Multicomponent long-wave--short-wave resonance interaction system: Bright solitons, energy-sharing collisions, and resonant solitons
We consider a general multicomponent (2+1)-dimensional long-wave–short-wave resonance interaction (LSRI) system with arbitrary nonlinearity coefficients, which describes the nonlinear resonance interaction of multiple short waves with a long-wave in two spatial dimensions. The general multicomponent LSRI system is shown to be integrable by performing the Painlevé analysis. Then we construct the exact bright multi-soliton solutions by applying the Hirota’s bilinearization method and study the propagation and collision dynamics of bright solitons in detail. Particularly, we investigate the head-on and overtaking collisions of bright solitons and explore two types of energy-sharing collisions as well as standard elastic collision. We have also corroborated the obtained analytical one-soliton solution by direct numerical simulation. Also, we discuss the formation and dynamics of resonant solitons. Interestingly, we demonstrate the formation of resonant solitons admitting breather-like (localized periodic pulse train) structure and also large amplitude localized structures akin to rogue waves coexisting with solitons. For completeness, we have also obtained dark one- and two-soliton solutions and studied their dynamics briefly.
pacs:05.45.Yv, 02.30.Jr, 02.30.Ik
Nonlinear wave interactions in physical systems lead to the formation of special nonlinear waves like solitons/soliton-like structures, shock waves, rogue waves, vortex solitons and so on (2); (1). The appearance of such nonlinear waves in almost every physical system motivated researchers from various disciplines of science to investigate their underlying remarkable dynamical behaviour in order to unearth non-trivial dynamical properties. Long-wave–short-wave resonance interaction (LSRI) is an interesting nonlinear interaction phenomenon that finds diversified applications namely in water waves, plasma physics, nonlinear optics, bio-physics and Bose-Einstein condensates (3); (4); (5). This LSRI process arises during the nonlinear interaction between low- frequency long waves (LWs) and high-frequency short waves (SWs). In fact, there occurs a resonance interaction between the long wave and short waves when the phase velocity of the LW matches exactly/approximately the group velocity of the SWs. The pioneering study of such resonant nonlinear wave interaction in the context of plasma physics was made by Zakharov (6). Kawahara et al. have investigated the nonlinear interaction between short- and long-capillary gravity waves (7). The energy exchange between a nonlinear electron-plasma wave and a nonlinear ion-acoustic wave through resonance interaction mechanism was studied in Ref. (8). At the same time, the LSRI phenomenon has been investigated independently by Benny (9) and by Yajima and Oikawa (10) to study the interaction of ion sound wave with the Langmuir wave. Since then, several theoretical and experimental works have been reported based on the LSRI phenomenon in various contexts.
The LSRI phenomenon arising due to the interaction between long gravity wave and capillary-gravity wave for finite-depth water was investigated in Ref. (11) by deriving a model equation and the solutions of the model equation were obtained (12). Later on, the resonant interaction of long and short internal waves in a three-layer fluid was studied experimentally (13). Apart from this, study on the resonant coupling between ultra-long equatorial wave and packets of short gravity wave was also carried out in Ref. (14). By using perturbation method, the one- and two-dimensional LSRI equations were obtained in Refs. (15); (16) for a two-layer fluid model and soliton (bright and dark type) solutions were constructed by applying the Hirota method (16).
The two-dimensional analog of two-component LSRI system was investigated by Ohta et al. (17), in which they have derived the governing equation for a physical setting describing the interaction of nonlinear dispersive waves of three channels. Also, they have obtained special multi-soliton solutions using the Hirota method in the Wronskian form and analyzed their interactions (17). The Painlevé analysis of the two-component LSRI equation studied by Ohta et al., has been carried out in (18) and there itself special dromion solutions have been obtained using the truncated Painlevé approach. Recently, the present authors have obtained more general multi-soliton solutions displaying a fascinating energy sharing (shape changing) collision in two-dimensional multicomponent LSRI equation (19). Very recently, the dynamics of bright soliton bound states of (2+1)-dimensional multicomponent LSRI system is investigated in detail in Ref. (20).
