Multicomponent coherently coupled and incoherently coupled solitons and their collisions
We consider the integrable multicomponent coherently coupled nonlinear Schrödinger (CCNLS) equations describing simultaneous propagation of multiple fields in Kerr type nonlinear media. The correct bilinear equations of -CCNLS equations are obtained by using a non-standard type of Hirota’s bilinearization method and the more general bright one solitons with single hump and double hump profiles including special flat-top profiles are obtained. The solitons are classified as coherently coupled solitons and incoherently coupled solitons depending upon the presence and absence of coherent nonlinearity arising due to the existence of the co-propagating modes/components. Further, the more general two-soliton solutions are obtained by using this non-standard bilinearization approach and various fascinating collision dynamics are pointed out. Particularly, we demonstrate that the collision among coherently coupled soliton and incoherently coupled soliton displays a non-trivial collision behaviour in which the former always undergoes energy switching accompanied by an amplitude dependent phase-shift and change in the relative separation distance, leaving the latter unaltered. But the collision between coherently coupled solitons alone is found to be standard elastic collision. Our study also reveals the important fact that the collision between incoherently coupled solitons arising in the -CCNLS system with is always elastic, whereas for the collision becomes intricate and for this case the -CCNLS system exhibits interesting energy sharing collision of solitons characterized by intensity redistribution, amplitude dependent phase-shift and change in relative separation distance which is similar to that of the multicomponent Manakov soliton collisions. This suggests that the -CCNLS system can also be a suitable candidate for soliton collision based optical computing in addition to the Manakov system.
pacs:42.65.Tg, 05.45.Yv, 42.81.Dp, 02.30.Ik
Journal Reference: J. Phys. A: Math. Theor. 44 (2011) 285211
Multicomponent solitons/solitary waves have attracted considerable attention in the field of nonlinear science as they display a rich variety of propagation and collision properties which are not possible in their single component counterparts [1, 3, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Such solitons appear in different areas of science like nonlinear optics [1, 2, 3], Bose-Einstein condensates , bio-physics , plasma physics , etc. Here, our main focus is on nonlinear optics. In this context, multicomponent temporal solitons can be formed when an optical pulse propagating through a multimode fiber due to a delicate balance between dispersion and Kerr nonlinearity . Multicomponent spatial solitons are self-trapped optical beams that result from an interplay between diffraction and nonlinearity .
Mathematically, the propagation and collision properties of multicomponent solitons/solitary waves arising in the field of nonlinear optics can be well described within the framework of multicomponent nonlinear Schrödinger (NLS) type equations [1, 2]. Especially, the short pulse propagation in polarization maintaining multimode birefringent fiber is governed by a set of multicomponent incoherently coupled NLS (ICNLS) equations . Similar set of ICNLS equations also arises in the context of partially incoherent beam propagation in Kerr type nonlinear media . These ICNLS equations involve the nonlinear couplings due to self-phase modulation (SPM) and cross-phase modulation (XPM) and depend only on the local intensities of the co-propagating fields, but insensitive to their phases.
In general cases, like pico-second pulse propagation in non-ideal low birefringent multimode fibers or beam propagation in weakly anisotropic Kerr type nonlinear media, the coherent effects due to the interaction of co-propagating fields should also be considered [1, 21]. To be specific, the propagation of coherently coupled orthogonally polarized waveguide modes in Kerr type nonlinear medium is governed by the following -component coherently coupled NLS (CCNLS) type equations :
where and are slowly varying complex amplitudes in each polarization mode, and are the propagation direction and transverse direction, respectively, is the degree of birefringence. Here, the incoherent and coherent couplings are represented by the parameters and , respectively. The above equation (1) also arises in the context of beam propagation in isotropic Kerr type nonlinear gyrotropic medium . In equation (1), the terms and correspond to four wave mixing (FWM) process which arise due to the coherent coupling between the co-propagating fields.
Generally, these CCNLS equations and also the ICNLS equations are non-integrable. However, these become integrable for specific choices which are of physical significance [23, 13, 24, 25]. In recent years, much attention has been paid to the integrable and non-integrable coupled ICNLS equations and many interesting phenomena have been explored . To be specific, the integrable -component Manakov type equations, with arbitrary , are well studied and it has been pointed out that these equations support bright optical solitons which undergo fascinating energy sharing collisions that have immediate technological applications in the context of collision based optical computing [4, 11, 12, 27, 28] and also in soliton amplification . Very recently, CCNLS equations have also been started to receive renewed attention due to their rich structure [13, 25, 9, 29]. Particularly, a set of physically interesting integrable -component CCNLS equations related to (1) is
where is the coupling coefficient. In gyrotropic nonlinear medium, the above equation (2) can be obtained for the following choice of the susceptibility tensor , with its components satisfying the relation . Equation (2) also describes the propagation of two optical pulses in an isotropic nonlinear Kerr medium when the components , and of the susceptibility tensor can be expressed as .
