-Dark-Dark Solitons in the Generally Coupled Nonlinear Schrödinger Equations
-dark-dark solitons in the generally coupled integrable NLS equations are derived by the KP-hierarchy reduction method. These solitons exist when nonlinearities are all defocusing, or both focusing and defocusing nonlinearities are mixed. When these solitons collide with each other, energies in both components of the solitons completely transmit through. This behavior contrasts collisions of bright-bright solitons in similar systems, where polarization rotation and soliton reflection can take place. It is also shown that in the mixed-nonlinearity case, two dark-dark solitons can form a stationary bound state.
Keywords: Coupled nonlinear Schrödinger equations, KP hierarchy, dark-dark solitons, function.
In studies of nonlinear wave dynamics in physical systems, nonlinear Schrödinger (NLS)-type equations play a prominent role. It is known that a weakly nonlinear one-dimensional wave packet in a generic physical system is governed by the NLS equation . Hence this equation appears frequently in nonlinear optics and water waves [2, 3, 4]. Recently, it has been shown that the nonlinear interaction of atoms in Bose-Einstein condensates is governed by a NLS-type equation as well (called Gross-Pitaevskii equation in the literature) . In these physical systems, the nonlinearity can be focusing or defocusing (i.e., the nonlinear coefficient can be positive or negative), depending on the physical situations  or the types of atoms in Bose-Einstein condensates . When two wave packets in a physical system or two types of atoms in Bose-Einstein condensates interact with each other, their interaction then is governed by two coupled NLS equations [2, 3, 6, 7, 8, 9, 10, 11]. The single NLS equation is exactly integrable . It admits bright solitons in the focusing case, and dark solitons in the defocusing case. Its bright -soliton solutions were given in , and its dark -soliton solutions can be found in . The coupled NLS equations are also integrable when the nonlinear coefficients have the same magnitudes [14, 15, 16]. In these integrable cases, if all nonlinear terms are of focusing type (i.e., the nonlinear coefficients are all positive), the coupled NLS equations are the focusing Manakov model which admits bright-bright solitons . If all nonlinear terms are of defocusing type (i.e., the nonlinear coefficients are all negative), the coupled NLS equations are the defocusing Manakov model which admits bright-dark and dark-dark solitons [17, 18, 19]. If the focusing and defocusing nonlinearities are mixed (i.e., the nonlinear coefficients have opposite signs), these coupled NLS equations admit bright-bright solitons [16, 20] and bright-dark solitons . Existence of dark-dark solitons in this mixed case has not been investigated yet.
Soliton interaction in these integrable generally coupled NLS equations is a fascinating subject. In the focusing Manakov model, an interesting phenomenon is that bright solitons change their polarizations (i.e. relative energy distributions among the two components) after collision . In the coupled NLS equations with mixed nonlinearities, energy can also transfer from one soliton to another after collision . In addition, solitons can be reflected off by each other as well . In the defocusing Manakov model, two bright-dark solitons can form a stationary bound state, a phenomenon which does not occur for scalar bright or dark solitons . All these interesting interaction behaviors can be described by multi-soliton solutions in the underlying integrable system. In the focusing Manakov model, -bright-bright solitons were derived in  by the inverse scattering transform method. In the mixed-nonlinearity model, two- and three-bright-bright solitons and two-bright-dark solitons were derived in [20, 21] by the Hirota method, and -bright-bright solitons were derived in  by the Riemann-Hilbert method. In the defocusing Manakov model, -bright-dark solitons were derived in , and degenerate two-dark-dark solitons were derived in , both by the Hirota method.
So far, progress on dark-dark solitons in the integrable generally coupled NLS equations is very limited. While dark-dark solitons in the defocusing Manakov model were derived in , we will show that their two- and higher-dark-dark solitons are actually degenerate and reducible to scalar dark solitons. In , the inverse scattering transform method was developed for dark solitons in the defocusing Manakov model. But, as we will show in this paper, their analysis can not yield general dark-dark solitons either due to their choices of the boundary conditions. To date, general multi-dark-dark solitons in the coupled NLS equations have never been reported yet (to our knowledge). As we will see, these general multi-dark-dark solitons are not easy to obtain due to non-trivial parameter constraints which must be met.