Generalizing the procedure given in Ref. (17) for three nonlinear dispersive waves, the propagation equation for multiple dispersive waves (say () waves) in a weak Kerr type nonlinear medium in the small amplitude limit can be obtained as shown in Ref. (21). The corresponding set of general non-dimensional multicomponent (2+1)D LSRI system governing the resonance interaction between multiple SWs (say ) with a LW is given by
where represents the -th SW, indicates the LW and the subscripts represent the partial derivatives with respect to the evolutional coordinate and the spatial coordinates ( and ). In the above (2+1)D LSRI system, “2” stands for the two spatial dimensions ( and ) and “1” stands for the evolutional coordinate “”. It should be noticed that in the waveguide geometry is the propagation direction, and denote the transverse coordinates. Here, and are real arbitrary parameters that determine the nature of higher dimensionality and the strength of nonlinear coupling of SWs, respectively. The nonlinearity coefficients in Eq. (1b), in particular, their signs play pivotal role in determining the dynamics of system. Physically, these coefficients can be related to the self-phase modulation (SPM) and cross-phase modulation (XPM) coefficients in the context of nonlinear optics. Especially in the -th SW component, the nonlinearity coefficients with () correspond to SPM (XPM) coefficients. In this work, we have clearly brought out the role of this parameter on the propagation and collision dynamics of bright solitons.
Interestingly, the one-component version () of the above system (1) can be derived from a set of two-dimensional coupled nonlinear Schrödinger equations (22); (23), describing the interaction of two-dimensional two waves propagating in different wave directions, when long-wave–short-wave resonance takes place, by using the approach of (17). This clearly indicates that system (1) is a natural generalization of the two-wave system to the ()-wave system in (2+1)-dimensions.
Very recently, we have derived the general ()-dimensional multicomponent Yajima-Oikawa system from a set of multiple coupled nonlinear Schrödinger equations when LSRI takes place and have shown that the system is integrable via Painlevé test (24). In addition to this, the bright multisoliton solutions and their interesting dynamics have been analyzed in Ref. (24). The present system is a two dimensional generalization of the multicomponent Yajima-Oikawa system with a change of sign before the second derivative term in (1a) and is of considerable physical importance. This clearly shows the physical significance of the considered system (1).
Additionally, motivated by the intriguing collision scenario of bright solitons in integrable multicomponent nonlinear systems with mixed type (focusing-defocusing) nonlinearities (25) we wish to explore such special features in the present system too. For this purpose, first we study the integrability property of the above general -component LSRI system. To the best of our knowledge, only a sub-system of (1) that can be reduced for , and has been shown to be integrable by Painlevé analysis (18). However, the integrability aspects of the present general (2+1)D -LSRI system (1) has not been investigated so far. Then we construct the multi-soliton solutions in Gram determinant form using Hirota’s method and analyze their different types of energy sharing interactions.
An interesting aspect of soliton studies is the formation and propagation of resonant solitons, which arise as a special case of multi-soliton solutions. The resonant soliton (RS) occurs when the phase-shift experienced by the colliding solitons becomes infinity or tends to become infinity (26); (28); (27). The difference between the resonance mechanism in LSRI process and in the RS is that the former arises when the group and phase velocities of the interacting nonlinear waves match each other while the latter appears for particular choice of soliton solution parameters, that is, the choice of the soliton parameters for which the phase-shift of the colliding waves (solitons) becomes infinity. For the RS, the amplitude reaches a maximum value and the nature of soliton energy switching becomes non-trivial. In general, RSs appear in integrable higher dimensional nonlinear systems which admit multi-soliton solution, as these RSs arise only as special cases of multi-soliton solutions describing the interaction of multiple solitons. These resonant solitons produce different types of interaction patterns such as Y-type, inverted Y-type and coupled (Y–inverted Y) type structures which may find applications in junction couplers. Especially, in the case of two-soliton resonance, the two colliding solitons may combine into a single soliton after collision (soliton fusion) or a single soliton may split up into two solitons after collision (soliton fission) or the two solitons collide, travel like a single soliton for certain distance and then separate into individual solitons (long-range interaction) (29); (28). So, it is our further interest to investigate the formation of such resonant solitons and their subsequent dynamics in the present system (1). Also, we construct the dark one- and two- soliton solutions of -LSRI system (1) and briefly discuss their propagation and collision dynamics for completeness. These dark solitons are interesting nonlinear objects which occur for asymptotically non-vanishing boundary conditions (30); (31); (32).
