Classification of Dark Solitons via Topological Vector Potentials
CClassification of Dark Solitons via Topological Vector Potentials
L.-C. Zhao , , Y.-H. Qin , and J. Liu , ∗ School of Physics, Northwest University, Xi’an 710127, China Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710127, China Graduate School, China Academy of Engineering Physics, Beijing 100193, China and CAPT, HEDPS, and IFSA Collaborative Innovation Center of theMinistry of Education, Peking University, Beijing 100871, China (Dated: June 19, 2020)Dark soliton is one of most interesting nonlinear excitations in physical systems, manifesting aspatially localized density “dip” on a uniform background accompanied with a phase jump acrossthe dip. However, the topological properties of the dark solitons are far from fully understood. Ourinvestigation for the first time uncover a vector potential underlying the nonlinear excitation whoseline integral gives the striking phase jump. More importantly, we find that the vector potential fieldhas a topological configuration in analogous to the Wess-Zumino term in a Lagrangian representa-tion. It can induce some point-like magnetic fields scattered periodically on a complex plane, eachof them has a quantized magnetic flux of elementary π . We then calculate the Euler characteristic ofthe topological manifold of the vector potential field and classify all known dark solitions accordingto the index. Introduction —Dark soliton is one of most commonlynonlinear excitation emerged in both quantum and clas-sical systems, including optics [1, 2], ultracold Bose-Einstein condensates [3, 4], polariton fluid [5–7], waterwaves [8] and the plasmas [9, 10]. The generation ofdark solitons are controllable and manipulatable in somesituations [11], allowing for many important applications[12], such as optical communication, observation of nega-tive mass effect [13], atomic matter-wave interferometers[14], quantum switches and splitters [4], etc..Dark soliton serving as an exact solution for nonlineardifferential equations has many implications in physics.If we stand on a moving soliton to investigate its behav-ior, the dark soliton (or anti-dark soliton) will representa kind of transmission waves that can pass through anonlinear potential well (or barrier for anti-dark soliton)without any reflection. More interestingly, in contrastto its bright counterpart, there is a phase jump (or shift)during the process depending on the soliton’s velocity. Inthe limit of zero velocity, i.e., for a stationary dark soli-ton, the phase jump usually takes π value in many cases[1]. It is thus reckoned that the π phase jump is the signalof topological excitation and the stationary dark soltionis in several respects the one-dimensional counterpart tovortices [15, 16]. However, recent studies indicates thatthe phase jump can be greater than π for a saturablenonlinear media [17, 18], for an anti-dark soliton it be-comes zero and tend to − π/ π/ ∗ Electronic address: [email protected] space to manifest some important physical effects suchas Aharonov-Bohm effect [21] or to demonstrate sometopological structures of vortex [15, 16, 22, 23], skyrmions[24, 25], and knots [26, 27]. It can also emerge in parame-ter space such as Berry phase theory or momentum spacesuch as topological energy band theory, to reveal bizarrevirtual particles and characterize new forms of mat-ter including topological insulators [28], Weyl fermionsemimetal [29], and even to promote the quantum com-puting [30, 31]. In this paper, we investigate the topolog-ical properties of dark solitons in a complex space to ad-dress the striking phase jumps. We obtain an area theo-rem that can associate the phase jumps with an area on aplane of the amplitude vs local phase of a soliton solution.With exploiting analytic extension of complex function,we uncover a topological vector potential in analogous tothe Wess-Zumino term [32], whose line integral gives thestriking phase jump. The vector potential correspondsto some point-like magnetic fields with magnetic flux ofelementary π . Our result indicates that, even though thedark soliton moving in real axis can not see any magneticfields, the point-like magnetic fields on the complex planecan actually affect the dark soliton’s