Multivalley dark solitons in multicomponent Bose-Einstein condensates with repulsive interactions
MMultivalley dark solitons in multicomponent Bose-Einstein condensates with repulsiveinteractions
Yan-Hong Qin , , Li-Chen Zhao , , , ∗ Zeng-Qiang Yang , and Liming Ling † School of Physics, Northwest University, Xi’an 710127, China NSFC-SPTP Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710127, China Department of Physics, School of Arts and Sciences,Shaanxi University of Science and Technology, Xi’an 710021, China and School of Mathematics, South China University of Technology, Guangzhou 510640, China (Dated: February 23, 2021)We obtain multivalley dark soliton solutions with asymmetric or symmetric profiles in multicom-ponent repulsive Bose-Einstein condensates by developing the Darboux transformation method. Wedemonstrate that the width-dependent parameters of solitons significantly affect the velocity rangesand phase jump regions of multivalley dark solitons, in sharp contrast to scalar dark solitons. Fordouble-valley dark solitons, we find that the phase jump is in the range [0 , π ], which is quite dif-ferent from that of the usual single-valley dark soliton. Based on our results, we argue that thephase jump of an n -valley dark soliton could be in the range [0 , nπ ], supported by our analysisextending up to five-component condensates. The interaction between a double-valley dark solitonand a single-valley dark soliton is further investigated, and we reveal a striking collision process inwhich the double-valley dark soliton is transformed into a breather after colliding with the single-valley dark soliton. Our analyses suggest that this breather transition exists widely in the collisionprocesses involving multivalley dark solitons. The possibilities for observing these multivalley darksolitons in related Bose-Einstein condensates experiments are discussed. PACS numbers: 03.75. Lm, 03.75. Kk, 05.45.Yv, 02.30.Ik
I. INTRODUCTION
Multicomponent Bose-Einstein condensates (BECs)provide a good platform for the investigation of vectorsolitons both theoretically and experimentally [1–3] dueto the abundance of intra- and interatomic interactions.Various vector solitons have been investigated in mul-ticomponent BECs with attractive or repulsive interac-tions [2–17]. The major theme of research on attractiveBECs is bright solitons [2, 6, 7, 9], while studies on darksolitons (i.e., bright-dark solitons) are considerably ham-pered by their background modulation instability [18].This characteristic makes it difficult to observe dark soli-tons experimentally in attractive BECs.In contrast, many more dark vector solitons havebeen experimentally observed in multicomponent repul-sive BECs, such as dark-dark solitons [4, 10, 14], dark-bright solitons [5, 8, 11, 13], dark-antidark solitons [12],dark-dark-bright solitons and dark-bright-bright solitons[15]. Very recently, experimental observations of the colli-sions of bright-dark-bright solitons were realized in three-component BECs with repulsive interactions [16]. Never-theless, the dark solitons in the abovementioned works re-fer mainly to single-valley dark solitons (SVDSs). There-fore, we aim to look for multivalley dark soliton (MVDS)solutions in repulsive BECs.In this work, we present the exact MVDS solutions ∗ Electronic address: [email protected] † Electronic address: [email protected] in multicomponent BECs with repulsive interactions byfurther developing the Darboux transformation (DT)method. The explicit soliton solutions admit an MVDSin one of the components and multihump bright soli-tons in the other components. In particular, the solitonwidth-dependent parameters have a considerable impacton both the velocity range and the phase jump of theMVDS. The phase jump of a double-valley (triple-valley)dark soliton can vary in the range of [0 , π ] ([0 , π ]).These characteristics of MVDS are distinct from those ofthe well-known scalar dark soliton, for which the widththat depends on both the velocity and the phase jumpcan be varied in the range [0 , π ][19–21]. Furthermore,we explore the collision dynamics of MVDSs. The in-teraction between two MVDSs reflects the density profilevariations only after a collision. Interestingly, one MVDScan transition to a breather after colliding with an SVDS;this can occur because the mixture of the effective ener-gies of solitons in the three components emerges duringthe collision process. This breather transition occurs ex-tensively in collision processes involving MVDSs. Thesefindings provide an important supplement for recent re-ports on nondegenerate vector solitons [22–25]. We ex-pect that more abundant MVDSs could exist in coupledBECs comprising more components and that the phasejump of an n -valley dark soliton could be in the range of[0 , nπ ].The remainder of this paper is organized as follows.In Sec.II, we introduce the theoretical model and presentthe double-valley dark soliton (DVDS) solutions in three-component repulsive BECs; we further show the densityprofiles and analyze the phase features of DVDSs. In a r X i v : . [ n li n . PS ] F e b Sec.III, we investigate the collision dynamics of DVDSsand report the striking state transition dynamics whenthey collide with SVDSs. In Sec.IV, we extend our studyto four-component repulsive BECs, where triple-valleydark solitons (TVDSs) can be obtained. Finally, the con-clusions and discussion are presented in Sec.V.