The rest of the article is organized as follows. In Sec. II, we briefly outline the integrability nature of the system by applying the Painlevé analysis and obtain the bright multi-soliton solutions by using the Hirota’s direct method (33). The dynamics of bright one-soliton is explained in Sec. III and two different types of energy sharing collisions of two bright solitons are discussed in Sec. IV. In Sec. V, we unearth the features of resonant solitons of -LSRI system (1). The dark soliton solutions of system (1) are given in Sec. VI and the results are summarized in the final section.
Ii Painlevé analysis and Multi-soliton solutions
The Painlevé analysis of Eq. (1) can be carried out in a standard way (34); (35) as done in Ref. (24) for the one-dimensional -component Yajima-Oikawa system, with arbitrary . For completeness, we briefly outline the main steps of the Painlevé analysis of system (1) in the Appendix. There we show that in fact the -LSRI system (1) admits () number of integer resonances and also admits sufficient number of arbitrary functions at each of those () resonances for arbitrary values of the real quantities with (, is a real constant, either positive or negative). Thus one can conclude that the -LSRI system (1) is integrable in Painlevé sense for arbitrary real nonlinearity coefficients with ’s being the same. The role of these nonlinearity coefficients in the formation and dynamics of solitons can be revealed by constructing the explicit multi-soliton solutions of (1) and by analyzing their underlying dynamics.
For this purpose, we obtain the bright multi-soliton solutions of the general -LSRI system (1), with , by applying Hirota’s bilinearization method (33). Equation (1) can be expressed in the following bilinear form:
by introducing the bilinearizing transformations to the dependent variables as and in Eq. (1). Here and are arbitrary complex and real functions, respectively, is a unknown constant and the Hirota’s -operators , , are defined as (33). For , Eqs. (2) admit bright soliton solutions with zero background, while for the general case, nonzero , (2) can admit bright-dark and dark-dark soliton solutions. In the present work, we restrict our study to a detailed investigation on the bright soliton dynamics and a brief discussion on the dark-dark soliton solutions. The results on the mixed (bright-dark) soliton solutions of -LSRI system (1) will be published as a separate paper.
The explicit form of bright -soliton solution, for arbitrary , is obtained by expressing the dependent variables and in terms of power series expansions as and , respectively, and by recursively solving the equations resulting from Eq. (2) at different powers of . We express the obtained bright -soliton solution in Gram determinant form as below:
|In Eq. (3c), and represent identity matrix and null matrix of dimensions () and (), respectively, and are square matrices of dimension () with elements|
The block-matrices , , and are of dimensions (), (), () and (), respectively and are defined as , , and , where , . Here , in which , and , , , are arbitrary complex parameters. Throughout this paper, and represent the component number and soliton number, respectively, while the symbols and appearing in the superscript indicate the transpose conjugate and transpose of the matrix, respectively. The proof of the above -soliton solution (3), which we have skipped here, can be easily done by expressing the bilinear equations (2), after substituting Eq. (3), in the form of Jacobi identity (for details see Ref. (19)).
The main difference between the soliton solution of present system to that of the solution given in Ref. (19) is the quantity and this plays vital role in defining the nature of soliton solution and in their collision dynamics as will be shown in the forthcoming sections. Particularly, the nature of solution (whether it is singular or non-singular) depends on the system parameters , which appear in , in addition to the soliton parameters , and . For various choices of parameters one can obtain solitons displaying different interesting dynamics. Hence the arbitrariness of nonlinearity coefficients , particularly their signs, gives an additional freedom resulting in rich soliton dynamics.
