Degenerate soliton solutions and their dynamics in the nonlocal Manakov system: II Interactions between solitons
In this paper, by considering the degenerate two bright soliton solutions of the nonlocal Manakov system, we bring out three different types of energy sharing collisions for two different parametric conditions. Among the three, two of them are new which do not exist in the local Manakov equation. By performing an asymptotic analysis to the degenerate two-soliton solution, we explain the changes which occur in the quasi-intensity/quasi-power, phase shift and relative separation distance during the collision process. Remarkably, the intensity redistribution reveals that in the new types of shape changing collisions, the energy difference of soliton in the two modes is not preserved during collision. In contrast to this, in the other shape changing collision, the total energy of soliton in the two modes is conserved during collision. In addition to this, by tuning the imaginary parts of the wave numbers, we observe localized resonant patterns in both the scenarios. We also demonstrate the existence of bound states in the CNNLS equation during the collision process for certain parametric values.
Keywords:Coupled nonlocal nonlinear Schrödinger equations Hirota’s bilinear method Soliton solutions
Finding new localized wave solutions, studying their dynamics in the nonlocal integrable equations and obtaining new integrable equations from the nonlocal reductions are active areas of research in the study of integrable systems. Recently in 1 (), Ablowitz and Musslimani introduced a reverse space nonlocal nonlinear Schrödinger (NNLS) equation to explain the wave propagation in a nonlocal medium. A special property associated with this equation is the existence of -symmetry when the self-induced potential obeys the -symmetry condition 2 (). The presence of nonlocal field as well as -symmetric complex potential make the nonlocal equations more interesting subject. Various recent studies have shown that the analysis of NNLS equation and its variant have both physical and mathematical perspectives 3b ()-8a (). Further only a few investigations on the dynamics of solitons in the coupled version of NNLS equation have been reported in the literature 57 ()-8a (). In particular, breathing one soliton solution is constructed for the following nonlocal Manakov equation through inverse scattering transform 8a (),
However the soliton shows singularity in a finite time at . The above equation is a vector generalization of reverse space NNLS equation. As we pointed out above Eq. (1) also possess self-induced potential and -symmetry since the later obeys the -symmetric condition. In the first part of accompanying work 8b (), we have constructed general soliton solution for Eq. (1) and its augmented version
by bilinearizing themt in a non-standard way. The obtained soliton solutions are in general non-singular 8b (). In the present second part we study the dynamics of obtained two-soliton solution which has been reported in the previous part 8b (). Before proceeding further, we first summarize the results presented in 8b ().
To construct soliton solution of Eq. (1), we also augment Eq. (2) given above in the bilinear process. Since, we have treated the fields and , , present in nonlocal nonlinearity as independent fields, to bilinearize them, we introduce two auxiliary functions, namely and in the non-standard bilinear process. By solving the obtained bilinear equations systematically, first we have derived the non-degenerate one soliton solution. From this soliton solution, we have deduced the degenerate one soliton solution and then the solutions which already exist in the literature. We have also constructed degenerate two-soliton solution for Eq. (1). As a continuation of the first part 8b () in the present work we study their dynamics.
We show that there exists three types of shape changing collisions in (1) for two specific parametric conditions, where the first shape changing collision is similar to the one that arises in the case of local Manakov equation 6a (). The second type of shape changing collision is similar to the one that occurs in the mixed CNLS equation 7a2 (). Besides these two collision scenario, we also observe a third type of collision which is a variant of the second type of shape changing collision and it has not been observed in any local 2-CNLS equation. By carrying out the asymptotic analysis for the fields and in a novel way, we deduce the conservation equation, expression for the phase shift and relative separation distances for all the three collisions. More surprisingly, in the new types of shape changing collisions, the difference in quasi-intensity of the two modes of a soliton before collision is not equal to the 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 solitons after collision in both the modes. In another type of collision the total quasi-intensity of individual solitons as well the total quasi-intensity of soliton before and after collision in both the modes are conserved. Finally, by tuning the imaginary part of the wave numbers we unearth a new type of localized resonant wave pattern that arises during first and second type of collision processes. We also demonstrate the existence of bright soliton bound states in the nonlocal Manakov equation.