II. DOUBLE-VALLEY DARK SOLITONS INTHREE-COMPONENT REPULSIVECONDENSATESA. Physical model and double-valley dark solitonsolutions
We note that it is difficult to obtain dark solitons withmore than a single valley in N ( N (cid:54) j , t + 12 q j , xx − ( | q | + | q | + | q | )q j = 0 , (1)where q j ( x, t )( j = 1 , ,
3) represents the mean-field wavefunctions of three-component repulsive BECs. Thismodel can also be used to describe the evolution of lightin defocusing nonlinear optical fibers [19, 20]. A recentexperiment on bright-dark-bright solitons in repulsiveBECs strongly supports the applicability of the aboveintegrable three-component Manakov model to three-component BECs with repulsive interactions [16]. Fortheir attractive counterpart, single-hump bright solitonsolutions [28, 30, 35, 36] and even multihump bright soli-ton solutions have been obtained in a multicomponentManakov system by the DT method [23], Hirota method[22, 24, 25] and other methods [37, 38]. Next, we system-atically seek the MVDS solutions in the multicomponentrepulsive Manakov model.The DT method is an effective and convenient wayto derive localized wave solutions [39–44]. Recently, itwas reported that SVDSs can be obtained through theDT method for multicomponent repulsive Manakov sys-tems [43]. In this paper, we further develop the DTmethod [43] to derive MVDS solutions in combinationwith the multifold DT for deriving nondegenerate brightsolitons [23]. We find that MVDSs can be derived byperforming a multifold DT with some special constraintconditions on the eigenfunctions of the Lax pair. Forexample, one DVDS can be obtained by performing atwo-fold DT with the spectral parameters written as λ j = ( ξ j + ξ j ) and adding some special constraint con-ditions to the eigenfunctions. The complex parameter ξ j = − v +i w j ( j = 1 ,
2) is introduced to simplify the soli-ton solution and facilitate the analysis of physical mean-ing of each parameter; the real part determines the soli-ton’s velocity, while the imaginary part is called a solitonwidth-dependent parameter. The detailed derivation ofexact DVDS solutions is given in Appendix A. For theDVDS solutions given in Eq. (A13), one DVDS is admit-ted in the first component, and one double-hump brightsoliton is allowed in the other two components (DBBS).By simplifying Eq. (A13), the exact general soliton solu-tions can be expressed as follows: q = N M e − i t (2a) q = − i2 w (cid:113) − v − w α ξ N M e i [ v x − (2+ v − w ) t ] (2b) q = − i2 w (cid:113) − v − w β ξ N M e i [ v x − (2+ v − w ) t ] (2c)with N = (cid:20) ξ ∗ ξ α e κ − κ + ξ ∗ ξ β e κ − κ + α β e κ + κ (cid:21) × ( w + w ) + ξ ∗ ξ ∗ ξ ξ ( w − w ) e − κ − κ ,N = ( w + w ) (cid:2) β ( w + w ) e κ +( w − w ) e − κ (cid:3) ,N = ( w + w ) (cid:2) β ( w + w ) e κ +( w − w ) e − κ (cid:3) ,M = (cid:2) α β e κ + κ + α e κ − κ + β e κ − κ (cid:3) ( w + w ) + ( w − w ) e − κ − κ ,κ = w ( x − v t ) , ξ = − v + i w ,κ = w ( x − v t ) , ξ = − v + i w , where α , β , v , w , and w are arbitrary real constants.Among them, the parameters w and w are two solitonwidth-dependent parameters ( w (cid:54) = w ). The parameter v is the soliton velocity, which must satisfy the con-straint condition v + w m <
1, where w m is the largerof the two width-dependent parameters, meaning thatthe soliton width-dependent parameters affect the veloc-ity range. In contrast, the width of a scalar dark solitondepends on the velocity, which cannot exceed the speedof sound [3, 19–21]. The parameters α and β are twofree parameters related to the relative valley values andthe center positions of the two valleys; therefore, theseparameters nontrivially contribute to the soliton profiles. B. Density profiles of double-valley dark solitons
The solution expressed in Eqs. (2) is generallyasymmetric but becomes symmetric when the twofree parameters satisfy the condition α = β = (cid:112) ( w − w ) / ( w + w ). Examples of asymmetric andsymmetric DBBS solutions are shown in Fig. 1 with solid FIG. 1: The density profiles of DVDSs. (a) shows an asym-metric DVDSs in component q . (b) and (c) show the den-sity profiles of asymmetric bright solitons in components q and q , respectively. (d-f) show the corresponding profiles forsymmetric solitons. The parameters are α = 0 . , β = 0 . α = β = 1 / √ v = 0 . w = 0 .
2, and w = 0 . lines and dashed lines, respectively. The plots from leftto right correspond to component q (red line), compo-nent q (green line) and component q (blue line). Forthe asymmetric case depicted in Figs. 1(a)-(c), the pa-rameters are v = 0 . w = 0 . v = 0 . α = 0 . β = 0 .
2. We can see that a DVDS emerges incomponent q , manifesting two spatially localized den-sity “dips” on a uniform background. An asymmetricdouble-hump bright soliton with one node is exhibitedin component q , and an asymmetric single-hump brightsoliton without a node presents in component q [45];these two bright solitons are quite different from the fun-damental dark-bright-bright soliton observed in [15, 16].As shown in Fig. 1(a), the two valleys of the asymmet-ric DVDS are not equal in either the width or the valleyvalues, in sharp contrast to dark solitons reported in pre-vious studies [2–5, 8, 10–17, 19, 20, 30–33, 43, 47]. Thischaracteristic is not observed for the stationary symmet-ric MVDS solution [46], which was reported to exist infocusing photorefractive media (similar to attractive mul-ticomponent BECs). The bright solitons that present incomponents q and q are similar to those obtained in atwo-component attractive (focusing) system [22–25].By setting the parameters α = β = √ w − w √ w + w and theother parameters the same as in the asymmetric case,the solution given in Eqs. (2) admits symmetric solitonprofiles, as shown in Figs. 1(d)-(f) (dashed lines). Theseprofiles are similar to those reported in previous studies[38, 46], but the total density can differ from the “sech-squared” form. Moreover, the DVDS is expressed by ex-ponential functions, which is distinctly different from theexpressions with associated Legendre polynomials [46].Moreover, the above DVDS solution is more general thanthose given in [38, 46] since it admits more free phys-ical parameters. The soliton velocity of the solution inEqs. (2) changes in the range of (cid:2) , (cid:112) − w m (cid:1) . In partic-ular, the interaction between the solitons can be inves- FIG. 2: The asymmetry degree vs moving velocity for DVDSs.The red solid line and blue dashed line correspond to theasymmetric and symmetric DVDSs, respectively. The asym-metry degree increases with increasing moving velocity forasymmetric DVDSs. However, that for symmetric DVDSs re-mains symmetric with varying velocity. The other parametersare the same as in Fig. 1. tigated analytically with further iterations by applyingthe developing DT method. When the soliton width-dependent parameters w and w are close to each other,both components q and q can show a symmetric (oran asymmetric) double-hump bright soliton. The char-acteristics of bright solitons are similar to the results in atwo-component attractive (focusing) system [22–25, 37].In fact, nondegenerate two-component bright soliton so-lutions can be reduced from the above three-componentsolution with the aid of the close relations between soli-ton solutions in attractive and repulsive systems [38].It is well known that the valley depth of an SVDS tendsto decrease with increasing moving velocity, and the sym-metric properties do not change. A similar characteris-tic is admitted by symmetric DVDSs. However, we findthat this characteristic no longer holds for asymmetricDVDSs, whose asymmetry can significantly change withvariation in the moving velocity, and the valley depthalso tends to become shallower with increasing movingvelocity. The asymmetry degree is defined as R ( v ) = I − I I + I , (3)where I and I are the values of the left and right valleysin space, respectively. We exhibit the asymmetry degreevs the moving velocity in Fig. 2 while keeping the otherparameters the same as in Fig. 1. The red solid curve(blue dashed line) corresponds to the asymmetric (sym-metric) DVDS. The asymmetry degree crucially dependson the velocity for asymmetric DVDSs, whereas R ( v ) isconstant for symmetric DVDSs. This is also a significantdifference between symmetric and asymmetric DVDSs. C. Phase properties of double-valley dark solitons
