Traveling dark-bright solitons in a reduced spin-orbit coupled system: application to Bose-Einstein condensates
TTraveling dark-bright solitons in a reduced spin-orbit coupled system:application to Bose-Einstein condensates
J. D’Ambroise, D. J. Franzeskakis, and P. G. Kevrekidis Department of Mathematics, Computer & Information Science,State University of New York (SUNY) College at Old Westbury,Old Westbury, NY, 11568, USA; [email protected] Department of Physics, National and Kapodistrian University of Athens,Panepistimiopolis, Zografos, Athens 15784, Greece Department of Mathematics and Statistics, University of Massachusetts,Amherst, MA, 01003, USA; [email protected]
In the present work, we explore the potential of spin-orbit (SO) coupled Bose-Einstein condensatesto support multi-component solitonic states in the form of dark-bright (DB) solitons. In the casewhere Raman linear coupling between components is absent, we use a multiscale expansion methodto reduce the model to the integrable Mel’nikov system. The soliton solutions of the latter allowus to reconstruct approximate traveling DB solitons for the reduced SO coupled system. For smallvalues of the formal perturbation parameter, the resulting waveforms propagate undistorted, whilefor large values thereof, they shed some dispersive radiation, and subsequently distill into a robustpropagating structure. After quantifying the relevant radiation effect, we also study the dynamicsof DB solitons in a parabolic trap, exploring how their oscillation frequency varies as a function ofthe bright component mass and the Raman laser wavenumber.
PACS numbers: 03.75.Mn, 03.75.Lm
I. INTRODUCTION
The subject of atomic Bose-Einstein condensates (BECs) has experienced numerous experimental and theoreticaldevelopments over the past two decades. These developments have been summarized not only in numerous books [1–6], but also in special volumes dedicated to the subject [7]. Within this theme, numerous more specialized topics haveemerged over time that have attracted considerable attention. One of the most recent and intensely investigated ones,concerns the realization of spin-orbit (SO) coupling for neutral atoms in BECs [8, 9] (see also Ref. [10] for theoreticalwork), as well as fermionic gases [11, 12]. In this context, there have been experiments exploring fundamentalphenomena; these include the phase transition from a miscible to an immiscible superfluid [8], the divergence of spin-polarization susceptibility during the transition from a non-magnetic to a magnetic ground state [9], the observationof
Zitterbewegung oscillations [13, 14], the demonstration of Dicke-type phase transitions [15], and the manipulationof the interplay between dispersion and nonlinearity to induce negative effective mass, dynamical instability andnonlinear wave formation [16], among others. At this stage, numerous works combining the efforts of experimentaland theoretical groups have summarized some of the principal developments in the field, both at the early stages [17–19], as well as more recently [20].On the other hand, there has been considerable progress towards exploring the dynamics of coherent nonlinearstructures, in the form of vector solitons, in multi-component repulsive BECs (including pseudo-spinor and spinorones) –cf. the recent review [21]. A principal structure that has been studied in numerous related experimental studieshas been the dark-bright (DB) soliton [22–26], the closely related (i.e., emerging from an SO(2) rotation) dark-darksoliton [27, 28] and, more recently, the so-called dark-antidark soliton [29] (below, we use the term “soliton” in aloose sense, without implying complete integrability [30]). An important characteristic of the DB soliton structure isthat the bright soliton component cannot be supported on its own in such repulsive BECs, yet it arises due to thewaveguiding/trapping induced by the dark soliton component of the DB soliton pair. It should also be mentionedhere that such states have been previously pioneered in nonlinear optics, where single and multiple ones such wereexperimentally realized in photorefractive crystals [31, 32]. Very recently, generalization of these structures in