Mixed solitons in () dimensional multicomponent long-waveshort-wave system
We derive a (2+1)-dimensional multicomponent long-waveshort-wave resonance interaction (LSRI) system as the evolution equation for propagation of -dispersive waves in weak Kerr type nonlinear medium in the small amplitude limit. The mixed (bright-dark) type soliton solutions of a particular (2+1)-dimensional multicomponent LSRI system, deduced from the general multicomponent higher dimensional LSRI system, are obtained by applying the Hirota’s bilinearization method. Particularly, we show that the solitons in the LSRI system with two short-wave components behave like scalar solitons. We point out that for -component LSRI system with , if the bright solitons appear in atleast two components, interesting collision behaviour takes place resulting in energy exchange among the bright solitons. However the dark solitons undergo standard elastic collision accompanied by a position-shift and a phase-shift. Our analysis on the mixed bound solitons shows that the additional degree of freedom which arises due to the higher dimensional nature of the system results in a wide range of parameters for which the soliton collision can take place.
The nonlinear interaction of multiple waves results in several new physical processes kiv (); akm (); scot (). It has been shown in the two layer fluid model that resonance between the long-wave component and short-wave component occurs when the phase velocity of the former matches with the group velocity of the latter oikawa (). This is a ubiquitous phenomenon which appears in hydrodynamics grim (), bio-physics boiti (), plasma physics zakh (), and in nonlinear optical systems kivol (). Though there are many studies on the long-waveshort-wave resonance interaction (LSRI) in one dimension zakh (); kivol (); yaji76 (); ma78 (); ma79 (); funa (), results are scarce for multicomponent higher dimensional LSRI system. In the context of nonlinear optics, the interaction of bright and small amplitude dark pulses in optical fiber is governed by the integrable Zakharov model kivol (); zakh ().
The resonance interaction of long-wave with short-wave was first investigated by Benney for capillary-gravity waves in deep water benny (). In this case simple interaction equations cannot be obtained, because for deep water waves there is no wave in the long wavelength limit. However, simple interaction equations can be deduced in a stable stratified fluid for oblique propagation of long and short-waves grim (). The single component two-dimensional LSRI equation for a two-layer fluid model has been derived in Ref. oikawa () by using the multiple scale perturbation method and bright and dark type one- and two-soliton solutions have been reported. In Ref. ohta7 (), Ohta et al derived an integrable two component analogue of the two-dimensional LSRI system as a governing equation for the interaction of the nonlinear dispersive waves by applying the reductive perturbation method. It is worth noting that there exists several articles oc1 (); oc2 (); rev () on this perturbation approach which is based on a consistent and mathematically rigorous expansion of the linear dispersion relation including the nonlinear optical response of the medium. It leads to a new equation for self-focusing of extremely focused short-duration intense pulses oc1 () and also to a general propagation equation for the pulse envelope of an electromagnetic field in an isotropic nonlinear dispersive medium oc2 () with all orders of dispersion, diffraction and nonlinearity. Very recently, the non-integrable three component Gross-Pitaevskii equations have been reduced to single component Yajima-Oikawa system by using multiple scale method nista (). In another recent work visi (), the one-dimensional integrable two-component Zakharov-Yajima-Oikawa equation has been derived using multiple scale method and special bright-dark one-soliton solutions have been reported.
In Ref. tklsri (), we have obtained general bright -soliton solution, for arbitrary , of the same integrable multicomponent ()D-LSRI system (see Eq. (12)) considered in the present paper. And also, the bright soliton bound states of the same system have been analyzed in Ref. epjsk (). Our earlier work tklsri () on bright soliton solutions of the multicomponent LSRI system shows that the role of long interfacial wave is to induce nonlinear interaction among the short-wave components resulting in non-trivial (shape-changing) collision behaviour characterized by energy exchange among the two short-wave components. As far as we know, for the first time in Ref. tklsri () we have identified the shape-changing/energy sharing collisions of bright solitons in a two dimensional integrable nonlinear system. This system will act as a potential candidate for realizing soliton collision-based computing and multi-state logic jak (); ste (); tkpre3 (). Now it is of interest to derive the general two dimensional -component equations describing the interaction of several short-wave packets with long-waves in a physical set up and to look for other types of multicomponent soliton solutions.
