . i . , k = 2+ i , ¯ k = − i with all other parameters remaining the sameas mentioned above. In these Figs. 3a and 3b, we observe that the intensity of soliton S is enhanced in the first mode and a similar change also occurs in the second modeas well. The intensity of soliton S is suppressed in both the modes. By comparingthe parameter values of Type-I and Type-II collisions, we can easily identify that theonly difference in them are signs in the real part of wave number. All other parame-ters remain the same. A simple sign change in the real parts of the above parameterscauses a dramatic change in the collision dynamics which in turn reveals the strongdependence of this process on complex parameters. The third type of shape chang-ing collision process is demonstrated in Figs. 4a and 4b, for the parametric values k = − . i . , ¯ k = 1+ i . , k = 2+ i , ¯ k = − i , α (1)1 = 1+ i , α (1)2 = 1 . i , α (2)1 = 0 . i , α (2)2 = 2 + i , β (1)1 = 1 − i , β (2)1 = − . − i , β (1)2 = − . − i and β (2)2 = 2 − i . In these figures also we observe that the quasi-intensity of soliton S issuppressed in both the modes. In contrast to this the quasi-intensity of soliton S isenhanced in q ( x, t ) mode and is suppressed in the q ( x, t ) mode. We note here thatall the parameter values are same for Type-II and its variant collisions except for thevalues of k and ¯ k . In all the shape changing collision process the quasi-intensity of itle Suppressed Due to Excessive Length 13 solitons in both the components get either enhanced or suppressed. This is becauseof energy exchange between the modes and the solitons as well.Finally, we note that in the second collision dynamics one may consider the soli-ton S as the signal whereas the soliton S as the pump wave (or energy reservoir).In this collision scenario the signals get enhanced or amplified without any use ofexternal amplification medium and without any creation of noise. From these resultswe conclude that one can use the focusing type nonlocal medium for simultaneouslyto amplify the signals and to construct the optical computer equivalent to Turing ma-chine in a mathematical sense [39]. One need not go to separately mixed focusing -defocusing type nonlinear medium for amplifying the signals. Fig. 4 (a) and (b) denote the resonant pattern appearing in Type-I collision which is demonstrated for thevalues k = 1 + i . , ¯ k = − i . , k = − i . , ¯ k = 2 + i . , α (1)1 = 1 + i , α (1)2 = 1 . i , α (2)1 = 0 . i , α (2)2 = 2 + i , β (1)1 = 1 − i , β (2)1 = − . − i , β (1)2 = − . − i and β (2)2 = 2 − i . Fig. 5 (a) and (b) represent the resonant pattern appearing in Type-II collision drawn for the parametervalues k = − i . , ¯ k = 1 + i . , k = 2 + i . , ¯ k = − i . , α (1)1 = 1 + i , α (1)2 = 1 . i , α (2)1 = 1 + i , α (2)2 = 2 + i , β (1)1 = 1 − i , β (2)1 = − − i , β (1)2 = 1 . − i and β (2)2 = 2 − i . A specific resonant behaviour has been observed during the interaction process inthe long wave-short wave resonance interaction system (LSRI system) [40]. The res-onant behaviour occurs exactly in the place at which the phase shift occurs duringthe collision process. In other words, the localized resonant patterns appear in theinteraction regime and it can be considered as an intermediate state. The resonancebehaviour was achieved by appropriately choosing the parameters. One can prolongthis intermediate state by fixing the phase shift as large as possible. One can alsoobserve such type of localized resonant behaviour in the present nonlocal Manakov system as well. The resonant behaviour appearing in the Type-I collision is demon-strated in Figs. 5(a)-5(b) for the parametric values k = 1 + i . , ¯ k = − i . , k = − i . , ¯ k = 2 + i . , α (1)1 = 1 + i , α (1)2 = 1 . i , α (2)1 = 0 . i , α (2)2 = 2 + i , β (1)1 = 1 − i , β (2)1 = − . − i , β (1)2 = − . − i and β (2)2 = 2 − i .The same resonant behaviour that one observes in the Type-II shape changing col-lision can be visualized Figs. 6(a) - 6(b) for the parametric values k = − i . , ¯ k = 1+ i . , k = 2+ i . , ¯ k = − i . , α (1)1 = 1+ i , α (1)2 = 1 . i , α (2)1 = 1+ i , α (2)2 = 2 + i , β (1)1 = 1 − i , β (2)1 = − − i , β (1)2 = 1 . − i and β (2)2 = 2 − i . Theresonant behaviours shown in Figs. 5 and 6 are obtained by changing the imaginaryvalues of the wave numbers which we have used to identify the shape changing colli-sion process. From these figures, we observe that the localized resonant pattern arisesin the phase shift regime. In addition to this, we point out that change in the imaginarypart of the wave numbers leads to a switching of the Type-I collision into Type-II col-lision. We also note that the resonant pattern appearing during the collision process isnot same as the one