SVDSs are well known to admit one phase jumpacross the dip, which varies in the range [0 , π ] [2, 3, 19–21]. Here, we characterize the phase jump propertiesof DVDSs. Generally, the phase value can be obtainedby calculating the argument φ ds ( x ) = Arg(q ds ) by drop-ping the momentum phase of the background. It shouldbe emphasized that the phase distribution in space can-not be directly obtained by calculating only the argu-ment based on the real and imaginary parts of the wavefunction. Importantly, the phase jump direction mustbe judged by the phase gradient flow ∂ x arctan (cid:104) Im(q ds )Re(q ds ) (cid:105) .Namely, positive (negative) phase gradient flow corre-sponds to a rising (falling) phase jump. Here, we definethe phase jump as (cid:52) φ ds = φ ds ( x → −∞ ) − φ ds ( x → + ∞ ). As an example, we plot the corresponding phasedistribution of Fig. 1(a) in Fig. 3(a). Remarkably, theDVDS is characterized by two phase jumps through twovalleys, in contrast to the SVDS with one phase jump[2–5, 8, 10–17, 19–21, 30–33, 43, 47]. Moreover, thephase jump value across a DVDS is larger than π in thiscase, which is also distinctive from the phase jump of aSVDS (which cannot exceed π ) [2–5, 8, 10–17, 19–21, 30–33, 43, 47]. We note that SVDSs can admit a phasejump greater than π in a saturable self-defocusing mate-rial [48, 49] but only one phase jump across the soliton.The phase distribution of a symmetric DVDS is simi-lar to that of an asymmetric DVDS, with only a smalldifference in the spatial position. Additionally, a brightsoliton with one node in component q always maintainsa π phase jump, where the abrupt phase change occursat the node, while a bright soliton without a node has nophase jump in component q .Furthermore, we find that the phase jump of a DVDSchanges with variation in the moving velocity. As an ex-ample, we display the variation in the phase jump of aDVDS with the velocity in Fig. 3(b), where the red solidline and the blue dashed line represent asymmetric andsymmetric DVDSs, respectively. The parameters are thesame as in Fig. 1 except for the velocity v . The phasejump value decreases dramatically with increasing veloc-ity. For this case, the velocity range is | v | ∈ [0 , √ / . π, π ]. Underthe limit of zero velocity, i.e., for a stationary DVDS, thephase jump takes a value of 2 π . Fig. 3(b) suggests thatthe change in the phase jump with velocity for a symmet-ric DVDS is identical to that for an asymmetric DVDS,and the parameters α and β do not affect the phasejump region. However, further analysis indicates thatthe soliton width-dependent parameters essentially de-termine the phase jump range. For example, we draw thevariation in the phase jump value with the soliton width-dependent parameter w in Fig. 3(c), where w is fixed tobe 0 . w , we can calculate the ve-locity region under the constraint condition w j + v < w ∈ (0 , . | v | ∈ [0 , √ / w ∈ (0 . , | v | ∈ [0 , (cid:112) − w ). Fig. 3(c)clearly shows the variation in the phase jump with anincrease in the width-dependent parameter w , alwaystaking a value of 2 π for a static DVDS. With an increase FIG. 3: The phase properties of a DVDS. (a) The phase distri-bution of an asymmetric DVDS corresponding to the densityprofile in Fig. 1(a). We can see that the DVDS admits twophase jumps across two valleys. (b) The phase jump vs mov-ing velocity for DVDSs. The red solid line shows the curvefor the asymmetric DVDS, while the blue dashed line showsthe curve for the symmetric DVDS. The other parameters forboth cases are identical to those in Fig. 1. The phase jumpdecreases gradually with increasing velocity. (c) The phasejump (in π unit) vs the soliton width-dependent parameterand velocity. The parameter w is fixed at 0 .
2. This demon-strates that the soliton width-dependent parameters signifi-cantly affect the phase jump and velocity region of DVDSs.The other parameters are α = 0 . β = 0 . in the width-dependent parameter w , the phase jumpregion decreases since the phase jump value increases atthe maximum velocity. With w = 0 .
2, the phase jumprange is [0 . π, π ]. For smaller soliton width-dependentparameters w and w , the phase jump value tends to bezero when the soliton’s velocity approaches the limit ofthe maximum velocity, meaning that DVDSs admit thelargest phase jump range of [0 , π ].We emphasize that the soliton width-dependent pa-rameters also affect the velocity region; as a result, themaximum speed of the DVDS can be much smaller thanthe speed of sound for DVDSs when choosing largerwidth-dependent parameters (see Fig. 3(c)). This is anotable characteristic for MVDSs that is absent for usualscalar SVDSs [19, 20]. For usual scalar SVDSs, the max-imum velocity is the speed of sound. For a bright-darksoliton in a two-component BEC with attractive interac-tions, the width and velocity are completely independentof each other [47], and there is no limit on the moving ve-locity. However, for dark-bright solitons in repulsive con-densates, a change in the soliton width also causes the ve-locity range to vary [47]. Our detailed analysis indicatesthat the relation between the soliton width and velocitysatisfies an inequality for MVDSs and dark-bright soli-tons [47], but this relation becomes an equality for usualscalar SVDSs [19, 20]. This characteristic of dark-brightsolitons has not been taken seriously in previous work [5].Nevertheless, we expect that the soliton width plays animportant role in the motion of dark-bright solitons inharmonic traps, in contrast to the motion of scalar darksolitons in harmonic traps [50]. Therefore, the effects ofthe soliton width should be considered when discussingthe motion of MVDSs in external potentials.Very recently, three-component vector solitons andtheir collisions were experimentally observed in BECswith repulsive interactions [16]. Motivated by these re-sults and the developed density and phase modulationtechniques in ultracold atomic systems [8, 15, 16, 51, 52],we analytically investigate the collision dynamics ofDVDSs by performing further iterations of the developedDT method. III. COLLISION DYNAMICS OFDOUBLE-VALLEY DARK SOLITONS