In recent years, much attention has been paid to investigate mixed (bright-dark) soliton dynamics in different dynamical systems including nonlinear optical systems and Bose-Einstein condensates kiv (); kivol (); tkpra8 (); ohta7 (); shep (); ohtadark (); nista (); visi () of coupled bright-dark solitons and to analyse their propagation properties and collision dynamics. In the present work, we derive the (2+1)-dimensional -component LSRI system governing the evolution of short-waves and long-waves (with ) in a nonlinear dispersive medium and reduce the system to an integrable system for a particular choice of the system parameters. Then by applying the elegant Hirota’s direct method to the integrable multicomponent LSRI system for a particular choice, we obtain the coupled bright-dark one- and two- soliton solutions. We will show that bright and dark parts of the mixed solitons in the two short-wave components case behave like scalar solitons whereas if we go for three or more short-wave components the multicomponent nature of the solitons will come into picture and one can observe interesting propagation and collision properties. It is straight-forward albeit lengthy procedure to extend the analysis to construct -soliton solution, with arbitrary .
The present paper is organized as below. The general -component LSRI system is derived by applying the multiple scale perturbation method in the next section. In section III, bilinear equations for the integrable (2+1)-dimensional multicomponent LSRI system are given. Sections IV and V deal with the mixed one- and two-soliton solutions of the multicomponent LSRI system. The collision dynamics of the solitons are discussed in section VI. Discussion on the mixed soliton bound states is presented in section VII. The final section is allotted for conclusion.
Ii The Model
To start with we obtain the general two-dimensional multicomponent evolution equation for the propagation of -dispersive waves in a Kerr type nonlinear medium (ex.: optical fiber, photo-refractive medium) by generalizing the approach developed in Refs. kivol (); ohta7 () for the two and three components case. The waves are assumed to obey the following weakly nonlinear dispersion relations
where and are the and components of the wave vector, and are the complex amplitude and angular frequency of the -th wave. Especially, in the physical setting of propagation of an incoherent self-trapped beam in a slow Kerr-like medium, the nonlinearity arising from the change in refractive index profile (say ) created by all incoherent components of the light beam can be expressed as , where is the intensity of the -th incoherent component, is the nonlinearity coefficient of the -th component and denotes the total number of components. This shows that we can very well have nonlinearities which are purely dependent only on intensities even for multicomponent systems. Such media will be appropriate to realize the type of dispersion relation considered in this paper. This type of system is known as incoherently coupled system in the context of nonlinear optics kiv (). The fundamental carrier wave is of the form . The most convenient way to derive the evolution equation for the amplitudes ’s is to Taylor expand the angular frequencies ’s around the and components of the wave vector of the carrier wave and , respectively, and the central frequency at , as below:
where , . In this Taylor expansion and in the following, the subscript ‘’ given in Eq. (2) as represents the fact that the quantity appearing inside the bracket is evaluated at and . In Eq. (2), , , , , and . Then by replacing , and by the operators , and , respectively and transforming to the moving co-ordinates , , , with the assumption that beyond a particular component (say ) all the derivatives , , are same and so also the derivatives , i.e., (say) and (say), and omitting the primes for simplicity of notation we get
Here the independent variables appearing in the suffixes after the comma denote partial derivatives with respect to that variables and the group velocities of the component along the and directions are and , respectively. Various quantities in the above equations (3) are defined as , , , , , , , .
We consider the case where the first -components are in the anomalous dispersion region and the remaining ()-components are in the normal dispersion regime. Following Ref. kivol (), the solutions of (3) are sought in the form
where , and ’s are assumed to take only small values.
In a similar manner, by incorporating (4) in (3b) and collecting the real and imaginary parts, we arrive at a set of two coupled equations. The resulting coupled equations can be grouped together to obtain the following equation by differentiating the imaginary part equation twice with respect to ‘’ and making use of the real part equation:
Equations (5) and (6) are general equations describing the two dimensional propagation of waves in the anomalous dispersion region and -waves in the normal dispersion region. For case with the system has two short-wave components and one long-wave component and coincides with the corresponding equations presented in Ref. ohta7 (). One can notice from the general form of equations (5) and (6) that for the same but with different value, (say ), there is another possibility which will have one short-wave component and two long-wave components, and ultimately the dynamics will be different from the case. Also, this systematic generalization to the component case is necessary to identify the way by which the additional wave components (modes) in the normal dispersion regime alter the governing equation for the three components case given in Ref. ohta7 ().
In the following, we investigate the cumulative effect of the small amplitudes ’s on the short-wave components by considering the superposition of these amplitudes involving only the sum of all the amplitudes and neglect all other combinations as they will be small due to the smallness of ’s. Particularly, we add all the -equations and define . By doing so we get
The dispersion relation for the linear excitation corresponding to the long-wave components is found as
where . Note that the velocity of the linear excitation depends upon the number of components and increases as we increase the number of components (modes) in the normal dispersion regime. Thus our systematic generalization to -component case shows that by altering the number of components in the normal dispersion region one can change the velocity of the pulse.