The outline of the paper is as follows. In section 2, by performing an asymptotic analysis, we investigate three types of shape changing collisions via intensity redistribution. 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 the observation of localized resonant pattern creation during the collision of degenerate two solitons. In this section, we also demonstrate the occurrence of bound states that occur between the interaction of degenerate two solitons. We present our conclusions in Sec. 4.
2 Asymptotic analysis of degenerate two bright nonlocal soliton solution: Shape changing or switching collision
Differing from the local case, we perform the asymptotic analysis on the degenerate two-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 8b () but differ in amplitudes and phases.
As far as Eqs. (1) and (2) are concerned, one can identify different types of shape changing collisions. In particular, here we point out three interesting cases, which we designate as Type-I, Type-II and a variant of Type-II collisions.
2.1 Type-I shape changing collision
We visualize Type-I collision for the following choice, namely
where , , , For this choice, the nonlocal solitons exhibit a shape changing collision similar to the one that occurs in the local Manakov equation 6a (). We call this type of collision as local Manakov type collision or Type-I collision. For the parametric restrictions (3) the solitons and are well separated initially. The variables and ’s in the two nonlocal solitons behave asymptotically as (i) , , , as (soliton 1 ()) and (ii) , , , as (soliton 2 ()). Here the variables, and are the real parts of the wave variables and and are equal to and , respectively.
2.2 Type-II shape changing collision and its variant
The nonlocal solitons exhibit another interesting collision scenario for the following parametric condition:
For this choice, the nonlocal Manakov equation admits a collision similar to the one that occurs in the local mixed CNLS equation 7a2 (). To differentiate this second type of collision from the earlier one which is pointed out in the previous paragraph we call this collision as nonlocal mixed CNLS like collision or Type-II collision. For the parametric restriction given in (2.2) the wave variables and ’s behave asymptotically as (i) , , , as (soliton 1 ()) and (ii) , , , as (soliton 2 ()).
For the same parametric restriction chosen in (2.2) we also come across another type of new shape changing collision which we call as variant of Type-II collision which has not been observed in any local -CNLS equation. Thus in the nonlocal Manakov equation, we observe three types of shape changing collisions whereas in the local Manakov equation we come across only one type of shape changing collision 6a (). We perform the asymptotic analysis for all the three types of shape changing collisions. Our results show that the asymptotic analysis carried out on Type-II and its variant collisions match with each other.
2.3 Asymptotic forms in Type-I shape changing collision
Now we can check that the parametric choice given in (3) for the Type-I collision leads to the following asymptotic forms.
(i) Before Collision: ()
In the limit , the two-soliton solution reduces to the following two independent one soliton solutions:
(a) Soliton 1: (, , ,
|where , , ,
, . Here, subscript () represents the modes and , respectively, and the superscript denotes the soliton 1 at . The parameters and denote the amplitude and phase of the soliton 1 in both the components before collision. The corresponding asymptotic analysis on the fields yields the following expression,
, , , ,
, . In Eq. (5b), ‘ hat’ corresponds to field .
(b) Soliton 2: (, , ,
|where , , , , . In the above, amplitude and phase of the soliton 2 before collision is represented by and , respectively. Here, the superscript denotes the soliton 2 before collision. In the same limit, the asymptotic expression of turns out to be|
|where , , , , .|
(ii) After Collision: ()
In this limit, , the two-soliton solution reduces to the following two one soliton solutions:
(a) Soliton 1: (, , ,
|where , , , , . Here, the quantities and define the amplitude and phase of the soliton 1 after collision. In the superscript of the above expressions denotes the soliton 1 at .|
|where , , , , .|
|(b) soliton 2: (, , ,|
|where , , ,
, . The amplitude and phase of the soliton 2 in the nonlinear Schrödinger field after collision is represented by and , respectively.
where , , , ,
, . One can find the explicit forms of the various constants which appear in the asymptotic forms given in Appendix A.