In this paper, we comprehensively analyze dark-dark solitons and their dynamics in the generally coupled integrable NLS equations. First, we show that these coupled NLS equations can be obtained as a reduction of the Kadomtsev-Petviashvili (KP) hierarchy. Then using -function solutions of the KP hierarchy, we derive the general -dark-dark solitons in terms of Gram determinants. These dark-dark solitons exist in both the defocusing Manakov model and the mixed-nonlinearity model. Recalling that bright-bright solitons exist in the mixed-nonlinearity model as well [16, 18], we see that the coupled NLS equations with mixed nonlinearities are the rare integrable systems which admit both bright-bright and dark-dark solitons. The dark-dark solitons obtained previously in [18, 19] for the defocusing Manakov model are only degenerate cases of our general solutions. Next, we analyze properties of these soliton solutions. For single dark-dark solitons, we show that the degrees of “darkness” in their two components are different in general. When two dark-dark solitons collide with each other, we show that energies in the two components of each soliton completely transmit through. This contrasts collisions of bright-bright solitons in these same equations, where polarization rotation, power transfer and soliton reflection can occur [14, 16, 20]. Thus dark-dark solitons are much more robust than bright-bright solitons with regard to collision. In the case of mixed focusing and defocusing nonlinearities, an interesting phenomenon is that two dark-dark solitons can form a stationary bound state. This is the first report of dark-dark-soliton bound states in integrable systems. However, three or more dark-dark solitons can not form bound states, as we will show in this paper.
We should mention that this KP-hierarchy reduction for deriving soliton solutions in integrable systems was first developed by the Kyoto school in the 1970s . So far, this method has been applied to derive bright solitons in many equations such as NLS, modified KdV, Davey-Stewartson equations [23, 24, 25]. This method has also been applied to derive -dark solitons in the defocusing NLS equation . But this reduction for dark-dark solitons in the generally coupled NLS equations is more subtle and has never been done before. In this paper, we will derive general -dark-dark solitons by this KP-hierarchy reduction and the grace of deep use of determinant expressions. Compared to the inverse scattering transform method  and the Hirota method , our treatment is much more clean, and the solution formulae much more elegant and general. Thus, the KP-reduction method has a distinct advantage in derivations of dark-soliton solutions.
2 The -dark-dark solitons
The generally coupled integrable NLS equations we investigate in this paper are
where and are real coefficients. This system is integrable [14, 15, 16]. Through and scalings, the nonlinear coefficients and can be normalized to be without loss of generality. When , this system is the focusing Manakov model which supports bright-bright solitons . When , this system is the defocusing Manakov model which supports bright-dark and dark-dark solitons [17, 18, 19]. When and have opposite signs, the system exhibits mixed focusing and defocusing nonlinearities. In this case, these equations support bright-bright solitons [16, 20], bright-dark solitons , and dark-dark solitons (as we will see below).
In this section, we derive the general formulae for -dark-dark solitons in the integrable coupled NLS system (1). The basic idea is to treat Eq. (1) as a reduction of the KP hierarchy. Then dark solitons in Eq. (1) can be obtained from solutions of the KP hierarchy under this reduction. For this purpose, let us first review Gram-type solutions for equations in the KP hierarchy [26, 27, 28].
where is the Hirota derivative defined by
is a complex constant, is an integer, and is a function of three independent variables . The Gram determinant solution of the above equations is given by
where the matrix element satisfies
and and are arbitrary functions satisfying
Before proving this lemma, several remarks are in order. The first equation in (2) is the bilinear equation for the two-dimensional Toda lattice (see p.984 of  and p.4130 of ), and the second equation in (2) is the lowest-degree bilinear equation in the 1st modified KP hierarchy (see p.996 of ). Since the two-dimensional Toda lattice hierarchy and modified KP hierarchies are closely related to the (single-component) KP hierarchy, all these hierarchies will be called the KP hierarchy in this paper. Regarding the parameter in the second equation in (2), it corresponds to the wave-number shift in  [see Eq. (10.3) there]. The bilinear equation with this parameter was not explicitly written down in , but can be found in  [see Eq. (N-3) there]. This parameter can be formally removed by the Galilean transformation for in (2). But for our purpose, it proves to be important to keep this parameter, as it will pave the way for the introduction of another similar parameter in Lemma 2 later. In that case, and can not be removed simultaneously by the Galilean transformation, and they are essential for the construction of non-degenerate dark-dark solitons in the generally coupled NLS system (1).
Here the bordered determinants are defined as
Using Lemma 1, we can obtain solutions to a larger class of equations in the KP hierarchy below.
Consider the following equations in the KP hierarchy,
where are complex constants, are integers, and is a function of four independent variables . The solution to these equations is given by the Gram determinant
where the matrix element is defined by
and , , , , are complex constants.
Proof. It is easy to see that functions , and satisfy the following differential and difference rules,
Next, we perform a reduction to the bilinear system (6) in the KP hierarchy. Solutions to the reduced bilinear equations are given below.
Assume that is a real function of real and and are complex functions of real and then the following bilinear equations
where , , and are real constants, and are complex constants, and the overbar ‘’ represents complex conjugate, admit the following solutions,
are complex constants satisfying the constraint
and are arbitrary complex constants.