Iii Bright one-soliton
In this section, the dynamics of one soliton appearing for different choices of strength of nonlinearities is explored in detail. For this purpose, we write the explicit form of bright one-soliton solution of -LSRI system (1), resulting for the choice in Eq. (3), as below:
where , , and . Here and in the following the subscripts and appearing in a particular complex parameter denote the real and imaginary parts of that complex parameter. The above bright one-soliton is characterized by () arbitrary complex parameters, , and . In addition to these soliton parameters, one can also tune the system parameters, namely, the nonlinearity coefficients (). The amplitudes (peak values) of soliton in the -th SW component and LW component are and , respectively. One can observe that the amplitude of LW soliton is independent of -parameters and . This shows the interesting possibility of controlling the SW soliton without altering the LW soliton by tuning and . It should be noticed that, in the present system the so-called line-solitons can propagate in two different planes, namely () plane and () plane for fixed and , respectively. The velocity of soliton in the () plane is while the soliton velocity in () plane is . One can also notice that the velocity of propagating soliton in the () plane can be altered without affecting the velocity of soliton in the () plane by mere tuning of the parameter. We would like to remark that the higher dimensionality coefficient especially affects the velocity of solitons in the () plane and shifts the position of solitons in the () plane. Here, in this paper, we investigate and explore several interesting points resulting from the arbitrariness of nonlinearity coefficients () for fixed parameter (say ).
One can observe that the nature of above one-soliton solution is determined by the quantity . Particularly, singular solutions result for and non-singular solutions result for . Such dependence of the existence of regular soliton on which in turn depends on the arbitrariness of , leads us to classify the soliton solutions of -LSRI system (1) into three cases, (i) positive nonlinearity coefficients (), (ii) negative nonlinearity coefficients ( and (iii) mixed-type coefficients (both positive and negative values of ), as in table 1.
|Case||Choice of||Choice of||Condition for||Amplitude of soliton in||Velocity of LW/SW soliton|
|regular soliton||-th SW comp.||LW comp.||in ()||in ()|
|()||(a) ; (or)|
In the above table, , and .
When all the nonlinearity coefficients are positive (), the numerator of the expression for (see below Eq. (4)) always takes negative values. Hence the regular (non-singular) soliton solution can be obtained for the choice either with or with , only for which the condition for non-singular solution (that is, ) is satisfied. In Fig. 1, we have shown the bright soliton of 2-LSRI system (Eq. (1) with ) propagating in the () plane for (top panels) and in the () plane for (bottom planes) for the choice , and . It is evident from Fig. 1, that the solitons propagate with different velocities in the () and () planes (respectively, and ). The quantities appearing in all the figures of this paper are adimensional. Additionally, one can also obtain similar type of solitons for and .
We have performed a direct numerical simulation of system (1) by using the split-step Crank-Nicolson method (36); (37) and plot the one-soliton propagation in Fig. 2 corresponding to the initial conditions of Fig. 1. Here we have considered the domain with , and
For negative values of nonlinearity coefficients (), the requirement for non-singular solution ( restricts and to be positive. It can be noticed that for a given set of parameters the amplitudes of the solitons in this case are the same as those of the previous case (). But there occurs a significant change in the velocity of solitons in the () plane and the soliton velocity in the () plane now becomes opposite to that of the case (see Table 1).
For mixed signs of nonlinearity coefficients (say, for and for ), all the soliton parameters (, and ) play crucial role in obtaining the non-singular soliton solution. We can have three different conditions (sub-cases) for achieving regular solutions: (a) , , (b) , , , (c) , , . Here the bright soliton in a particular SW component can have the same amplitude for the three subcases (a), (b) and (c), which is different from the previous two cases (i) and (ii), for a given set of soliton parameters (, and ) with fixed magnitude of . However, the soliton velocities for the choices (a) and (b) in case (iii) are the same as that of case (i) while that of the choice (c) in case (iii) is the same as the velocity of case (ii) (see Table 1).