three-components, in the form of dark-dark-bright and bright-bright-dark solitons, were experimentally observed in F = 1spinor BECs [33] (see also Ref. [34] for relevant theoretical predictions).Vector solitons have also been studied in the context of SO coupled BECs. In particular, in the one-dimensional(1D) setting, solitonic structures of the bright [35, 36] or dark [37, 38] types –as well as gap solitons [39–41] in BECsconfined in optical lattices– were predicted to occur, and their dynamics was studied [42]. In fact, the presence ofSO coupling enriches significantly the possibilities regarding the structural form of solitons, as well as their stabilityand dynamical properties. Examples include the prediction of structures composed by embedded families of bright,twisted or higher excited solitons inside a dark soliton, that occupy both energy bands of the spectrum of a SOcoupled BEC and performing Zitterbewegung oscillations [38], or the possibility for effective negative mass bright a r X i v : . [ n li n . PS ] O c t (dark) solitons that can be formed in SO coupled BECs with repulsive (attractive) interactions [43] (note that such achange of sign in the effective mass, i.e., the possibility of “dispersion management” for SO coupled BECs was laterdemonstrated in the experimental work of Ref. [16]). In addition, the existence of quasiscalar soliton complexes dueto a localized SO coupling-induced modification of the interaction forces between solitons [44], or of freely movingsolitons in spatially inhomogeneous BECs with helicoidal SO coupling [45], was also reported. We also note in passingthat, in higher-dimensional settings, stable solitons –composed by mixed fundamental and vortical components– werefound in free 2D [46] and 3D [47] space; these structures are supported by the attractive cubic nonlinearity, withoutthe help of any trapping potential (see also Ref. [48] and Refs. [49, 50] for relevant work in dipolar BECs, as well asthe recent review [51]).In the present work, motivated by the developments in the study of DB solitons in pseudo-spinor condensates,we will attempt to identify such structures in SO coupled BECs. In the relevant process, there is a nontrivial“impediment”, namely a Raman coupling –represented by a Rabi-type linear coupling, of strength Ω R – between thetwo SO-coupled BEC components. Such a coupling, is known to result in population exchange between components,such that the difference of the two condensate populations oscillates at frequency 2Ω R [52]. This naturally enforces asimilar background state between the two components, which would not allow the formation of the DB soliton state(recall that the dark soliton is formed on top of a background wave, while the bright one assumes trivial boundaryconditions [53]).Here, we will thus study the problem under the assumption that the Rabi coupling is absent. In fact, the only termthat we will consider as acting will be the one associated with the wavenumber of the Raman laser, which couples thetwo components imposing in the relevant vector nonlinear Schr¨odinger (NLS) [Gross-Pitaevskii (GP) equation in theBEC context] a Dirac-like coupling –cf., e.g., Ref. [35]. Such a “reduced SO coupled system”, constitutes a vectorialNLS model that also finds applications in nonlinear optics, where it describes the interaction of two waves of differentfrequencies in a dispersive nonlinear medium [53, 54]. Our approach in this effort will be based on developing amultiscale expansion technique, to transform the original nonintegrable model to another, integrable one, facilitatingthe study of the former on the basis of the connection to the latter. Such multiscale expansion methods are usuallyemployed in studies on the existence, stability and dynamics of solitons both in nonlinear optics and BECs (see thereviews [53, 55] and [56, 57] respectively, and references therein). Here, our perturbative approach reveals that, in thesmall-amplitude limit, DB soliton solutions of the reduced SO system do exist, and can be well approximated by thesoliton solutions of the completely integrable Mel’nikov system [58, 59]. Our analytical predictions will then be testednumerically, with and without a trap (in the