Proof. In Lemma 2, if one assumes are real, are pure imaginary, are integers, and then we have
where is 1 when and 0 otherwise, then
Under the above reduction, the solution (7) for can be rewritten as
Thus if satisfies the constraint
Using , this equation gives
Differentiation of (23) with respect to gives
The first two equations of (18) are just
which is just
Now we transform the bilinear equations (11) in Theorem 1 into a nonlinear form. To do so, we set
The first bilinear equation in (11) is
which can be further rewritten as
The second bilinear equation in (11) is just
In the same way, from the third bilinear equation in (11) we have
Eqs. (36) are then transformed into
which has N-dark-dark soliton solutions as
The N-dark-dark soliton solutions for the generally coupled NLS equations (1) are
are real constants, are complex constants, and these constants satisfy the following constraints
These solitons are dark-dark solitons, i.e., both and components are dark solitons, because it is easy to verify that
where and are phase constants. Thus the and solutions approach constant amplitudes and at large distances. When and , which correspond to self-focusing nonlinearities for both and components in Eqs. (1), the constraints (41) can not be satisfied, thus dark-dark solitons can not exist as expected. When and , which correspond to self-defocusing nonlinearities for both and components, dark-dark solitons can exist as Ref.  shows. A new phenomenon revealed by Theorem 2 is that, when and have opposite signs, which correspond to mixed focusing and defocusing nonlinearities in the and equations, the constraints (41) can still be satisfied, hence dark-dark solitons can still exist. This phenomenon will be demonstrated in more detail in the next section. Interestingly, when and have opposite signs, Eqs. (1) also admit bright-bright solitons . Thus Eqs. (1) with opposite signs of and are the rare equations which support both dark-dark and bright-bright solitons.
where and are the real and imaginary parts of . Then Eq. (41) becomes
Solving this equation, we find that can be obtained explicitly as
Here and are all free parameters as long as the quantity under the square root of (44) as well as the whole right hand side of (44) are non-negative. If , we will see that the soliton solution (39) would be singular. Thus in this paper, we will always take to avoid this singularity.
We would like to make four remarks here. The first remark is on the above derivation of dark solitons through KP-hierarchy reduction. This derivation is non-trivial. To better understand it, we can split it into two parts. One part is the reduction of the bilinear equations (11) of the generally coupled NLS equations (1) from the KP-hierarchy equations (6). The other part is the reduction of the soliton solutions to the bilinear equations (11) from the -solutions (7) of the KP-hierarchy equations (6). In the first part, when we impose on the -functions the conjugation constraint [see (15)]
and the linear constraint [see (22)]
with being real, then one can readily verify that the KP-hierarchy equations (6) reduce to the bilinear equations (11) of the coupled NLS equations (1). In the second part, in order for the -functions (7) to satisfy the conjugation constraint (45), it is sufficient to require [see (15)]
A sufficient condition for (47) to hold is that
are real, and are pure imaginary. These conditions are the same ones we imposed at the beginning of the proof of Theorem 1. Under these conditions, the -solutions (7) of the KP-hierarchy equations (6) then reduce to the solutions (12) for the bilinear equations (11) of the coupled NLS equations (1). In order for the -functions (7) to satisfy the linear constraint (46), by rewriting these -functions as (19) and inserting them into this linear constraint, we then get the parameter constraint (21), which is equivalent to the parameter constraint (41) in Theorem 1. This splitting of the earlier derivation of dark solitons into these two parts helps to clarify this derivation and make it more understandable.
The second remark is on the solution form (39) of dark solitons in the generally coupled NLS equations (1). It is known that the NLS equation of focusing type is a reduction of the two-component KP hierarchy (see , page 966 and 999), and the NLS equation of defocusing type is a reduction of the single-component KP hierarchy . It is also known that solutions to the single-component KP hierarchy can be expressed as single Wronskians [26, 31, 32], and solutions to the two-component KP hierarchy can be expressed as double Wronskians . Thus -bright solitons in the focusing NLS equation can be expressed as double Wronskians [34, 35], and -dark solitons in the defocusing NLS equation can be expressed as single Wronskians . These Wronskian solutions can also be expressed as Gram-type determinants [26, 27, 31, 36, 37]. For the vector generalization (1) of the NLS equation, in order to obtain its -bright-soliton solutions, one should increase the number of components, and take (1) as a reduction of the three-component KP hierarchy. Thus -bright solitons in (1) can be expressed as three-component Wronskians (or the corresponding Gram-type determinants ). But to obtain -dark solitons in Eqs. (1), one should increase copies of independent variables to and in the single-component KP hierarchy [see Eqs. (6)], thus -dark solitons in Eqs. (1) can still be expressed as single Wronskian (or the corresponding Gram determinant) as we have done above.