appear in the higher dimensional integrable systems [40]. In thelocal Manakov case, one does not observe such behaviour and this occurs only due tothe manifestation of nonlocal nature of the system. We point out that the same type ofresonant pattern also appears in the variant of Type-II shape changing collision also. Fig. 6 (a) and (b) are the parallel propagation of soliton occur in bound state for the parameter values k = 0 . . i ¯ k = − . . i , k = 0 . . i , ¯ k = − . . i , α (1)1 = 1+ i , α (1)2 = 1+ i , α (2)1 = 0 . i , α (2)2 = 3 + i , β (1)1 = 1 − i , β (2)1 = − . − i , β (1)2 = − − i and β (2)2 = 3 − i . Fig. 7 (a) denote a novel double hump breathing type bound state occur in the first mode q and (b)denotes a single hump breathing type bound state occur in mode q . The Figs. (a) and (b) drawn for thevalues k = 1 + 0 . i ¯ k = − . i , k = 1 + 2 . i , ¯ k = − . i , α (1)1 = 1 + i , α (1)2 = 1 + i . , α (2)1 = 0 . i , α (2)2 = 3+ i . , β (1)1 = 1 − i , β (2)1 = − . − i , β (1)2 = − − i . and β (2)2 = 3 − i . . Multi-soliton bound states exist in both integrable and non-integrable systems.When the velocity of the solitons are equal then the solitons bind together to form itle Suppressed Due to Excessive Length 15 the bound states [41]. The bound states can be in different possible forms such asparallel solitons, composite solitons and so on. The parallel soliton bound state existsif the central position of the solitons are different whereas the composite solitons existwhen the central positions of the solitons are same [41]. However, these bound statesolitons are unstable against small perturbations and they separate into individualsolitons which propagate with their identities after some time.In the following, we illustrate the existence of bound states in the nonlocal Man-akov system (1). As we pointed out in section III. B, the soliton velocities are ruledby the parameters k jR , ¯ k jR , j = 1 , , and the central position of the solitons aregoverned by ∆ R k I +¯ k I ) and ∆ R k I +¯ k I ) respectively. To explore the parallel solitonbound state, we fix the parametric values as k = 0 . . i ¯ k = − . . i , k = 0 . . i , ¯ k = − . . i , α (1)1 = 1 + i , α (1)2 = 1 + i , α (2)1 = 0 . i , α (2)2 = 3 + i , β (1)1 = 1 − i , β (2)1 = − . − i , β (1)2 = − − i and β (2)2 = 3 − i . Theoutcome is displayed in Figs. 7a and 7b. This type of bound state also exhibits oscil-latory behaviour as shown in Figs. 8a and 8b for the parameter values k = 1 + 0 . i ¯ k = − . i , k = 1 + 2 . i , ¯ k = − . i , α (1)1 = 1 + i , α (1)2 = 1 + i . , α (2)1 = 0 . i , α (2)2 = 3 + i . , β (1)1 = 1 − i , β (2)1 = − . − i , β (1)2 = − − i . and β (2)2 = 3 − i . . As evidenced from this figure one may observe a novel doublehump bound soliton state in the q component and a single hump soliton bound statein the q component. We point out that the parallel propagation of bound state and theoscillation occurring in the amplitude of the bound state are controlled by the imag-inary parts of the wave numbers which appear in the central position of the solitons.We also note here that the oscillations occurring in the amplitude of bound state areusually controlled by the amplitude parameters in any other local integrable systems. In this part of the work, we have brought out the nature of degenerate soliton col-lisions in the nonlocal Manakov system. In particular, we have brought out threedifferent types of shape changing collisions for two different parametric conditions.Interestingly one of them does not exist in the case of local Manakov equation. Wehave also explained the changes which occur in the quasi-intensity, phase shift andrelative separation distance during both the types of energy sharing collisions. Wehave noticed that in the Type-II and its variant shape changing collisions the differ-ence in the energy of a soliton in the two modes is not preserved during the collisionprocess. However, the total energy of a soliton in the two modes is conserved in theType-I shape changing collision. We have also demonstrated the occurrence of local-ized resonant pattern and bound state solutions in the CNNLS equation. Our studygives a better understanding of nonlocal soliton collision in the PT -symmetric ar-rays of wave guide systems where the medium exhibits nonlocal nonlinearity. Nextwe plan to investigate the non-degenerate soliton solutions and their interaction dy-namics in some detail. Acknowledgements
The work of MS forms part of a research project sponsored by DST-SERB, Govern-ment of India, under the Grant No. EMR/2016/001818. The research work of MLis supported by a SERB Distinguished Fellowship and also forms part of the DAE-NBHM research project (2/48 (5)/2015/NBHM (R.P.)/R&D-II/14127).