The above analysis shows that the parameters can be profitably used for controlling the dynamics of solitons in the SW components in -LSRI system (1). It clearly indicates that the propagation characteristics of solitons can be tuned by suitably altering the system parameters which will find application in controlling soliton and also in pulse shaping in the context of nonlinear optics.
Iv Bright two soliton collisions
with , , , and , . The multicomponent nature of the system results in fascinating collision dynamics of solitons. The arbitrariness of the nonlinearity coefficients , particularly, their signs play pivotal role in the collision dynamics displayed by the bright solitons in -LSRI system, as mentioned in the introduction. The soliton collisions can be well understood by performing an asymptotic analysis (38); (25); (39); (19); (40) and we do not present the corresponding mathematical details here, for brevity. It is instructive to mention that the present system results in different collisional behavior in the () and also in the () planes as the solitons admit different velocities in those planes as will be shown below.
A careful asymptotic analysis shows that the change in the amplitude of a given soliton (say -th soliton) after collision can be related to that of before collision, in the -th SW component through the relation
where the transition amplitudes ’s are defined as and . Here and , where , takes the form as given below Eq. (5). When the transition amplitudes become unimodular (that is, ) there occurs an elastic collision. This is possible only for the choice . However, in a general setting, the bright solitons undergo energy sharing (shape-changing or energy exchange) collisions as . Importantly, for the solitons in system (1), the nature of energy switching is determined by the strength (sign) of the nonlinearity coefficients () in addition to the soliton parameters. Additionally, the solitons (say and ) experience a phase-shift ( and ) given by
where , and . On the other hand, the solitons appearing in the LW component undergo only elastic collision with a phase-shift (7) for all the choices of polarization parameter . The phase-shift experienced by the colliding solitons results in a change in the relative separation distance between the two solitons. This can be defined as the difference between the relative separation distances between the solitons before collision () and after collision () and its exact form is obtained as
The soliton collision scenario and the bound solitons of system (1) with and have been studied in detail by the present authors in their earlier works (19); (20). A detailed study on the dynamics of bright soliton bound state of the present system for arbitrary nonlinearities can also be carried out as done in Ref. (20). Below we discuss the collision scenario of bright solitons for different choices of .
Case (i): Positive nonlinearity coefficients ()
The energy sharing collision scenario of two bright solitons for this case in the () plane (() plane) is shown in the top (bottom) panels of Fig. 3. In the () plane, the amplitude of soliton is enhanced (suppressed) in the () component after collision, while the change in the amplitude of soliton is just opposite to that of in a given SW component. Both the colliding solitons experience a phase-shift as give by (7). The switching nature of soliton intensity (energy) during collision in the () plane is reversed as compared with the collision scenario in the () plane. But, the LW solitons emerge unaltered after collision only with a phase-shift in both the () and () planes. In the () plane, solitons can undergo both head-on and overtaking collisions, whereas in the () plane they are restricted to undergo only overtaking collisions since the condition for regular soliton requires and to be positive or negative simultaneously, which means that both the solitons should propagate in the same direction but with different speeds. Notice that a given soliton (either or ) experiences an opposite kind of energy switching in two SW components. In this collision process, the energy of solitons in individual SW (and LW) component and also the total energy of solitons among all the SW components are conserved. We refer to such a collision process as a type-I energy-sharing collision. Such a type-I energy-sharing collision has previously been observed in multicomponent Manakov type system (39) and also in multicomponent Yajima-Oikawa system (24).
The elastic collision of solitons in the () as well as in the () planes are depicted in Fig. 4 for the choices and , by keeping the other parameters fixed as in Fig. 3. Here, the amplitudes of both solitons, and remain unaltered after collision in all the three components (2 SWs and 1 LW components). But they suffer a phase-shift after collision.