The third remark we make is on comparison of the KP-hierarchy reduction method and the inverse scattering method for deriving dark-soliton solutions. As is well known, the inverse scattering method is another way to derive soliton solutions. For bright solitons, the inverse scattering method (or its modern Riemann-Hilbert formulation) is a powerful way to derive such solutions (see [38, 39] for instance). Recently, bright-bright -solitons in a very general class of integrable coupled NLS equations were easily derived by this method , and Eqs. (1) are special cases of such general equations. But for dark solitons, the inverse scattering method is more difficult due to non-vanishing boundary conditions, which create branch cuts and other related intricacies in the scattering process . In , the inverse scattering transform analysis was developed for the defocusing Manakov equations [ in (1)] with non-vanishing boundary conditions. But in their analysis, the boundary conditions (42) were taken such that [see their equation (2.3)] (actually was taken there, but the case of can be reduced to the case of through Galilean transformation). When , one can see from our general formula (39) that and are simply proportional to each other, thus their inverse scattering analysis could only obtain degenerate dark-dark solitons which are reducible to scalar dark solitons in the defocusing NLS equation. In order to derive the more general dark-dark solitons (39) with , the inverse scattering method would be even more complicated than that in . Comparatively, the KP-hierarchy reduction method we used above is free of these difficulties, and is thus a simpler method for deriving dark-soliton solutions.
Our last remark is on dark solitons in an even more general coupled NLS equations
where are real constants as in (1), and is a complex constant. If , (49) reduces to (1)). This more general coupled NLS system (49) is also integrable. Its Lax pair as well as -bright-bright solitons are given in . To explore dark-dark solitons in this system, we look for solutions with the following large-distance asymptotics [as in (39)]
where are non-zero complex constants, and are real constants. Inserting this asymptotic solution into (49), we see that due to the -terms, Eqs. (49) can hold only if , and . Based on the previous solutions (39), this would imply that the and components of dark-dark solitons in the general system (49) must be proportional to each other, thus are equivalent to scalar dark solitons in the defocusing NLS equation. Except these trivial dark-dark solitons, Eqs. (49) do not admit other dark-dark solitons of the form (50) when . This is a dramatic difference between the cases of and in Eqs. (49). Whether the general system (49) admits dark-dark solitons with background asymptotics different from (50) is still unclear.
3 Dynamics of dark solitons
In what follows, we investigate the dynamics of single-dark-soliton and two-dark-soliton solutions in the generally coupled NLS equations (1). In the analysis of these solutions, and will be treated as arbitrary parameters. In the illustrations of solutions in the figures, we will pick
which correspond to mixed focusing and defocusing nonlinearities. The reason for this choice is that dark solitons under such mixed nonlinearities have never been studied before. We will show that under these mixed nonlinearities, some novel phenomena (such as existence of two-dark-soliton bound states) would arise. Soliton dynamics under other and values, such as in the defocusing Manakov equations where , would also be briefly discussed when appropriate.
3.1 Single dark solitons
and are complex constants satisfying
or equivalently, is given by formula (44), where . This soliton would be singular if , i.e., . Thus we will require below to avoid singular solutions. It is easy to see that the intensity functions and of these dark solitons move at velocity . In addition, they approach constant amplitudes and respectively as . As varies from to , the phases of the and components acquire shifts in the amount of and , where
i.e., and are the phases of constants and respectively. Without loss of generality, we restrict , i.e., . At the center of the soliton where , intensities of the two components are
These center intensities are lower than the background intensities and , thus these solitons are dark solitons. Notice that the center intensities of the and solutions are controlled by their respective phase shifts and , thus these phase shifts dictate how “dark” the center is. This general single dark-dark soliton (52)-(53) has been derived for the defocusing Manakov model before by the Hirota method in [18, 17]. In particular, a parameter constraint similar to (54) was given in . If , then , hence . In this case, the and components are proportional to each other, and have the same degrees of darkness at the center. This soliton is equivalent to a scalar dark soliton in the defocusing NLS equation, thus is degenerate. It is noted that the single-dark-dark soliton derived in  [see Eq. (5.8) there] corresponds to this degenerate type of dark-dark solitons. To illustrate, we take
which satisfy the constraint (54). Intensities of the solution (52)-(53) are displayed in Fig. 1(a). This soliton is stationary, and both its and components are black (with zero intensity) at the soliton center.
Non-degenerate single-dark-dark-solitons in Eqs. (1), however, are such that . The and components in these solitons are not proportional to each other, thus are not reducible to scalar single dark solitons in the defocusing NLS equation. Since , , thus . This means that the and components in these non-degenerate solitons have different degrees of darkness at its center. To illustrate, we take
which also satisfies the constraint (54). Here the value is obtained from the formula (44) with the plus sign and . Intensities of this soliton are displayed in Fig. 1(b). This soliton is also stationary. At its center, the component is black, but the component is only gray. This type of non-degenerate single dark-dark solitons in the coupled NLS system (1) has not been obtained before (to our knowledge).