The collision dynamics of DVDSs mainly include twocases: (i) a collision between a DVDS and an SVDS, (ii) acollision between two DVDSs. We investigate these twotypes of collisions based on two dark soliton solutionsderived by the three-fold and four-fold DT (see detailsin Appendix A). A collision between two DVDSs usuallymakes each soliton’s profile vary, similar to a collision be-tween nondegenerate bright solitons [23]. However, thecollision between a DVDS and an SVDS demonstrates astriking state transition process: the DVDS transformsinto a breather after colliding with the SVDS, and thelatter does not admit any oscillation, just a profile vari-ation.We first study the interaction between a DVDS andan SVDS based on the exact solution Eq. (A23) by im-plementing a three-fold DT with the introduced com-plex parameters ξ = − v + iw , ξ = − v + iw (gen-erating a DVDS in component q ) and ξ = − v + iw (generating an SVDS in component q ) (see AppendixA for the detailed calculation process). A typical exam-ple of the striking state transition process is illustratedin Fig. 4(a1). The asymmetric DBBS is evidently trans-formed into a breather, whose density evolution admits aperiodic oscillation behavior. In contrast, the fundamen-tal DBBS (moving toward the left) maintains a solitonstate with slight profile variation. Such a state transitioninduced by a collision is not discussed in the previousliterature [4, 5, 8, 19, 20]. Figs. 4(a2) and (a3) showthe density evolution of bright solitons in components q and q , respectively, demonstrating that this breathingbehavior also emerges after the interaction.To understand the state transition phenomenon, wefurther analyze the exact solution given by Eq. (A23)by using the asymptotic analysis technique [26, 53, 54].Our analysis suggests that the state transition is inducedby the mixture of effective energies of the solitons in thethree components during the collision process. Before thecollision (within the limit t → −∞ , with v > v ), the FIG. 4: The collision dynamics of DVDSs. (a1): Collisionbetween one DVDS and an SVDS. There is a striking statetransition process in which a DVDS transitions to a breatherafter the collision, whereas the SVDS does not breath andexperiences only slight profile variation. (a2) and (a3) showsimilar state transition dynamics in the other bright solitoncomponents. (b1): Collision between two DVDSs. For thiscase, there is no state transition occurrence for the DVDSs.(b2) and (b3) display similar collision dynamics in the otherbright soliton components. The parameters are v = 0, w =0 . w = 0 . v = − . w = 0 . α = 1, β = 1, and α = β = 1 for (a1-a3). The parameters are v = − . w = 0 . w = 0 . v = 0, w = 0 . w = 0 . α = 1, β = 1, α = 1, and β = 1 for (b1-b3). DVDS-related vector soliton takes the following asymp-totic forms: q i = f e − ν + f e ν + f e − ν + f e ν m e − ν + m e ν + m e − ν + m e ν e − i t , (4a) q i = g ( g e − κ + g e κ ) m e − ν + m e ν + m e − ν + m e ν e i ϕ , (4b) q i = h ( h e − κ − h e κ ) m e − ν + m e ν + m e − ν + m e ν e i ϕ . (4c)After the collision (in the limit t → + ∞ ), the asymp-totic analysis expressions for the vector soliton take thefollowing forms: q f = δ (cid:0) δ e − i θ + δ e i θ (cid:1) + δ ( δ e ν + δ e − ν )+ δ ( δ e ν + δ e − ν ) (cid:37) ( (cid:37) e − i θ + (cid:37) ∗ e i θ )+ (cid:37) ( (cid:37) e − ν + (cid:37) e ν + (cid:37) e ν + (cid:37) e − ν ) e − i t , (5a) q f = e i ϕ ζ ( ζ e − κ + ζ e κ ) + e i ϕ ζ ( ζ e − κ + ζ e κ ) (cid:37) ( (cid:37) e − i θ + (cid:37) ∗ e i θ )+ (cid:37) ( (cid:37) e − ν + (cid:37) e ν + (cid:37) e ν + (cid:37) e − ν ) , (5b) q f = e i ϕ ς ( ς e − κ + ς e κ ) + e i ϕ ς ( ς e − κ + ς e κ ) (cid:37) ( (cid:37) e − i θ + (cid:37) ∗ e i θ )+ (cid:37) ( (cid:37) e − ν + (cid:37) e ν + (cid:37) e ν + (cid:37) e − ν ) . (5c) In Eqs. (4) and Eqs. (5), ν = κ + κ , ν = κ − κ , θ = ϕ − ϕ ,κ = w ( x − v t ) , ϕ = v x −
12 ( v − w + 2) t,κ = w ( x − v t ) , ϕ = v x −
12 ( v − w + 2) t. In addition, the expressions for f j , g j , h j , m j , δ j , ζ j , ς j ,and (cid:37) j are given in Appendix B, and they are all complexconstants related to ξ j ( j = 1 , , E ∗ j = dφ j dt (where φ j is the phase of the wave function).We know that the effective energy of the soliton in com-ponents q , q and q is E ∗ = − E ∗ = − ( v − w + 2)and E ∗ = − ( v − w + 2), respectively. For Eqs. (5),we can see that the effective energies of bright solitoncomponents mix and emerge in the DVDS componentafter the collision process. The breathing behavior ob-viously originates from the energy mixing term in e ± iθ .Namely, the effective energy difference of bright solitonsdetermines the oscillation period, and the period is T = 2 π | E ∗ − E ∗ | = 4 π | w − w | . (6)The oscillation period is identical among the three com-ponents. Based on this discussion, we also revisit thecollision dynamics of nondegenerate bright solitons [23]and find that an effective energy mixture can also emerge.This means that the abovementioned breather behavioris also observable during the collision between a nonde-generate bright soliton and a degenerate bright soliton.Next, we investigate the interaction between twoDVDSs based on the exact solution given in Eq. (A24)by implementing a four-fold DT with the parameters ξ = − v + iw , ξ = − v + iw (which generates oneDVDS), ξ = − v + iw , and ξ = − v + iw (which gen-erates another DVDS) (see Appendix A for the detailedcalculation process). For this case, a typical example ofthe density distribution is displayed in the second panelof Fig. 4, for which (b1), (b2) and (b3) correspond tocomponent q , component q and component q , respec-tively. As shown in Fig. 4(b1), the collision between thetwo DVDSs causes their profiles to change, accompaniedby a phase shift. However, for this case, there is no statetransition occurrence for either DVDS, which is dramat-ically different from the collision dynamics between theDVDS and SVDS described in Fig. 4(a1). Moreover, theeffective energy mixture no longer emerges in this case.The inelastic collision dynamics for the other two brightsoliton components (see Figs. 4(b2) and (b3)) are simi-lar to the those for the two nondegenerate bright solitonsthat collide in [23]. IV. TRIPLE-VALLEY DARK SOLITONS INFOUR-COMPONENT REPULSIVECONDENSATES
We now extend our discussion to four-componentBECs with repulsive interactions, which are described by the following repulsive four-component Manakov model:iq j , t + 12 q j , xx − ( | q | + | q | + | q | + | q | )q j = 0 , (7)where q j ( x, t )( j = 1 , , ,