Case (ii): Negative nonlinearity coefficients ()
Bright solitons in 2-LSRI system with negative nonlinearity coefficients () admit similar type of collision behavior as that of the positive nonlinearity case (). Here also the solitons undergo both head-on and overtaking collisions in the () plane and there occurs only overtaking collision in the () plane, involving energy-sharing nature of type-I or elastic nature of soliton amplitudes accompanied by a phase-shift. The only difference is that the signs of both and have to be fixed as positive for obtaining regular solitons (non-singular solution) and hence the direction of overtaking collision in the () plane gets reversed.
Case (iii): Mixed-type nonlinearity coefficients
For mixed type nonlinearity coefficients (, and , ) the bright solitons display a different type of energy sharing collision of solitons. We have shown such an energy sharing collision in Fig. 5.
It can be noticed that the amplitude of soliton gets suppressed while that of soliton gets enhanced after collision in both SW components. This same type of energy switching of solitons among all SW components results due to a special kind of soliton energy conservation. Here the energy in individual component and also the difference of energy between the SW components are conserved, rather than the conservation of total energy among the components as in the cases (i) and (ii). We refer to such collision process as type-II energy sharing collision. This type-II energy-sharing collision of solitons has been observed in a particular one-dimensional integrable CNLS system (39) and also recently in multicomponent Yajima-Oikawa system (24) with focusing-defocusing (mixed) type of nonlinearities. To the best of our knowledge, for the first time we report such type-II energy sharing collision in higher-dimensional system. The solitons in the LW component emerge elastically after collision with a phase-shift given by Eq. (7). Thus, for the mixed type nonlinearity coefficients the nature of switching of intensities (energy) for a given soliton is same in all the number of SW components in the -LSRI system (1), while in a given component the colliding solitons and experience an opposite kind of energy switching. This enables the present system (1) to achieve amplification of a particular soliton in all the components after collision with other soliton. Typical type-II energy-sharing collisions in the () plane (() plane) are shown in the top (bottom) panels of Fig. 5.
We have also noticed that multi-soliton collision, collision involving more than two solitons, which takes place in a pair-wise manner. So, based on the above two-soliton collision scenario one can easily investigate the features of a multi-soliton collision of system (1).
V Resonant Solitons
The general -soliton solution (3) features special localized structures, namely resonant solitons, in addition to the standard interacting solitons. These resonant solitons can be achieved by appropriately choosing the soliton parameters such that the phase-shifts due to collision become infinity, i.e. [see Eq. (7)]. Thus the resonant soliton is a localized wave that appears in the interaction regime and can be viewed as an intermediate state during soliton interaction. The reason for the existence of such a long-living intermediate state is that as the phase-shifts of colliding solitons approach infinity, the span of the interaction regime also extends to infinity before the solitons get well separated. This is due to the fact that the change in the relative separation between the solitons [see Eq. (8)] becomes infinity. In such a case, it is possible to achieve large amplitude localized wave structures (the resonant soliton) for infinite distance/time.
The present -LSRI system (1) supports RS for particular choice of soliton parameters, , . We find from Eq. (7) for the phase-shifts that the RS is possible when , which results in infinite magnitude for . This can be obtained by setting and for this choice the two colliding solitons approach each other only asymptotically and forms a RS which is very distinct from the standard interacting solitons. This RS in the LW component exhibits localized periodic structures in the () plane, similar to the breathers on a zero background (41), and we refer to these structures as resonant-breathers (RBs) in analogy with the standard breather in soliton theory [see Figs. 6c–6e]. The localization of these RBs can be either in ’ or in ’’ coordinate depending on the values of and , .