latter, we will examine the oscillatory dynamics of the DB solitons), aswell as within or outside the range of expected validity of the theory.Our presentation will be structured as follows. In section II, we will introduce the model and present our analyticalresults based on the multiscale expansion method. Section III, is devoted to the presentation of numerical results,where we will examine in particular the range of validity of our analytical approximations. Finally, in section IV, wewill summarize our findings and present our conclusions, suggesting also possible directions of extension of the presentprogram towards future studies. II. MODEL AND ITS ANALYTICAL CONSIDERATIONA. Mean-field model for spin-orbit coupled condensates
We consider a quasi-1D SO coupled BEC, confined in a trap with longitudinal and transverse frequencies, ω x and ω ⊥ , such that ω x (cid:28) ω ⊥ . In the framework of mean-field theory, and in the case of equal contributions of Rashba [60]and Dresselhaus [61] SO couplings (as in the experiment [8]), this system is described by the energy functional [8, 10]: E = u † H u + 12 (cid:0) g | u | + g | v | + 2 g | u | | v | (cid:1) , (1)where u ≡ ( u, v ) T , and the condensate wavefunctions u and v are the two pseudo-spin components of the BEC.Furthermore, the single particle Hamiltonian H in Eq. (1) reads: H = 12 m (ˆ p x + k L ˆ σ z ) + V ( x ) + Ω R ˆ σ x + δ ˆ σ z , (2)where ˆ p x = − i (cid:126) ∂ x is the momentum operator in the longitudinal direction, m is the atomic mass, and ˆ σ x,z are thePauli matrices. The SO coupling terms are characterized by the following parameters: the wavenumber k L of theRaman laser which couples the two components, the strength of the coupling Ω R , and a possible energy shift δ due todetuning from the Raman resonance; below, our analysis will be performed in the case of δ = 0 (see also discussionbelow).Here, it should be noted that upon expanding the squared term in the first part of the single atom Hamiltonian (2),it is clear that a term ∼ k L ˆ p x ˆ σ z appears, which describes the velocity mismatch between the two components, equalto 2 k L . The physical origin of this term is due to the fact that the two Raman laser beams couple atoms havingdifferent velocities. Such a velocity mismatch between different field components is also typical in nonlinear optics,e.g., in the case of interaction between two waves of the same polarization but of different frequencies in a nonlineardispersive medium [54] (see also Ref. [53] and discussion below).In addition, the external trapping potential V ( x ), is assumed to be of the usual parabolic form, V = (1 / mω x x .Finally, the effective 1D coupling constants g ij are given by g ij = 2 (cid:126) ω ⊥ α ij , where α ij are the s-wave scattering lengths.Below, we will present results for repulsive ( α ij >
0) interatomic interactions; furthermore, since in the typical caseof Rb atoms [8] the ratios of the scattering lengths are α : α : α = 1 : 0 .
995 : 0 . α ≈ α ≈ α = α .It is also important to discuss here the versatility of the Hamiltonian (2) with respect to the different parameters.The SO coupling is characterized by a strength k L , which only depends on the laser wavelength λ L and the relativeangle between the counter-propagating beams; thus, by changing the geometry of the lasers, one can control the SOinteractions. Additionally, the Rabi oscillation frequency Ω R depends on the laser beam intensity, which can alsobe controlled, while the energy difference δ can be easily tuned by changing the relative frequency of the counter-propagating lasers. Thus, unlike the SO coupling in condensed matter and electron systems where such a couplingis an intrinsic property of the material [60, 61], in the context of BECs this coupling can be accurately controlled bydifferent external parameters [17–19].Using Eq. (1), we can obtain the following dimensionless equations of motion: iu t = (cid:18) − ∂ x − i (cid:15)∂ x + V ( x ) + | u | + | v | (cid:19) u + Ω R v, (3) iv t = (cid:18) − ∂ x + i (cid:15)∂ x + V ( x ) + | u | + | v | (cid:19) v + Ω R u, (4)where subscripts denote partial derivatives, while energy, length, time and densities are measured in units