Similarly we can calculate the asymptotic forms of Type-II and its variant collisions. However, to avoid too many details, we do not present their explicit forms, but only demonstrate numerically the typical cases.
From the above asymptotic forms of solitons and , we conclude that a definite intensity redistribution has occurred among the modes of the nonlocal solitons which can be identified from the amplitude changes in the solitons and . During the collision process, the phases of the solitons have also changed. The conservation of total energy (or intensity) of the solitons is yet another quantity which characterizes these three shape changing collisions. The conservation of energy which occurs in the Type-II and its variant collisions is entirely different from the collision in the local mixed CNLS equation. In order to show the intensity redistribution among the modes of the nonlocal solitons from the asymptotic forms, we calculate the explicit expressions of the amplitudes and phases of the solitons. The obtained expressions of all the quantities which appear from asymptotic forms are given in the Appendix A.
2.4 Intensity redistribution
In this subsection, first we demonstrate how the intensity redistribution and conservation of energy occur between the solitons in the Type-I, Type-II and variant of Type-II collisions. To demonstrate this, we begin our analysis with the asymptotic forms obtained in the previous sub-section.
2.4.1 Intensity redistribution in Type-I collision
In Type-I collision, the analysis reveals that the amplitudes of the solitons and are changing from and to and , , respectively, due to collision. Similarly the amplitudes of the fields , are also changing, during the evolution process, from and to and , . Here, ’s are polarization vectors of the th soliton. This is because of the energy sharing interaction that occurs between them.
In Type-I collision, the quasi-intensity (quasi-power) of the soliton in the first mode shares with the soliton in mode and the same kind of intensity sharing occurs between the modes of the soliton also. This in turn confirms that the intensity redistribution occurs in between the modes. Even though the intensity redistribution occurs among the solitons that are present in the modes and the total energy of the individual solitons is conserved which can be confirmed from
where the explicit forms of , are given in Appendix A. In the above, subscripts denote the modes while superscripts represent the soliton number. The above conservation form, reveals the fact that the total quasi-intensity of the individual solitons is conserved. In the local case, the total energy of each soliton is calculated by adding the absolute squares of the amplitudes of the individual modes of the solitons 6a (). Even though the amplitudes of both the co-propagating solitons are 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. This can be verified by the following conservation form
Eq. (8) confirms that the total energy of the solitons and before collision is equal to the total energy of the solitons after collision.
The change in amplitude of each one of the solitons in both the components can be evaluated by introducing the transition amplitude by , where is the amplitude of the -th soliton in the -th component after collision and is the amplitude of the soliton in the corresponding mode before collision. To calculate the intensity exchange among the modes of the solitons we multiply the transition amplitude by the transition amplitude of field , where and are the amplitudes of the solitons of the field of each mode after and before collision respectively. This definition also differs from the local Manakov case in which we multiply by its own complex conjugate transition element , to get 7a.
The intensity exchange between the solitons and due to Type-I collision is defined by
where all the quantities in the expression (9) are given in the Appendix A. By suitably fixing the parameters, we can make the right hand side of the above expression to be equal to one. For this special parametric choice, we can come across a pure elastic collision (or shape preserving collision). For all other parametric values there occurs a 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 by suitably fixing the values of the parameters. For this case, the intensity of any one of the solitons in one of the modes becomes zero.