4) denote the four componentfields in BECs. By applying the three-fold DT with λ j = 1 / ξ j +1 /ξ j ) and ξ j = − v + iw j ( j = 1 , , q and triple-humpbright solitons in the other three components. The exactsoliton solutions can be expressed in the following form(we do not provide the explicit solution process here forbrevity): q = 1 ξ ξ ξ N M e − i t , (8a) q = − i 2 w ξ α (cid:113) − v − w N M e i [ v x − (2+ v − w ) t ] , (8b) q = − i 2 w ξ β (cid:113) − v − w N M e i [ v x − (2+ v − w ) t ] , (8c) q = − i 2 w ξ γ (cid:113) − v − w N M e i [ v x − (2+ v − w ) t ] , (8d) with κ =2 w ( x − v t ) , κ = 2 w ( x − v t ) , κ = 2 w ( x − v t ) ,N = ξ ξ ∗ ξ ∗ α η e κ + ξ ∗ ξ ξ ∗ β η e κ + ξ ∗ ξ ∗ ξ γ η e κ + η (cid:104) ξ ξ ξ α β γ e κ + κ + κ + ξ ∗ ξ ξ β γ e κ + κ + ξ ξ ξ ∗ α β e κ + κ + ξ ξ ∗ ξ α γ e κ + κ (cid:105) + ξ ∗ ξ ∗ ξ ∗ η ,N = (cid:104) ρ + ρ β e κ + ρ γ e κ + η β γ e κ + κ (cid:105) e κ / ,N = (cid:104) ρ + ρ α e κ − ρ γ e κ + η α γ e κ + κ (cid:105) e κ / ,N = (cid:104) ρ − ρ α e κ − ρ β e κ + η α β e κ + κ (cid:105) e κ / ,M = (cid:104) α β e κ + κ + α γ e κ + κ + β γ e κ + κ (cid:105) η + η + η α e κ + η β e κ + η γ e κ + η α β γ e κ + κ + κ . In the expressions listed above, η = ( w − w ) ( w − w ) ( w − w ) , η = ( w + w ) ( w − w ) ( w + w ) , η =( w + w ) ( w − w ) ( w + w ) , η = ( w − w ) ( w + w ) ( w + w ) , η = ( w + w ) ( w + w ) ( w + w ) , ρ = ( w − w )( w − w )( w − w ) , ρ = ( w + w ) ( w − w )( w + w ) , ρ = ( w − w )( w + w ) ( w + w ) , ρ = ( w − w )( w − w ) ( w − w ), ρ = ( w + w ) ( w + w ) ( w − w ), and ρ = ( w − w ) ( w − w )( w − w ).The parameters v , w , w , w , α , β , and γ are real con-stants that co-determine the profile and position of theTVDS. The parameters w , w , and w are three soli-ton width-dependent parameters ( w (cid:54) = w (cid:54) = w ). Theparameter v is the soliton velocity, which should satisfythe constraint v + w j < j = 1 , , α , β and γ are three free parameters associated with thecenter positions and relative values of the three valleys.The profile of the TVDS is generally asymmetric but be-comes symmetric for certain values of the parameters α , β , and γ , which is similar to the above three-component FIG. 5: The density profile (a) and phase distribution (b)of an asymmetric TVSD. Three phase jumps across the threedensity valleys can be observed. The parameters are v = 0 . w = 0 . w = 0 . w = 0 . α = , β = , γ = 1. case. With the arbitrary setting of these parameters, thesolution given in Eqs. (8) shows a TVDS in component q , a triple-hump bright soliton with two nodes in com-ponent q , a bright soliton with one node in component q and a bright soliton without nodes in component q .The bright solitons in the last three components are simi-lar to the solitons reported in three-component attractiveBECs [23]. In this section, we mainly discuss the TVDSin the first component.The density profile and phase distribution of an asym-metric TVDS are displayed in Fig. 5. Fig. 5(a) depictsthe density distribution of an asymmetric TVDS. Thedensity profile of the TVDS is somewhat similar to thoseof stationary multisoliton complexes on a background[55, 56]: both are formed as the special nonlinear su-perpositions of pairs of bright and dark solitons. How-ever, the derivation method for the TVDS is distinctive.Consequently, our developed DT method can be used toderive more general MVDS solutions with moving veloc-ities and investigate their collisions analytically throughfurther iterations. Fig. 5(b) demonstrates that the TVDSfeatures three phase jumps across three valleys. The soli-ton width-dependent parameter w j also affects the phasejump region of the TVDS. As an example, we demon-strate the variation in the phase jump value with thechanges in the soliton width-dependent parameters ( w and w ) and the velocity in Fig. 6. The parameter w isfixed to be 0 .
2. For the static TVDS solution, the phasejump is 3 π . The phase jump value achieves the mini-mum value when the velocity tends toward the limit ofthe maximum velocity. With w = 0 .
2, the phase jumprange of the TVDS is [0 . π, π ], but the phase jumprange of the TVDS can vary within [0 , π ] by further de-creasing the value of w . The phase jumps of triple-humpbright solitons in component q , component q , and com-ponent q are always 2 π , π and 0, respectively. Moreover,the parameters w j can also vary the soliton velocity re-gion for a TVDS. The velocity range is [0 , (cid:112) − w m ),where w m is the largest of the three width-dependentparameters. These characteristics are similar to those ofthe DVDS case, which could hold for dark solitons withmore valleys.Moreover, the collision dynamics of TVDSs include FIG. 6: The phase jump (in π unit) of a TVDS with vary-ing velocity and soliton width-dependent parameters w and w . The width-dependent parameters significantly affect thephase jump and velocity region. w is fixed to be 0 .
2. Theother parameters are α = 1 / β = 1 / γ = 1, and t = 0 . three main cases: a collision between a TVDS and anSVDS, a collision between a TVDS and a DVDS, anda collision between two TVDSs. A breather transitionalso emerges for the TVDS in the first and second cases,whereas in the third case, while the soliton profiles simi-larly vary after the collision, there is no breather behav-ior. The oscillation behavior is much more abundant inthese cases than in the cases for the above DVDS. Wewill systemically discuss their properties in the futurewith the further development of analysis techniques. V. CONCLUSION AND DISCUSSION
We obtain exact DVDS and TVDS solutions by furtherdeveloping the DT method. Their velocity and phasejump characteristics are characterized in detail, and inparticular, we demonstrate that changes in the soli-ton width-dependent parameters have considerable influ-ences on the velocity and phase jump ranges. This find-ing indicates that the effects of the soliton width shouldbe considered when studying the motion of MVDSs inexternal potentials. The collision dynamics of the DVDSand TVDS are also discussed. The collisions involvingDVDSs are discussed in detail, and we report a strik-ing state transition process in which a DVDS transitionsinto a breather after colliding with an SVDS due to themixture of the effective energies of the soliton states inthe three components. Furthermore, our analyses sug-gest that breather transitions exist widely in the collisionprocesses involving MVDSs.Our discussion can be extended to N -component cou-pled systems. The ( N − N − N − , ( N − π ]. This ar-gument is supported by our calculation up to a five-component case. Nevertheless, further study is neededto learn how to express the analytical solution for arbi-trary N -component coupled systems. The collision prop-erties between MVDSs and degenerate vector solitons areexpected to be much more abundant than those of previ-ously reported vector soliton collisions [5, 16, 35, 42, 47].In particular, the state transition between the breatherand MVDS could have some important hints for solitonstate manipulation fields.Recently, three-component vector solitons and theircollisions were experimentally observed in BECs withrepulsive interactions [16]. This finding indicates thatsoliton dynamics could be quantitatively described bythe above integrable repulsive three-component Manakovmodel. Therefore, we discuss the possibilities to observeDVDSs in three-component repulsive BECs in combina-tion with well-developed quantum engineering techniques[8, 15, 16, 51, 52]. Let us consider quasi-one-dimensionalelongated BECs of Rb. The different components arethe magnetic sublevels m F = 0 , ± F = 1 hyper-fine manifold. In the beginning, all atoms are preparedin the m F = 0 state. One can use spatial local controlbeams to transfer atoms from the initial state m F = 0to m F = ±
1. With knowledge of the density and phasegiven by the solution of Eqs. (2), one could transfer theatom and imprint phase on them simultaneously to ap-proach the initial state for the DVDS and bright solitonstates in the corresponding components. Our numericalsimulations suggest that these soliton states are robustagainst small deviations and noise, suggesting it is pos-sible to observe these solitons experimentally.