of (cid:126) ω ⊥ , a ⊥ (which is equal to (cid:112) (cid:126) /mω ⊥ ), ω − ⊥ , and α , respectively, and we have also used the transformations k L → a ⊥ k L = (cid:15)/ R → Ω R / ( (cid:126) ω ⊥ ). Finally, the trapping potential in Eqs. (3)-(4) is now given by V ( x ) = (1 / x , whereΩ = ω x /ω ⊥ (cid:28) R = 0. Furthermore, at a first stage of our analysis,we will assume, to a first approximation, that the trapping potential can also be neglected. In such a case, i.e., for V ( x ) = 0, we may introduce a Galilean transformation, x → x + ( (cid:15)/ t , and thus use an effectively co-traveling frame.In that frame, and under the above assumptions, the equations of motion (3)-(4) become: iu t + 12 u xx − (cid:0) | u | + | v | (cid:1) u = 0 , (5) i ( v t − (cid:15)v x ) + 12 v xx − (cid:0) | u | + | v | (cid:1) v = 0 . (6)Here, it should be mentioned again that the above system finds also applications in the context of nonlinear optics:this system of two incoherently coupled NLS equations describes the evolution of two co-propagating, and interact-ing, slowly-varying electric field envelopes, u and v , of different frequencies, in a weakly dispersive and nonlinearmedium [53, 54]. Here, the group-velocity mismatch (cid:15) between the two components stems from the fact that, due tothe presence of dispersion, the refractive index takes different values for the two different frequencies, which resultsin different group velocities for the two components. Finally, it is noted that the presence of the external potentialwould also be relevant to nonlinear optics, as it may account for a possible parabolic transverse spatial profile of themedium’s refractive index. B. Multiscale analysis and the Mel’nikov system
We now proceed to study analytically the system (5)-(6) using a multiscale expansion method. Here, the wavenum-ber of the Raman laser coupling, parametrized by (cid:15) , will be used as a formal small parameter. Our aim is to findapproximate DB soliton solutions for the SO coupled BEC, using the pseudo-spinor system in the absence of theexternal potential and linear coupling, as our (perturbative) starting point.Let us assume that the u -component carries a dark soliton, while the v -component is a bright soliton. Pertinentboundary conditions for the unknown fields u and v are thus | u | → | u | and | v | → | x | → ±∞ , where the arbitrarycomplex constant u denotes the background amplitude of the dark soliton component. Then, we seek for solutionsof Eqs. (5)-(6) in the form: u ( x, t ) = u ρ ( x, t ) / exp[ iφ ( x, t )] , (7) v ( x, t ) = q ( x, t ) exp[ iCx − i (cid:0) | u | + C / (cid:1) t ] , (8)where the real functions ρ and φ , the complex function q , as well as the constant C (which will be designated as aspeed in what follows) will be determined below. Note that the above mentioned boundary conditions now imply that ρ → | q | → | x | → ±∞ .Substituting Eqs. (7)-(8) into Eqs. (5)-(6), and separating real and imaginary parts in Eq. (5), we obtain the system: φ t + | u | ρ + | q | + 12 φ x − ρ − / ( ρ / ) xx = 0 (9) ρ t + ( ρφ x ) x = 0 , (10) iq t + i ( C − (cid:15) ) q x + (cid:15)Cq + 12 q xx − (cid:2) | u | ( ρ −
1) + | q | (cid:3) q = 0 . (11)We now expand the density and phase of the dark component, as well as the wavefunction of the bright component, ρ, φ, q , in powers of the small parameter (cid:15) as follows. ρ = 1 + (cid:15)ρ (1) + (cid:15) ρ (2) + · · · , (12) φ = −| u | t + (cid:15) / φ (1) + (cid:15) / φ (2) + · · · (13) q = (cid:15)q (1) + (cid:15) q (2) + · · · , (14)where the functions ρ ( j ) , φ ( j ) and q ( j ) ( j = 1 , , . . . ) depend on the slow variables X = (cid:15) / ( x − Ct ) and T = (cid:15) / t .Upon substituting the above expansions into the system (9)-(11), we obtain the following results. First, at theleading-order of approximation [i.e., at orders O ( (cid:15) ) and O ( (cid:15) / )], Eqs. (9) and (10) lead to the self-consistent deter-mination of the constant C and to an equation connecting the unknown functions ρ and φ , namely: C = | u | , φ (1) X = Cρ (1) . (15) C effectively represents the speed of sound, i.e., the velocity of linear waves propagating on top of the continuous-wavebackground of amplitude u . To the next order of approximation [i.e., at orders O ( (cid:15) ) and O ( (cid:15) / )], Eqs. (9) and (10)lead to the following nonlinear equation: ρ (1) T + 3 C ρ (1) ρ (1) X − C ρ (1)