Apart from the 2-component CCNLS equations, -component CCNLS equations with are also of special physical interest and have been derived under different physical contexts. Particularly, the spatial evolution of mutually guided four wave mixing states in medium is governed by -component CCNLS type equation . It has also been shown that the co-propagation of two optical pulses in birefringent fiber can be described by 4-component CCNLS equations . In ref. , the dynamics of spinor Bose-Einstein condensates has been investigated by considering a set of integrable 3-coupled CCNLS type equations  and novel polar and ferromagnetic solitons have been reported. Now it is of interest to investigate the integrable multicomponent CCNLS equations which are closely associated with the near-integrable or non-integrable systems appearing in nonlinear optics.
Being motivated by these reasons, we consider the following integrable -component generalization of (2) describing the simultaneous propagation of -optical fields in Kerr type nonlinear media.
The above system has been studied in ref. by applying the Hirota’s direct method but trivial soliton solutions with less number of parameters only have been reported due to the restricted bilinearization of (3). It also should be noticed that the information regarding the coherent and incoherent contributions from the co-propagating fields are lost completely if the two-soliton solution is constructed by a linear superposition as pointed out in refs.[13, 25]. So it is of importance to obtain correct bilinear equations of system (3) which will result in more general soliton solutions with interesting properties. In ref. , Kanna et alhave considered a 2-component integrable model which can be reduced from (2) by redefining as and reported novel solitons with variable profiles and classify them as degenerate (solitons with same intensity in both components) and non-degenerate (solitons with different intensity in two components) solitons. But study on the present system (3) suggests that a broader classification of solitons of the general -component system (3), with arbitrary , can be made based on the presence and absence of the coherent nonlinearity, and the degenerate and non-degenerate solitons reported in  appear as their sub-cases, which will be discussed in the following sections.
The aim of the present work is three-folded. First, to obtain the correct bilinear equations of system (3) and to construct exact one- and two-bright soliton solutions of (3). Next, to analyse the collision dynamics of solitons in the 2-component and 3-component CCNLS equations and to bring out their salient features. Finally, to generalize the results to the -component case.
This paper is set out as follows. The correct bilinear equations of (3) are obtained and the solitons are classified in a systematic way in section 2. The bright one- and two-soliton solutions of -component and -component CCNLS equations are obtained in section 3 and in section 4, respectively. Then the results are generalized to arbitrary -component case in section 5. Section 6 deals with the collision dynamics of the solitons. Final section is allotted for conclusion.
2 Non-standard bilinearization and classification of solitons of integrable multicomponent CCNLS system (3)
Hirota’s direct bilinearization method is one of the powerful techniques to construct soliton solutions of integrable nonlinear evolution equations . In this section, we construct the bilinear equations of the CCNLS system (3) by applying the Hirota’s direct method . A new type of bilinearization procedure has been developed by introducing an auxiliary function for the Sasa-Satsuma higher order nonlinear Schrödinger equations in ref.  by Gilson et al. By adopting this technique, here we obtain correct bilinear equations of system (3) resulting in more general bright soliton solutions which display the effects of both intensity and phase dependent nonlinearities. By performing the bilinearizing transformation
to equation (3) with the introduction of an auxiliary function , we arrive at the following set of bilinear equations:
where and . Here and are complex and real functions, respectively, denotes the complex conjugate, and are the well known Hirota’s -operators  which are defined as
The above set of equations (5) can be solved by introducing the following power series expansions for , , and
where is the formal power series expansion parameter. The resulting set of linear partial differential equations after collecting the terms with the same powers in , can be solved recursively to obtain the forms of , , and .
It can be inferred from the above bilinear equations (5) that when the auxiliary function “” becomes zero, the contribution from the coherent coupling vanishes and the above bilinear equations reduce to that of integrable -component Manakov system. We notice that for equation (5c) results in the condition , which ultimately restricts the energy sharing of a given field/soliton among all its components. In the following, we obtain explicit conditions on soliton parameters for which “” becomes zero and we refer the soliton arising for this choice, , as incoherently coupled soliton (ICS) as the contribution from the coherent nonlinearity is absent. The ICS results due to the interplay between dispersion/diffraction and the nonlinearity arising due to SPM and XPM effects. However the general bilinear equations (5) with non-vanishing auxiliary function “”involve the effect of coherent coupling also. Hence, we designate the soliton resulting for the general choice, , as coherently coupled soliton (CCS). These CCSs are formed due to the contribution from the dispersion/diffraction and the combined nonlinear effect resulting from SPM, XPM and four wave mixing process.
3 Bright soliton solutions of 2–component CCNLS equations
In this section, the bright one- and two-soliton solutions of the 2-component CCNLS equations (2) are obtained by applying the non-standard type of Hirota’s bilinearization method explained in the previous section. We present the results for the and cases explicitly in order to emphasize the additional features of the case ( represents the components). Here onwards we designate the -component -soliton solution as () soliton solution for convenience.