AppendixA. Amplitude and phase forms obtained from asymptotic analysis
The explicit expression for the amplitudes and phases of the solitons 1 and 2 beforeand after collision ( t → ±∞ ) obtained from the asymptotic analysis of Type-I col-lision are given below: The amplitude and phase of the soliton 1 S before collisionare A − j = α ( j )1 Γ / , ˆ A − j = β ( j )1 Γ / , θ − jR = ln | Γ || κ | . (14a)The amplitude and phase of the soliton 2 S before collision are A − j = ( κ ¯ ̺ ) / (cid:18) ( − j k β (3 − j )1 ν − ¯ k α ( j )2 Γ + ¯ k α ( j )1 Γ (cid:19) ( Γ κ ̺ ) / (cid:18) ̺ Γ Γ − ̺ ν ν − ̺ Γ Γ (cid:19) / , (14b) ˆ A − j = ( κ ̺ ) / (cid:18) − k β ( j )2 Γ + k β ( j )1 Γ + ( − (3 − j ) ¯ k α (3 − j )1 ν (cid:19) ( Γ κ ¯ ̺ ) / (cid:18) ̺ Γ Γ − ̺ ν ν − ̺ Γ Γ (cid:19) / , (14c) θ − jR = ln | ¯ ̺ ̺ ( ̺ Γ Γ − ̺ ν ν − ̺ Γ Γ ) || Γ κ κ κ | , (14d)The amplitude and phase of the soliton 1 S after collision are A j = ( κ ¯ ̺ ) / (cid:18) ( − j k β (3 − j )2 ν − ¯ k α ( j )2 Γ + ¯ k α ( j )1 Γ (cid:19) ( Γ κ ̺ ) / (cid:18) ̺ Γ Γ − ̺ ν ν − ̺ Γ Γ (cid:19) / , (14e) ˆ A j = ( κ ̺ ) / (cid:18) − k β ( j )2 Γ + k β ( j )1 Γ + ( − (3 − j ) ¯ k α (3 − j )2 ν (cid:19) ( Γ κ ¯ ̺ ) / (cid:18) ̺ Γ Γ − ̺ ν ν − ̺ Γ Γ (cid:19) / , (14f) θ jR = ln | ¯ ̺ ̺ ( ̺ Γ Γ − ̺ ν ν − ̺ Γ Γ ) || Γ κ κ κ | . (14g) itle Suppressed Due to Excessive Length 17 The amplitude and phase of the soliton 2 S after collision are A j = α ( j )2 Γ / , ˆ A j = β ( j )2 Γ / , θ jR = ln | Γ || κ | . (14h)To verify the non-conservation and conservation relations (10) and (11) for Type-IIshape changing collision and its variant, one has to use the expressions of amplitudesand phases of the solitons before collision given in Eqs. (14a)-(14d) for calculatingthe quantities A l + j and ˆ A l + j . Similarly to calculate the quantities A l − j and ˆ A l − j forshape changing collision and its variant, one has to use the expressions of amplitudesand phases of the solitons after collision given in Eqs. (14g)-(14h). Conflicts of interest
The authors declare that they have no conflict of interest.
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