ACKNOWLEDGMENTS
This work is supported by the National Nat-ural Science Foundation of China (Contract No.12022513,11775176,12047502), the Major Basic ResearchProgram of Natural Science of Shaanxi Province (GrantNo. 2018KJXX-094), Scientific research program of Edu-cation Department of Shaanxi Provience (18JK0098) andScientific Research Foundation of SUST (2017BJ-30).
Appendix A: Derivation of the double-valley darksoliton solution of equation (1)
The N -component repulsive BEC can be described bythe following N -coupled Manakov model [16, 43]:i q t + 12 q xx − q † qq = 0 , (A1)where q = ( q , q , ...q N ) T , which admits the following Lax pair: ψ x = U ( λ ; Q ) ψ, (A2a) ψ t = V ( λ ; Q ) ψ, (A2b) with U = (i λσ + i Q ) , (A3a) V = (cid:20) i λ σ + i λQ −
12 (i σ Q − σ Q x ) (cid:21) , (A3b)where Q = (cid:34) − q † q 0 N × N (cid:35) , σ = (cid:34) × N N × − I N × N (cid:35) , As mentioned in the main text, multivalley dark solitonscan be obtained in an N ( N >
2) component system. Asan example, we will take the DVDS solution derivationprocess in the three-component case to introduce the cal-culation method for MVDSs.To obtain the DVDS solutions, we use the followingseed solutions: q = e − i t , q = 0 , q = 0 . (A4)First, we need to solve the Lax pair equation (A2) withthe above seed solutions. We use the following gaugetransformation: S = diag(1 , e it , , , Which converts the variable coefficient differential equa-tion into a constant-coefficient equation. Then, we canobtain ψ ,x = U ψ , (A5a) ψ ,t = (cid:18) i2 U + λU + i2 λ (cid:19) ψ , (A5b)where U = i λ − − λ − λ
00 0 0 − λ , In the following, we consider the property of U . Wecan obtain the characteristic equation of matrix U at λ = λ j = a j + i b j :det(i τ j − U ) = ( λ j + τ j ) ( τ j − λ j + 1) = 0 . (A6)The eigenvalues of (A6) are τ j = − (cid:113) λ j − , τ j = τ j = − λ j , τ j = (cid:113) λ j − . (A7)To obtain the vector solution of (A2), we further diago-nalize the matrix U . Then, we obtain φ x = (cid:101) U φ, φ = H − Sψ, (A8a) φ t = (cid:18) i2 (cid:101) U + λ j (cid:101) U + i2 λ j (cid:19) φ, (A8b)where the transformation matrix H can be expressed asthe following form: H = λ j + 1 + τ j λ j + 1 + τ j λ j + 1 − τ j λ j + 1 − τ j . Thus, we have the vector solution for (A8): φ j = φ j φ j φ j φ j = c j exp i[ τ j x + ( λ j +2 λτ j − τ j ) t ] c j exp i[ τ j x + ( λ j +2 λτ j − τ j ) t ] c j exp i[ τ j x + ( λ j +2 λτ j − τ j ) t ] c j exp i[ τ j x + ( λ j +2 λτ j − τ j ) t ] . (A9) The coefficients c j , c j , c j , and c j are arbitrary com-plex parameters. Then, the special solution of Lax pair(A2) at λ j can be obtained based on ψ j = S − Hφ j : ψ j = ψ j ψ j ψ j ψ j = (1+ λ + τ j ) φ j +(1+ λ + τ j ) φ j [(1+ λ − τ j ) φ j +(1+ λ − τ j ) φ j ] e − it φ j φ j . (A10) We can see from (A7) that the algebraic equation (A6)has a pair of opposite complex roots, i.e., τ j = − τ j . Toobtain the DVDS, we pick only one of them in specialsolution (A10). Namely, we need to let φ j = 0 (i.e.,the coefficient c j = 0) or φ j = 0 (i.e., the coefficient c j = 0). In this paper, we choose φ j = 0. Then, theabove special solutions ψ j for spectral problem (A2) at λ j are re-expressed as follows: ψ j = ψ j ψ j ψ j ψ j = (1 + λ + τ j ) φ j [(1 + λ − τ j ) φ j ] e − it φ j φ j . (A11) Next, we need to perform a two-fold DT using the specialsolutions (A11) to derive the DVDS solutions. First, weperform the first-step iteration by applying the DT in [43]with λ = a + i b and constrain the eigenfunction ψ = φ = 0, i.e., the coefficient c = 0 (or the eigenfunction ψ = φ = 0, i.e., the coefficient c = 0): ψ [1] = T [1] ψ, T [1] = I + λ ∗ − λ λ − λ ∗ ψ ψ † Λ ψ † Λ ψ ,Q [1] = Q + ( λ − λ ∗ ) (cid:34) σ , ψ ψ † Λ ψ † Λ ψ (cid:35) , (A12)where Λ = diag(1 , − , − , −
1) and a dagger denotes thematrix transpose and complex conjugate. For the second-step iteration, we employ ψ , which is mapped to ψ [1] = T [1] | λ = λ ψ with λ = a + i b , and we constrain theeigenfunction ψ = φ = 0, i.e., the coefficient c = 0 (or the eigenfunction ψ = φ = 0, i.e., the coefficient c = 0): ψ [2] = T [2] ψ [1] , T [2] = I + λ ∗ − λ λ − λ ∗ ψ [1] ψ [1] † Λ ψ [1] † Λ ψ [1] ,Q [2] = Q [1] + ( λ − λ ∗ ) (cid:20) σ , ψ [1] ψ [1] † Λ ψ [1] † Λ ψ [1] (cid:21) . (A13)Furthermore, we require that the solitons’ velocities inthese two iteration processes be equal. Only if the aboveiterative processes and velocity requirements are met canthe first component of solution Q [2] be the DVDS solu-tion. Then, we need to analyze the relationship betweenthe soliton velocity and spectral parameter λ j .We find that the solitons’ velocities can be obtained bycalculating the following: (cid:101) v j = [Im( τ j ) − b j ] Re( τ j ) − [Im( τ j ) + 3 b j ] a j Im( τ j ) + b j , (A14)and