Next we apply the multiple scale approximation method to derive the two-dimensional multicomponent LSRI system as in Ref. ohta7 (). We re-scale the variables , , , and as
|At the order of , we notice from Eq. (5) that all the group velocities of the short-wave components along the direction are the same and their magnitudes are equal to the phase velocity of the long-wave component ‘’ (i.e. ). This is the condition for resonant interaction between long-waves and short-waves. Equation (5) reduces to the following set of coupled equations at the order of after replacing by and rescaling of the variables as defined in equation(10),|
Equation (11) is the multicomponent LSRI system in ()-dimensions which is non-integrable in general. By suitably choosing the constants ’s, ’s, , , , and , along with the assumption that there is no group velocity delay between the short-wave components, we arrive at the following ()-component (2+1)-dimensional LSRI system for the -dimensional propagation of dispersive waves in weak Kerr-like nonlinear media,
In Eq. (12), the subscripts denote partial derivatives with respect to those independent variables. As mentioned in the introduction, we have obtained more general bright -soliton solution, with arbitrary , of Eq. (12) tklsri (). In reference tklsri (), we have expressed the bright -soliton solution of (12) in Gram determinant form and explicitly proved that the general multisoliton solution indeed satisfies the bilinear equations. We have also pointed out in the same work that for the two short-wave components case (), the bright soliton solutions reported by Ohta et al in Ref. ohta7 () follow as special cases of our general multi-soliton solutions tklsri (). As the -component LSRI system (12) admits -soliton solution, for arbitrary tklsri (), the system can be integrable hirotabook (); hiet (). The study on the other integrability aspects of Eq. (12) is under progress and will be published elsewhere.
Iii Hirota’s Bilinearization Method for the ( Multicomponent LSRI system
There are several efficient analytical tools to construct various types of localized structures for nonlinear evolution equations, which include inverse scattering transform method, Hirota’s bilinearization method, Darboux transformation method, Lie symmetry analysis, tanh method, etc. By performing the bilinearizing transformations using Hirota’s direct method hirotabook (); bull (), we construct soliton solutions of Eq. (12) in this paper. In Ref. gramm (); jh (), an extension of Hirota’s bilinear formalism (i.e. multilinear operator) that can encompass any degree of multilinearity has been presented. Using this generalization of Hirota’s method, propagation of a monochromatic laser beam coupled to its second and third harmonics in a nonlinear medium has been studied by V. Cao Long et al. long1 (); long2 (). Recently, a bilinearization procedure with a set of generalized bilinear differential operators different from the standard Hirota’s operators, having nice mathematical properties has been proposed ma1 (). Apart from this, in Ref. ma2 (), it has been pointed out that by employing Lie symmetry approach to the one-dimensional scalar nonlinear Schrödinger equation and by performing a direct search various exact new interesting solutions can be obtained. The Lie algebraic structure of system (12), specifically for has been discussed in Ref. lie (). Indeed, it will be an interesting future direction to compute the Lie symmetries of the multicomponent system (12) for . In this connection, we may also mention that in the past Lie symmetries of certain (2+1) dimensional systems have been constructed by first finding the symmetries of a given (2+1) dimensional system and then reducing it to a (1+1) dimensional system, which on identifying its own Lie symmetries can be reduced to ordinary differential equations. In certain cases, the (2+1) dimensional evolution equations also lead to the identification of infinite dimensional Kac-Moody-Virasoro algebras symmetry (). Apart from the above one can also construct the various interaction solutions of the present system using the Wronskian technique as done in Ref. wronskian () for the KdV equation. The multicomponent system (12) will admit a richer solution structure that may comprise bright solitons, bright-dark solitons, dark solitons, dromions, rational solutions, periodic solutions, elliptic function solutions, and so on.
The present work specifically deals with the study of interesting bright-dark (mixed) solitons of Eq. (12), comprising of bright parts and dark parts, such that . These solitons are usually referred as “symbiotic” solitons as the bright part cannot be supported in a stand-alone fashion and exists only due to the presence of its dark counterpart. These bright-dark solitons are of much theoretical and experimental interest and significant attention has been paid to investigate these intriguing vector solitons as pointed out in the introduction. In the following, we will employ the Hirota’s direct method to construct such coupled bright-dark (mixed) soliton solutions for the system (12) which can find application in various frontier areas like nonlinear optics, water waves and Bose-Einstein condensates.
To construct the mixed type soliton solutions, we perform the bilinearizing transformations, , , , , and , , where ’s and ’s are arbitrary complex functions of and while is a real function. The resulting bilinear equations are
where and , -s are the standard Hirota’s bilinear operators hirotabook (), stands for complex conjugation and is a constant yet to be determined. One can have bright solitons for the choice in ohta7 (); tklsri () and the bright soliton collision dynamics of system (12) has been discussed in Ref. tklsri (). However for non-vanishing ‘’ values, the system can admit coupled bright-dark and dark-dark type soliton solutions. In this paper, we focus only on mixed (bright-dark) solitons corresponding to mixed type boundary conditions, that is, , , .