We have shown such resonant solitons (breathers) in Fig. 6 for the positive nonlinearity coefficients (). Here the two colliding solitons form a zero-amplitude resonant state in the interaction regime of two SW components. However, in the LW component, the resonant soliton looks similar to the breather on a zero background with periodic oscillations in its amplitude. An interesting physical process to be noticed in the formation of resonant soliton in the interaction regime is that the energy of SW components completely disappears and reappears in the LW component, thereby resulting in a resonant soliton with large amplitude localized structure with periodic oscillations in that LW component. This is a consequence of total energy conservation of the system (1).
When and , in addition to , we get the resonant breather (Fig. 6c) and this will be localized along coordinate for (Fig. 6d). For the choice and , the resonant breather will be localized in the coordinate (Fig. 6e). It is also possible to localize the resonant-breather both in ’’ and ’’ coordinates by tuning the and parameters suitably. Particularly, for and (or and ) we have non-trivial resonant-breather co-existing with solitons as shown in Fig. 6f. This localized structure looks akin to a rogue wave but in a zero-background co-existing with regular solitons. This localized structure is a special feature of the (2+1)D LSRI system (1) and has been predicted here for the first time to the best of our knowledge. It will be relevant to point out that the co-existence of rogue wave in the non-zero background with dark-bright soliton has been observed in a two-component vector NLS system with focusing nonlinearity (42).
|Case||Choice of||Choice of||Choice of||RS/RB in () plane of the LW component||Figure No.|
|(i)||RB:- infinite length breather||6c|
|(ii)||RB:- breather localized in||6d|
|(iii)||RB:- breather localized in||6e|
|RB:- single excited structure localized in both ‘’ and ‘’ (can be viewed as rogue wave in zero background co-existing with solitons)||6f|
|(v)||QRB:- breather of finite span||7a|
|(vi)||QRB:- breather along but of finite duration||7b|
|(vii)||QRB:- breather along but of finite duration||7c|
Noticeably, for , the two colliding solitons form a quasi-resonant breather (QRB) in which the intermediate interaction regime exists only for a finite duration and after that it splits into two individual solitons. Such quasi-resonant breathers are shown in Fig. 7. Here we can obtain the resonant breather for a finite length and can be localized either in or . But we can not obtain the rogue-wave-like structure co-existing with solitons for this quasi-resonant choice. The main difference between Figs. 7 and 6 is that the span of the resonant breather is finite in the former while it extends up to infinity in the latter. We have summarized the above details on resonant solitons (breathers) in Table 2.
One can also demonstrate the existence of resonant solitons in the () plane similar to that of the () plane. Interestingly, in the () plane, the RS exhibits constant amplitude wave structure with oscillating side-bands (see Figs. 8a–8d) and the number of sidebands depends upon the absolute difference between and (that is, ). When , the sidebands disappear and we get the RS with a single peak wave structure (see Figs. 8e–8f) and the resulting collision scenario depicted in Figs. 8e and 8f look like the process of soliton fission and fusion respectively. Unlike in the case of resonant solitons in the () plane, here in the () plane the resonant soliton can only be localized along the direction.
A similar analysis on the dynamics of resonant solitons for other choices of arbitrary nonlinearity coefficients can be performed by extending the above type of investigation. One can also perform a detailed analysis on the dynamics of multi-soliton resonance in view of the above discussion and can explore different (web-like) structures.
Vi Dark soliton solutions
For completeness, in this section we obtain dark-soliton solutions of the -LSRI system (1) using the Hirota’s bilinearization method (30); (31); (32). For this purpose, let us consider the constant parameter () appearing in the bilinear equations (2) to be non-zero. This will lead us to obtain the dark solitons in an asymptotically non-vanishing limit, i.e. and when . To construct the -dark soliton solution, one has to consider the form of power series expansion for and as , , and . Considering the length of the paper, we explicitly construct the one- and two- soliton solutions in this section and one can extend the present algorithm straightforwardly to obtain the -dark soliton solution for arbitrary .
vi.1 Dark one-soliton solution
To construct the dark one-soliton solution (), we terminate the power series expansion as , , and . Then by recursively solving the resulting set of bilinear equations after substituting the above power series into Eq. (2), we obtain the one-dark soliton solution as