XXX + 12 C (cid:16) | q (1) | (cid:17) X = 0 . (16)On the other hand, to the leading-order of approximation [i.e., at order O ( (cid:15) )], Eq. (11) yields the equation: q (1) XX − | u | ρ (1) q (1) + 2 Cq (1) = 0 . (17)Equations (16)-(17) constitute the so-called Mel’nikov system [58, 59], which is apparently composed of a KdV equationwith a self-consistent source, which satisfies a stationary Schr¨odinger equation. This system has been derived in earlierworks to describe dark-bright solitons in nonlinear optical systems [62], in Bose-Einstein condensates [63, 64] and,more, recently, in nematic liquid crystals [65]. The Mel’nikov system is completely integrable by the inverse scatteringtransform [66], and possesses the exact soliton solution: ρ (1) ( X, T ) = 2 C sech ξ, ξ ≡ µ (cid:18) X − λ C T (cid:19) , (18) q (1) ( X, T ) = Q C sech ξ exp (cid:18) − i C (cid:15) / t (cid:19) , (19)with real parameters µ and λ given by: µ = − C, λ = − | Q | µ − µ . (20) FIG. 1: The solution (21)-(22) is propagated according to Eqs. (5)-(6) for the values (cid:15) = 0 .
005 and µ = Q = 1, u = − C = 0 . | u ( x, t ) | and | v ( x, t ) | , respectively. The traveling DBsoliton can be clearly discerned with no visible radiation in this example. The speed of the center of mass roughly agrees withthe theoretical prediction of | C | in this case. The above results can now be used for the construction of an approximate SO coupled DB soliton solution ofEqs. (5)-(6). This solution, which is valid up to order O ( (cid:15) ), reads: u ( (cid:15) ) ( x, t ) = u (cid:16) (cid:15)C sech ξ (cid:17) exp[ − iC t − i ( (cid:15) / µ/C ) tanh ξ ] , (21) v ( (cid:15) ) ( x, t ) = (cid:15)Q C sech ξ exp[ iCx − i (cid:16) (cid:15) / / (8 C ) + 3 C / (cid:17) t ] . (22)It is clear that the field u has the form of a density dip on top of the background wave, with a phase jump at thedensity minimum, and thus is a dark soliton; on the other hand, the field v has the sech-shaped form, and it representsa bright soliton.It is important to notice here that the relevant waveform crucially depends on the laser wavenumber, i.e., theperturbative parameter (cid:15) . In the limiting case of (cid:15) = 0, the DB soliton degenerates back into the uniform equilibriumstate of the system. Additionally, it is important to highlight that the state that we prescribe here is a genuinelytraveling state. The speed | C | is fully determined by the amplitude of the associated background through | C | = | u | ,while it is also subject to higher-order corrections [ λ(cid:15)/ (4 C ), i.e., of O ( (cid:15) )], given the definition of the co-traveling framevariable ξ . III. NUMERICAL RESULTSA. DB soliton dynamics in a homogeneous background
In order to examine the robustness of the identified wave structures in the previous section, we now turn todirect numerical simulations. Firstly, we initialize the solution at t = 0 according to Eqs. (21)-(22); that is, we set u ( x,
0) = u ( (cid:15) ) ( x,
0) and v ( x,
0) = v ( (cid:15) ) ( x, t according to the dynamical equations (5)-(6) for the values µ = Q = 1. Values of (cid:15) > (cid:15) the solitons emit larger amounts of radiation. InFig. 1, for (cid:15) = 0 . (cid:15) , FIG. 2: Similar to Fig. 1, but for (cid:15) = 0 . we notice that for the larger value of (cid:15) = 0 . R (cid:63) ( (cid:15), t ) = (cid:90) x ( t ) x ( t ) | φ ( (cid:15) ) ( x, t ) | dx/ (cid:90) x ( t ) x ( t ) | φ ( (cid:15) ) ( x, | dx. (23)where R (cid:63) has the following meaning: R (cid:63) = R B refers either to the bright component for φ ( (cid:15) ) = v ( x, t ), or R (cid:63) = R D refers to the dark component with φ ( (cid:15) ) = u ( x, t ). The window [ x ( t ) , x ( t )] remains a fixed length L centered atthe bright or dark soliton peak for all t -values as the soliton propagates according to Eqs. (5)-(6). Hence, the ratio R (cid:63) effectively measures the change in mass (occurring through a process of emission) realized in the vicinity of thesoliton, and gives a measure of the potential “distortion” of the initial condition, as transcribed from our Mel’nikovsystem reconstruction.For the bright ratio R B ( (cid:15) , t ) with (cid:15) fixed, the window length L is computed at t = 0 to be the minimal one,such that ≥