This procedure can be very well applied to construct the dark-dark soliton solutions also. Here for convenience we consider the first ‘’ short-wave components to be comprised of bright parts of the mixed solitons and the remaining components to exhibit dark parts of the mixed solitons. To construct the mixed soliton solutions we expand the variables ’s, ’s and as power series expansions in a standard way tkpra8 (); hirotabook (). After solving the resultant set of equations recursively we can obtain the explicit forms of ’s, ’s and and hence the multisoliton solutions can be constructed.
Iv Multicomponent mixed type one-soliton solution
The mixed one-soliton solution of (12) with -bright and -dark parts can be obtained by restricting the power series expansions as , , and by solving the resulting equations, after their substitution into the bilinear equations (13) at various powers of recursively. The mixed one-soliton solution can be expressed in the following standard form,
|The various quantities appearing in the above equations are given below:|
In equations (14) and in the following the suffixes and of a particular quantity denote the real and imaginary parts of that quantity, respectively. Also ’s, , , and are complex parameters, while and , , are real parameters. The above solution is non-singular for the condition . The amplitude (peak value) of the bright part of the mixed soliton is and that of -th dark part of the mixed soliton is . The speed of the soliton is . It can be noticed that both parts of the soliton have the same central position . But their phases are different. In fact, the phase of the dark component has two contributions, one from the background carrier wave and the other from . The quantity indeed determines the darkness of the dark soliton. It is interesting to notice that the bright and dark parts of the mixed soliton of the LSRI system with more than two short-wave components display several interesting features in contrast to the case of just two short-wave components as will be shown. To elucidate the understanding of such behaviour we present the explicit forms of one- soliton solutions for the two and three short-wave components and analyse them in the following subsections. For brevity, in the following we refer to mixed -soliton solution with -bright parts and -dark parts as (b-d) mixed soliton solution.
iv.1 Two short-wave components (, ) case
This case admits only a simple type of bright-dark pair in which the bright part of mixed soliton appears in the first component and the dark part of the mixed soliton in the remaining component or vice-versa. The one-soliton solution for this case can be expressed as
The amplitude of the bright part is independent of the parameter , but it is influenced significantly by the background carrier wave (). Such a mixed soliton at and is depicted in Fig. 1 for the parametric choice , , , , , .
iv.2 Three short-wave components () case
Next we consider Eq. (12) with . For this case the mixed soliton can be split up into bright and dark parts among the three short-wave components in two different ways. One corresponds to the (b-d) mixed soliton case where bright parts are in the and components while the dark part of mixed soliton appears in the component. The other possibility is a (1b-2d) mixed soliton case in which the bright part of the mixed soliton appears in the component while the dark parts are split among the remaining components and .
iv.2.1 (2b-1d) mixed one-soliton solution
The one-soliton solution for this case where the bright parts appear in the and components while the third component comprises of the dark part of the mixed soliton can be written from (14) as
where , , , , , and and are as defined in eqn. (14). takes the same form as in eqn. (15c) with the above redefinition of . Here one can observe that the -parameters appear explicitly in the amplitude of the bright soliton. The (2b-1d) one-soliton solution is characterized by twelve real parameters , , , , , , , , and and is restricted by the condition for non-singular solutions. Such type of (2b-1d) bright one-soliton is shown in Fig. 2(a) at and for the parameters , , , , , , and . One can also tune the intensity of bright parts without altering the depth of the dark part of the mixed soliton by suitably choosing the parameter as can be evidenced from Fig. 2(b) which is drawn for same parameter value as that of Fig. 2(a) except for . The soliton appearing in the long-wave component looks similar in both the cases and so we do not present it here.
iv.2.2 (1b-2d) mixed one-soliton solution
This case corresponds to the appearance of the bright part of the mixed soliton in the component while its dark part appears in the and components. The corresponding mixed one-soliton solution is
where , , , , , and and are as in Eq. (14). This solution is characterized by five complex parameters , , , and and four real parameters and , along with the condition . It can be observed from the above solution that in contrast to the (2b-1d) case, here the amplitudes of the bright and dark parts cannot be controlled by the parameters. For illustrative purpose, in Fig. 3 we have shown the (1b-2d) mixed one-soliton solution for the parameters , , , , , , , , and at and .
V Multicomponent mixed type two-soliton solutions
It is a straightforward but lengthy procedure to construct the two-soliton solution. We obtain the mixed two-soliton solution of system (12) by restricting the power series expansion for ’s, ’s and as , , , , and following the standard procedure tkpra8 (). The explicit form of (b-d) mixed two-soliton solution is given below.