3.1 Bright (2,1) soliton solution
To obtain bright one-soliton solution of system (2), we restrict the power series expansion (6) as , , , . By substituting these series expansions into (5) and after recursively solving the equations resulting at like powers of , we obtain the following one-soliton solution.
|The auxiliary function is found to be|
Here, , , and ’s are complex parameters. Throughout this paper, the real and imaginary parts of a parameter are represented by the subscripts R and I, respectively.
(i) Bright (2,1) ICS:
The (2,1) ICS results for the vanishing auxiliary function (). We find from (5fgd) that the condition for to be zero is . This (2,1) ICS always exhibits the standard “sech” type profile and can be expressed as
where , , , and . For this case, either or and correspondingly the solitons in and components are related as or . Ultimately, the intensity profiles of these ICSs are same in both the components (that is, ). One can also refer these equal intensity solitons in both components as degenerate (2,1) ICSs and are characterized by two complex parameters and (or ). These solitons behave as standard NLS solitons during propagation. The amplitude of soliton in the -th component is . The velocity and central position of soliton in both components are and , respectively, and can be tuned by altering either or (or ). Such an incoherently coupled soliton is depicted in figure 1 for the parameters , , , and .
(ii) Bright (2,1) CCS:
The () CCS solution can be obtained for non-zero auxiliary function (). We require , for non-vanishing and the corresponding () CCS solution is found to be
where , , , , , , , and . The (2,1) CCS can exhibit both equal and non-equal intensities in both components, which may be referred as degenerate and non-degenerate (2,1) CCSs, respectively. Thus the degenerate and non-degenerate solitons obtained by one of the authors and co-workers in ref.  can be deduced as sub-cases of coherently coupled solitons of system (2) discussed here. Generally, these CCSs admit double hump profiles. The existence of coherent coupling is reflected by such kind of distinct profiles. One can also obtain perfect “sech” type soliton profile when the parameters are chosen suitably. This can be achieved for the condition , which makes . Here represents the amplitude (peak value of the envelope) of soliton in -th component and and are the velocity and the central position of the soliton in both components, respectively. These (2,1) CCSs are characterized by three complex parameters , and . The non-degenerate (2,1) CCS having double hump profile in component and special flat-top profile in component is depicted in figure 2 for the parameters , , , and . Such flat-top type solitons have been reported in non-integrable complex Ginzburg-Landau equations . Thus from our above analysis we observe that one can switch from coherently coupled soliton to incoherently coupled soliton and vice-versa by tuning the polarization parameters (’s) suitably.
3.2 Bright (2,2) soliton solution
The bright two-soliton solution of system (2) is obtained by restricting the power series expansion (6) as , , , . Then, by solving the resultant set of linear partial differential equations, we get the bright two-soliton solution as
|and the auxiliary function is given by|
Here, , . Various other quantities appearing in the above equation (10) can be obtained from the Appendix by substituting . The above general two-soliton solution is characterized by six complex parameters and .
4 Bright soliton solutions of 3-component CCNLS equations
We obtain the exact bright one- and two-soliton solutions of three component CCNLS equations in this section by applying the non-standard Hirota’s bilinearization method described in section 2.
4.1 Bright (3,1) soliton solution
The bright (3,1) soliton solution of CCNLS equation (3) with , can be obtained by terminating the power series expansion (6) as , , , and by solving the set of partial differential equations arising at like powers of . Then the (3,1) soliton solution can be written as
|The auxiliary function is found to be|
The above bright one-soliton solution can also be classified into the following ICS and CCS as in the previous section.
(i) Bright (3,1) ICS:
The (3,1) ICS appears for the choice , and in its explicit form it reads as
where , , , and . Unlike in the two-component case, here the degenerate ICS is not at all possible. These (3,1) ICSs admit either non-degenerate (completely different intensity profiles in all the three components) or partially degenerate (same intensity profiles in any two of the components) soliton profiles. These solitons are equivalent to the -component Manakov type solitons  and are characterized by three arbitrary complex parameters. Such non-degenerate type ICS arising for the parametric choice , , , , and is shown in figure 3.
(ii) Bright (3,1) CCS:
The (3,1) CCSs appear for the choice . The exact form of this soliton can be obtained by rewriting the one-soliton solution (5fgjka) as
where , , , , , , , and . These (3,1) CCSs admit both single hump and double hump profiles. In fact, we obtain perfect ‘sech’ type (3,1) CCSs for specific choice of parameters satisfying the relation , which ultimately makes in the above equation (5fgjkm). The CCSs can also have degenerate intensity profiles in addition to non-degenerate profiles. A typical degenerate and also a non-degenerate type CCSs are shown in figure 4 and figure 5 for the parametric choices , , , , and and , , , , and , respectively.
4.2 Bright (3,2) soliton solution
The bright (3,2) soliton solution of system (3) with , can be obtained as in the two-component case by restricting the power series expansion (6) and by allowing to run from 1 to 3. The two-soliton solution is found to be
|and the auxiliary function|