the inverse soliton width is (cid:101) w j = − [Im( τ j ) + b j ] . (A15)Moreover, the spectral parameters should satisfy the con-straint condition[ b j − Im( τ j )] + [ a j − Re( τ j )] − > . (A16)With the spectral parameter λ j = a j + i b j , the velocityor width of the soliton cannot be directly determinedby the real or imaginary part of the spectral parameter.This greatly increases the difficulty of taking the equalvelocity in the two-fold DT (A12)-(A13) with differentspectral parameters. Accordingly, we express the realand imaginary parts of the spectrum parameter by thevelocity (cid:101) v j and width (cid:101) w j . By combining (A14)-(A16)and performing further calculations, we obtain a j = − (cid:101) v j (1 + (cid:101) v j + (cid:101) w j )2( (cid:101) v j + (cid:101) w j ) , b j = (cid:101) w j (1 − (cid:101) v j − (cid:101) w j )2( (cid:101) v j + (cid:101) w j ) , (A17)and the constrain condition is represented as (cid:101) v j + (cid:101) w j < . (A18)Under this representation, one can guarantee that thesolitons’ velocities in the first-step iteration (A12) andthe second-step iteration (A13) remain exactly equal, i.e., (cid:101) v = (cid:101) v . In other words, by performing the iterations in(A12) and (A13) and setting the spectral parameters un-der the conditions (A17) and (A18), the first componentof Q [2] is the DVDS solution.As we have mentioned above, the velocity expressionplays a crucial role in the derivation process of multi-valley dark solitons. However, for the spectral parameterwith the form λ j = a j +i b j , we must take the square rootof a complex number to obtain the physical parameters,namely, the velocity and width. To facilitate this analy-sis, we employ a more direct approach. Interestingly, we0 FIG. A1: The relation between the velocity and spectralparameter expressed by different representations. (a): Thespectral parameters are written as λ = ( ξ + ξ ) and ξ = − v +i w . For this case, there is a linear relationship be-tween the velocity and the real part of the spectral parameter;namely, v is just simply the velocity of the soliton. (b): Thespectral parameter is written as λ = a + i b . This clearlyshows a nonlinear relation between the velocity and spectralparameter. Therefore, we introduce the parameters ξ j to sim-plify our soliton solution, which is also much more convenientfor discussing the soliton’s physical properties. The white linecorresponds to w (cid:54) = 0 or b (cid:54) = 0. note that if we describe the spectral parameter with theform λ j = ( ξ j + ξ j ) with an arbitrary complex parame-ter ξ j = − v j + i w j , the solution can be simplified greatly.Moreover, the physical meanings of the parameters v j and w j are much clearer than those of the parameters a j and b j , as shown in Fig. A1. Then the eigenvalues ofEq.(A6) can be re-expressed as τ j = − (cid:20) ξ j − ξ j (cid:21) , τ j = − (cid:20) ξ j + 1 ξ j (cid:21) ,τ j = − (cid:20) ξ j + 1 ξ j (cid:21) , τ j = 12 (cid:20) ξ j − ξ j (cid:21) . To simplify the solution forms, we can set the coefficients φ j , φ j , and φ j to c j = 1 , (A19) c j = ξ j + 1 ξ j (cid:113) − | ξ j | α j , (A20) c j = ξ j + 1 ξ j (cid:113) − | ξ j | β j , (A21)where α j and β j are arbitrary constants. Then, the spe-cific solution (A11) of spectrum problem (A2) at λ j canbe re-expressed as the following form: ψ j = ψ j ψ j ψ j ψ j = (1 + ξ j ) φ j (1 + ξ j ) φ j e − it φ j φ j . (A22)Then, we can obtain the DVDS solution in the firstcomponent of Q [2] and the bright solitons in the othertwo components by performing the following two-fold DT processes. For the first-step DT (A12), the spectral pa-rameters are λ = ( ξ + 1 /ξ ) and ξ = − v + i w ,and the eigenfunction is constrained by ψ = φ = 0,i.e., α = 0 (or the eigenfunction ψ = φ = 0, i.e., β = 0). For the second-step DT (A13), the spectralparameters are λ = ( ξ + 1 /ξ ) and ξ = − v + i w ,and the eigenfunction is constrained by ψ = φ = 0,i.e., β = 0 (or the eigenfunction ψ = φ = 0, i.e., α = 0). The simplified solution for Q [2] (A13) is pre-sented in Eq. (2), where β = 0 , α = 0, α and β arenonzero constants. This indicates that the parameter v is the velocity, and w and w are the width-dependentparameters for DVDSs. Therefore, the physical meaningsof the parameters v j and w j are much clearer than thoseof the parameters a j and b j (A17), as shown in Fig. A1.To study the collision dynamics of DVDSs, further it-erations are needed. For example, we can investigatethe collision between a DVDS and an SVDS by per-forming a third-step DT with λ = ( ξ + 1 /ξ ) and ξ = − v + i w . We employ ψ , which is mapped to ψ [2] = ( T [2] ψ [1]) | λ = λ with ψ [1] = ( T [1] ψ ) | λ = λ : ψ [3] = T [3] ψ [2] , T [3] = I + λ ∗ − λ λ − λ ∗ ψ [2] ψ [2] † Λ ψ [2] † Λ ψ [2] ,Q [3] = Q [2] + ( λ − λ ∗ ) (cid:20) σ , ψ [2] ψ [2] † Λ ψ [2] † Λ ψ [2] (cid:21) . (A23)For this case, we constrain the eigenfunctions ψ (cid:54) = 0and ψ (cid:54) = 0, i.e., α (cid:54) = 0 and β (cid:54) = 0, and the eigen-functions ψ and ψ are the same as in the first-stepand second-step DT. A typical example of this case isshown in Figs. 4(a1)-(a3), demonstrating a striking col-lision process for which a DVDS is transformed into abreather after colliding with an SVDS.Naturally, by performing a fourth-step DT, one caninvestigate the interaction between two DVDSs. Beforeperforming this iteration, the eigenfunctions of ψ in thethird-step DT (A23) should be made to satisfy the con-straint conditions where the eigenfunction ψ = φ = 0,i.e., β = 0 (or the eigenfunction ψ = φ = 0,i.e., α = 0). Then, we employ ψ , which is mappedto ψ [3] = ( T [3] ψ [2]) | λ = λ = ( T [3] T [2] T [1] ψ ) | λ = λ .Then, the two DVDS solutions can be obtained as fol-lows with the spectral parameters λ = ( ξ + 1 /ξ ) and ξ = − v + i w , and we can constrain the eigenfunc-tion ψ = φ = 0, i.e., α = 0 (or the eigenfunction ψ = φ = 0, i.e., β = 0): Q [4] = Q [3] + ( λ − λ ∗ ) (cid:20) σ , ψ [3] ψ [3] † Λ ψ [3] † Λ ψ [3] (cid:21) . (A24)In the four-component case, the TVDS can also be ob-tained by performing the above developed DT with three-fold iterations . We do not show the detailed derivationfor this process herein. The simplified solution for TVDSis presented in (8).In general, the n -fold Darboux matrix can be con-1structed in the following form: T n = I + Y n M − n ( λ I − D n ) − Y † n Λ , (A25)with Y n = [ | ψ (cid:105) , | ψ (cid:105) , · · · , | ψ n (cid:105) ] = (cid:34) Ψ Ψ (cid:35) , D n = diag ( λ ∗ , λ ∗ , · · · , λ ∗ n ) , M n = (cid:18) (cid:104) ψ i | ψ j (cid:105) λ ∗ i − λ j (cid:19) ≤ i,j ≤ n . Ψ is a 1 × n matrix, and Ψ is an N × n matrix. TheB¨acklund transformation between the old potential func-tions and the new functions is expressed as follows: q [ n ] = q + 2Ψ M − Ψ † . (A26) Appendix B
The explicit expressions for f j , g j , h j , and m j of theasymptotic expressions in Eqs. (4) are f = ξ ∗ ξ ∗ ξ ∗ ( w − w ) ξ ξ ξ ( w + w ) σ σ , f = ξ ∗ ξ σ σ α β ,f = ξ ∗ ξ ∗ ξ ξ β σ σ , f = ξ ∗ ξ ∗ ξ ξ α σ σ ,g = 2 α w p v − i w [2( v − v ) w + i σ ] ,g = w − w w + w σ , g = σ β ,h = 2 β w p v − i w [2( v − v ) w − i σ ] ,h = w − w w + w σ , h = σ α ,m = ( w − w ) ( w + w ) σ σ , m = σ σ α β ,m = σ σ α , m = σ σ β ,p = (cid:112) − | ξ | , p = (cid:112) − | ξ | ,ξ = − v + i w , ξ = − v + i w , ξ = − v + i w ,σ = ( v − v ) +( w − w ) , σ = ( v − v ) +( w − w ) ,σ = ( v − v ) +( w + w ) , σ = ( v − v ) +( w + w ) ,σ = ( v − v ) +( w − w ) , σ = ( v − v ) +( w − w ) ,σ = ( v − v ) +( w + w ) , σ = ( v − v ) +( w + w ) . The explicit expressions for δ j , ζ j , ς j , and (cid:37) j of theasymptotic expressions in Eqs. (5) are R j = | ξ j | , S j = 1 + ξ j , Λ ij = ξ i ξ ∗ j − , Ξ ij = ξ i − ξ ∗ j , Γ j = ( ξ j + ξ ∗ j )(1 + | ξ j | ) , G j = Ξ ij Λ ij , ( i, j = 1 , , δ = p p α β R R | Λ | G G G | ξ | | ξ | ξ ξ ,δ = Ξ | ξ | G G , δ = Ξ | ξ | G G ,δ = R R | Λ | Λ Λ | ξ | | ξ | ξ ξ , δ = | ξ − ξ | ξ ξ ,δ = − α β | Ξ | ξ ∗ ξ ∗ | G | | G | (cid:2) G G G − (2 | ξ | | S | +2 | ξ | | S | − Γ Γ )(2 | ξ | | S | +2 | ξ | | S | − Γ Γ ) (cid:3) ,δ = R R Λ Λ | G | | ξ | | ξ | ξ ξ ,δ = α | ξ | | S | + 2 | ξ | | S | − Γ Γ ξ ∗ ξ | G | ,δ = β | ξ | | S | + 2 | ξ | | S | − Γ Γ ξ ξ ∗ | G | ,ζ = p α R R G Λ ξ ξ ∗ ξ | ξ | G ,ζ = ξ ∗ − ξ ∗ | ξ | ξ G Λ (cid:2) ξ Γ − S | ξ | (cid:3) ,ζ = 8 β | G | ( λ − λ ∗ )( λ ∗ − λ ) (cid:2)(cid:0) λ | ξ | − Γ (cid:1)(cid:0) | S | | ξ | + 2 | S | | ξ | − Γ Γ (cid:1) + G G (cid:3) ,ζ = p β R R Ξ | Λ | Λ G G | ξ | | ξ | ξ G , ζ = ξ ∗ − ξ ∗ ,ζ = α Ξ [ | ξ | ξ ∗ S Λ + | ξ | S Γ − ξ | ξ | R − | ξ | | ξ | S ( ξ + ξ ∗ )] / [ ξ | G | ] ,ς = p α R R G G Λ ξ ξ ∗ ξ | ξ | G , ς = ξ ∗ − ξ ∗ | ξ | ξ ∗ ξ G Λ ,ς = 16 β | G | ( λ ∗ − λ )( λ ∗ − λ ) (cid:2) S | ξ | ξ ∗ /ξ − ξ ∗ S Γ + | ξ | R (cid:3) ,ς = − p β ( ξ ∗ − ξ )8 | ξ | | ξ | ξ ξ G R R | Λ | G Λ Ξ ,ς = ( ξ ∗ − ξ ∗ )(2 S | ξ | − ξ Γ ) ,ς = α Ξ (cid:2) − ξ G G − (2 S | ξ | − ξ Γ )(2 | ξ | R + 2 | ξ | R − Γ Γ ) (cid:3) / (2 | G | ) ,(cid:37) = p p α β R R G G | Λ | | ξ | | ξ | | G G | ,(cid:37) = G G G Ξ ,(cid:37) = R R Λ Λ | Λ | | ξ | | ξ | , (cid:37) = | ξ − ξ | ,(cid:37) = − α β | Ξ | | G | | G | (cid:2) − (2 | ξ | R + 2 | ξ | R − Γ Γ )(2 | ξ | R + 2 | ξ | R − Γ Γ ) + G G G (cid:3) ,(cid:37) = β | Ξ | (cid:2) | ξ | R + 2 | ξ | R − Γ Γ (cid:3) / (2 | G | ) ,(cid:37) = α | Ξ | (cid:2) | ξ | R + 2 | ξ | R − Γ Γ (cid:3) / (2 | G | ) . [1] P. G. Drazin and R. S. Johnson, Solitons: an introduction (Cambridge University,1989). [2] P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-Gonz´alez,
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