motion with assign-ing a phase variation. We then have made a topologicalclassification for all known dark solitions according totheir Euler characteristic of the vector potential field. Area theorem and topological vector potential —A darksoliton is a spatially localized density “dip” on top ofa finite uniform background, accompanied with a phasejump through the dip [1–4]. A finite phase step emergesat soliton center when the dark soliton is stationary. Fora moving dark soliton, there is a continuous phase shiftacross the soliton. The solution for a dark soliton canbe depicted explicitly by a complex function ψ ( x, v, t ),where x is the spatial coordinate, v is the soliton’s mov-ing velocity, and t is the evolution time. If we choose thesoliton’s center as the reference to investigate the darksoliton’s evolution, the soliton will be always stationary a r X i v : . [ n li n . PS ] J un FIG. 1: (coloronline)(a) The area in the density-phase planefor a dark soliton solution. (b) The phase jump vs movingvelocity for a scalar dark soliton. The red solid line is givenby the area theorem, blue dashed line is WKB approximation,and the green dots are exact dark soliton solution. See textfor details. but its background admits a uniform density flow witha velocity v . In this frame, the dark soliton can be ex-pressed as an eigenstate solution of form ¯ ψ ( x, v ), withthe boundary conditions: lim x →−∞ ¯ ψ ( x, v ) = √ Ie ivx , andlim x → + ∞ ¯ ψ ( x, v ) = √ Ie ivx + i ∆ φ ds , where I is the backgrounddensity, ∆ φ ds is the phase jump across the dark soliton.The local phase of the wave function ¯ ψ ( x, v ) can be writ-ten as φ ( x ) = φ ds ( x ) + vx , where vx term is from theextended background flow and the φ ds ( x ) denotes thephase of the localized dark soliton solution. This analy-sis implies that a plane wave can propagate from −∞ to+ ∞ without any reflection through an effective quantumwell induced by a dark soliton, the total phase jump canbe expressed as ∆ φ ds = (cid:82) + ∞−∞ dφ ds ( x ) dx dx. The stationary wavefunction ¯ ψ ( x, v ) can be viewedas an one-dimensional steady flow, satisfying the flowconservation of | ¯ ψ ( x, v ) | dφ ( x ) dx = Iv . We calcu-late (cid:82) + ∞−∞ ¯ ψ ∗ ( − i∂ x ) ¯ ψdx = (cid:82) + ∞−∞ | ¯ ψ ( x, v ) | dφ ( x ) dx dx = (cid:82) + ∞−∞ Ivdx , and then we have ∆ φ ds = (cid:82) + ∞−∞ ( dφ ( x ) dx − v ) dx = (cid:82) + ∞−∞ (1 − | ¯ ψ ( x, v ) | /I ) dφ . The above formulaimplied that the the total phase shift of the dark solitonexactly corresponds to an area on the amplitude-phaseplane. We term it as area theorem and sketch it in Fig. 1(a). Its validity has been verified by our numerical simu-lations as shown in Fig. 1 (b). The area theorem is rathergeneral, not only applicable to the simple scalar darksoltion (or anti-dark soliton) as discussed, but also othercomplicated vector solitons, such as dark-bright soliton[33–35], spin soliton [36], magnetic soliton [37] and dark-bright-bright soliton [38]. The area on the plane of am-plitude vs. phase in fact corresponds to a classical canon-ical action, which has a close relation to the Aharonov-Anandan phases of nonadiabatic evolutions [39].The area theorem can help us to uncover the topo-logical properties of the dark solitons. We introduce afunction F [ z ], which is the analytic extension of F [ x ] =(1 − | ¯ ψ ( x, v ) | /I ) dφ ( x ) /dx , with replacing x by z = x + iy . The phase jump of dark soliton can be de-scribed by an integral of the vector potential A alongreal x coordinate, which is introduced by considering acircle integral in the complex plane, i.e., (cid:72) C F [ z ] dz = (cid:72) C ( u [ x, y ] + iw [ x, y ])( dx + idy ) = (cid:72) ( u [ x, y ] , − w [ x, y ]) · d r + i (cid:72) ( w [ x, y ] , u [ x, y ]) · d r = (cid:72) A · d r , where A = u [ x, y ] e x − w [ x, y ] e y , and d r = dx e x + dy e y . From thearea theorem, we see that F [ z ] might have some singu-larities of z N = x N + iy N ( N is an integer) on complexplane corresponding to the divergence of flow velocityor the zero point of the density amplitude. Accordingto the Cauchy integral formula, a meromorphic functioncan be expressed in terms of these singularities, that is, F [ z ] = F ( ∞ ) − πi (cid:82) Γ f ( w ) w − z dw = (cid:80) N