99% of the power of the initial profile is contained within the window. That is, the bright spot in the v -component is contained within the window. For the dark ratio R D ( (cid:15) , t ) with (cid:15) fixed, the window length L iscomputed at t = 0 to be the minimal one, such that ≥
1% of the power of the initial profile is contained within thewindow. Thus, the dark spot in the u component is contained within the window too. In the top and middle plotsof Fig. 3 we see that the power within these windows decreases for both the dark and bright soliton components asradiation leaks out of the window for increasing t . The radiation is more prominent for larger values of (cid:15) , and it isessentially non-existent for a small enough value such as (cid:15) = 0 . (cid:15) for the fixed values of L = 30 and t = 400. The radiation in the bright solitoncomponent is more prominent than that in the dark component according to this measure. This may be attributedto the topological nature (and hence additional robustness) of the dark soliton. FIG. 3: In the top and middle panels R D ( (cid:15) , t ) and R B ( (cid:15) , t ) are plotted as a function of t , for (cid:15) = 0 .
005 (blue solid line), (cid:15) = 0 .
075 (green line with x-symbols), (cid:15) = 0 .
145 (red dotted line), (cid:15) = 0 .
215 (cyan dash-dotted line), and (cid:15) = 0 . R D ( (cid:15), t = 400) (solid line) and R B ( (cid:15), t = 400) (dashed line) are plotted as a functionof (cid:15) for L = 45, showing that the radiation is more significant in the bright- rather than in the dark-soliton component. B. DB soliton dynamics in the trap
We now consider the dynamics in the presence of a parabolic trap, namely: V ( x ) = (1 / x , as discussed inSec. II. Here, the presence of the trap results in the loss of the invariance with respect to spatial translations, whichsuggests the consideration of the original system, in the form of Eqs. (3)-(4), for Ω R = 0.Before “embedding” the DB soliton into our numerical scheme, we first identify stationary solutions usingan initial guess based on the Thomas-Fermi approximation [1, 2] for the dark soliton component, namely u ≈ e − iµ t max (cid:16)(cid:112) µ − V ( x ) , (cid:17) (where µ is the chemical potential of the u -component), while the bright componentis absent, i.e., v = 0. A Newton-Raphson based continuation in (cid:15) then allows us to numerically converge to thedesired “background” (stationary) solution u = u T F ( x ) , v = 0 , for the full set of equations (3)-(4) (for Ω R = 0). Here, u T F is the true numerical solution to the problem, ratherthan the approximate analytical one. Now, embedding within this background, our approximate analytical solutionof Eqs. (21)-(22) from the multiscale expansion argument of Section II, gives the approximate dark-bright solitonsolutions u = u ( (cid:15) ) u T F , v = v ( (cid:15) ) , in the presence of the trap V ( x ).These dark-bright soliton structures oscillate over time within the trapped condensate. A typical example of theresulting oscillation is shown in Fig. 4. Here, it should be noted that we find that there are rather small-amplitudewavepackets that escape from the core of the “imprinted” waveform and reach the boundary of the domain, subse-quently becoming back-scattered. To avoid such a spurious effect, over the course of the admittedly long oscillationperiods of the DB soliton, we include in the numerical simulations an absorbing layer that is activated outside theregion of the background solution width. This layer absorbs excess radiation in order to minimize interference withthe DB soliton structure upon potential back-scatter.The oscillation frequency ω of the dark-bright solitary wave is tracked as a function of the atom number of thebright soliton component, N b = (cid:90) x ( t ) x ( t ) | v | dx, (24) FIG. 4: Density contour plot showing the oscillation of a DB soliton in a parabolic trap according to Eqs. (3)-(4) (for Ω R = 0),for parameter values (cid:15) = 0 .
1, Ω = 0 . Q = 4 . u = 1, µ = √
2, and µ = 0 . ω/ Ω as a function of N b for various (cid:15) values. The prediction of Ref. [69] [cf. Eq. (25)]is plotted as the red dashed line. where [ x ( t ) , x ( t )] is a moving window centered at the bright soliton. The resulting normalized (to the trap frequency)oscillation frequency ω/ Ω is compared in Fig. 5 to the prediction of Busch and Anglin [69], for a regular (two-component) dark-bright soliton in the absence of spin-orbit coupling: ω = Ω √ − N b (cid:113) µ + ( N b ) . (25)Interestingly, it is found that the theoretical prediction of Eq. (25) not only provides a fair approximation to thefrequency of oscillation, but, in fact, one that is progressively more accurate for higher values of (cid:15) . IV. DISCUSSION AND CONCLUSIONS
In the present work, we have explored the possibility of the intensely studied spin-orbit (SO) coupled BECs to bearstructures that are prototypical in pseudo-spinor condensates, namely dark-bright (DB) solitons. In fact, we havestudied a “reduced spin-orbit coupled system”, corresponding to the absence of the linear Raman coupling (whosefeasibility is worth further experimental consideration) between components. The reason for this assumption was thatthe presence of such a linear coupling results in the onset of population exchange between components, which makesimpossible the formation of states with different boundary conditions (this was also confirmed in our simulations–results were not shown here). The studied model, apart from SO-coupled BECs, finds also applications in nonlinearoptics, describing the interaction between two electric field envelopes of different frequencies in a dispersive nonlinearmedium.In this limit of zero Raman coupling, we have been able to provide a systematic perturbative method for theconstruction of dark-bright solitons. This method relies on the asymptotic reduction of the nonintegrable reduced SOsystem to the completely integrable Mel’nikov system. For the latter, exact analytical soliton solutions do exist, andcan be used to reconstruct DB solitons for the reduced SO system.The validity of our analytical approximations, and the robustness of the DB solitons, were explored via numericalcomputations. It was found that the solitons persist not only in the limit of sufficiently small value of the perturbationparameter (cid:15) (where the accuracy of the perturbation theory and the control of the associated error guarantees itsrelevance), but also even in the case where (cid:15) is not that small. Finally, we investigated the DB soliton dynamics inthe more realistic setting where a parabolic trap is present. It was found that DB solitons oscillate in the trap witha frequency proximal to that predicted for “regular” DB solitons in the absence of the SO coupling.This study opens numerous veins for future research. On the one hand, it is worthwhile to further explore thedynamics of DB solitons from the point of view of, e.g., their complex interaction dynamics (see for some recentexamples, the work of [70]), and how the presence of the spin-orbit coupling could modify that. On the otherhand, one could envision generalizations of the relevant structures, in the presence of spin-orbit coupling, to higherdimensions. These include the study of baby-Skyrmions –otherwise known as filled-core vortices or vortex-brightsolitons– in two spatial dimensions [71–73] (see recent relevant work in Ref. [74]), and even to true skyrmions in threespatial dimensions [75].
Acknowledgments.
P.G.K. and D.J.F. gratefully acknowledge the support of the “Greek Diaspora Fellowship Program” of StavrosNiarchos Foundation. P.G.K. also acknowledges the support of the NSF under the grant PHY-1602994. Constructivediscussions with D. E. Pelinovsky at the early stages of this work are also kindly acknowledged. [1] L. P. Pitaevskii and S. Stringari,
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