2.4.2 Intensity redistribution in Type-II collision and its variant
In Type-II collision process also the amplitude of the solitons changes in both the fields and . In this collision scenario, the quasi-intensity of soliton is enhanced in both the modes while the quasi-intensity of soliton is suppressed. This collision scenario is entirely different from the one that occurs in the Type-I collision. A remarkable feature of the Type-II collision is that the total energy of individual solitons is not conserved, so that
From Eq. (10), we infer that the difference in quasi-intensity of soliton in both the modes before collision is not equal to the same after collision. This is also true for the soliton as well. In the local mixed CNLS equation the energy difference turns out to be the same before and after collision 7a2 (). The free parameters that appear in the degenerate nonlocal two soliton solution (28a)-(28c) given in 8b () do allow the similar kind of shape changing collision as the one happens in the case of local mixed CNLS equation.
In Type-II shape changing collision, the total intensity of the solitons and in both the components before collision is equal to the the total intensity of the solitons and after collision, that is
The intensity exchange between the solitons in the Type-II collision can also be calculated by defining the transition matrices. In this case the transition matrices are defined by , . A special case in which the right hand side becomes one produces shape preserving elastic collision.
In addition to the above Type-II collision, we also observe a variant of it. In the variant of Type-II collision, the intensity of soliton is suppressed in both the modes whereas the intensity of soliton is suppressed in mode and is enhanced in mode. This collision scenario is entirely different from the previous collision processes and has not been encountered in any local -CNLS equation. The variant of Type-II collision also obeys the non-conservation and conservation relations (10) and (11), respectively.
We recall here that Eq. (1) corresponds to three different equations, namely nonlocal version of (i) Manakov equation, (ii) defocusing CNLS equation and (iii) mixed CNLS equation, depending upon the sign of . It is noted that the shape changing collision that occurs in the local Manakov system differs from the one that occurs in the local mixed CNLS system 7a2 (). For example, the shape changing collision that occurs in the mixed coupled NLS equation can be viewed as an amplification process in 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 that does not exist in the local Manakov case 6a (). Very surprisingly, the nonlocal Manakov 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 that occurs in the mixed coupled NLS equation. This type of collision has not been observed in any other -dimensional nonlocal integrable system.
In Figs. 2, 3 and 4, we have demonstrated the shape changing collisions that occur in (1) for . The local Manakov type shape changing collision that occurs in the system (1) is illustrated in Figs. 2a-2b whereas in Figs. 3a-3b the shape changing collision that occur in (2) as in the case of mixed CNLS equation is shown. A variant of Type-II intensity switching collision is illustrated in Figs. 4a-4b. These three figures also reveal that besides the change in amplitudes, changes also occur in phase shift and relative separation distances. In the following, we calculate these changes.
2.5 Phase shifts
During the collision process, another important quantity, namely the phase is also being altered. The phase identifies essentially the position of the solitons. The change in phase can be calculated from the expressions already obtained. The initial phase of the soliton (=) changes to . Similarly, the initial phase of the soliton (=) changes to . Therefore, the phase shift suffered by the soliton in both the modes during collision is
|Similarly the phase shift suffered by the soliton is|
|From the above two phase shift expressions of solitons and , we find that|
In the Type-II and its variant shape changing collisions, the phase shift suffered by the solitons and is equal to the phase shift suffered by the soliton and in the Type-I collision, respectively. In the Type-II and its variant collisions, the phase shift of the solitons one and two is related by the same relation given in (12c). From this relation, we infer that in both the collision processes the soliton gets phase shifted opposite to the soliton . We also find that the phase shifts not only depend on the amplitude parameters and , but also on the wave numbers , . This is similar to the local Manakov equation and mixed CNLS equation.
2.6 Relative separation distances
The changes which occur in phases of both the solitons in turn cause a change in their relative separation distances during both the collision process. The relative separation distance is nothing but the distance between the positions of the solitons after and before collision 7a. We denote them by , where is equal to the position of soliton minus the position of soliton after collision (at ) and is equal to the position of soliton minus the position of soliton before collision (at ), that is
where and denote the positions of and at , respectively, whereas and are the positions of and at , respectively. Their explicit forms can be obtained from the phase shifts of the solitons which turns out to be
|The total change in relative separation distance is given by|