Res [ F [ z N ]] z − z N , where Γdenotes the closed curves encircling the singularities sep-arately and Res [ F [ z N ]] is the residue [40]. Our physicalobservation indicates that these singularities emerge peri-odically and in pairs on the complex plane, which meansthat Res [ F [ z N ]] can be expressed as ± Ω / πi . Accord-ingly, the vector potential A is derived as A = (cid:88) N ± Ω[( x − x N ) e y − ( y − y N ) e x ]2 π [( x − x N ) + ( y − y N ) ] . (1)Here, A term takes the form of Wess-Zumino topologicalterm, i.e., a closed differential 1-form and its differen-tial 2-form is zero in the Lagrangian representation [32].This implies that corresponding magnetic filed will bezero everywhere in whole complex plane except for thosesingular points. That is, B = e z (cid:88) N ± Ω δ [ r − r N ] . (2)The ± represent the magnetic flux direction.It is interesting to compare with the Aharonov-Bohmeffect [21] that predicts a topological phase when an elec-tron moving on a close path around a solenoid. A darksoliton solution moving on real axis can not see thosemagnetic fields scattered on the complex plane, howeverit will acquire a phase jump due to the presence of thevector potential. In this sense, the phase jump for thedark soliton can be viewed as 1D counterpart to the fa-mous Aharonov-Bohm phase. However, the phase shiftusually does not simply equal to the magnetic flux, be-cause the integral path for dark soliton is not a closedpath around these singularities. Moreover, we find thatthe flux has a quantized magnetic flux of elementaryΩ = π , corresponding to a monopole with a charge 1 / Topological properties of scalar dark solitons —We firstdemonstrate our theory with a simple scalar soliton.The scalar nonlinear Schr¨odinger equation (NLS) of form i ∂ψ∂t = − ∂ ψ∂x + | ψ | ψ , has wide applications in non- FIG. 2: (a) The topological vector potential A for a scalardark soliton. (b) The corresponding magnetic field B and (c)the phase jump induced by the magnetic field. The solitonspeed is set to be v = 0 . c s . (d) The vector potential A for ananti-dark soliton [19]. (e) The corresponding magnetic filed B and (f) the phase jump induced by the magnetic field. Thesoliton speed is set to be v = 0 . c s . linear fiber [1], water wave [8], plasma [9], and Bose-Einstein condensate [4]. It has a dark soliton solutionwith a uniform flowing background, ¯ ψ ( x, v ) = ( − iv + √ − v tanh[ √ − v x ]) e ivx , with an eigenvalue of µ =1 + v /
2. According to area theorem, we can calcu-late the phase jump of the dark soliton from follow-ing integral expression, ∆ φ ds = (cid:82) + ∞−∞ (1 − | ¯ ψ | ) dφdx dx = (cid:82) + ∞−∞ v ( − v ) cosh ( √ − v x ) + v − dx . The results are shown inFig. 1 (b). It indicates that the area theorem can pre-cisely predicts the phase jump .The phase variation of dark soliton can be also under-stood from the WKB approximation. The amplitude ofdark soliton serves as an effective quantum well poten-tial. The classical action is (cid:82) + ∞−∞ ( (cid:112) k ( x ) − v ) dx , where k ( x ) = µ − | ¯ ψ | . In WKB approximation, the actionshould correspond to a quantum phase. The results areshown in Fig. 1 (b). Interestingly, the WKB phase agrees FIG. 3: (a) The phase distribution of the dark soliton phaseand its corresponding magnetic filed B on the complex planefor a dark-bright-bright soliton for which dark soliton admitssymmetric double-valley. The soliton speed is v = 0 . c s . Thesingularity locations are ( ± . , ± .
35) with a period of π onthe y-axis. The subgraphs show the phase variations inducedby the singular magnetic fields on the two separate lines, re-spectively. (b) The phase distribution of dark soliton and itscorresponding magnetic filed B for a dark-bright-bright soli-ton for which dark soliton admits asymmetric double-valley.The singularity locations are ( − . , ± . − . , ± . . , ± . . , ± . . , ± .
41) with a periodof 2 π on the y-axis. The soliton speed is v = 0 . c s . Thesubgraphs show the phase variations induced by the singularmagnetic fields on the five separate lines, respectively. well with the area theorem in the limit when the soliton’svelocity tends to sound speed, while they deviate dramat-ically in the low velocity limit. This can be understoodfrom the quantum-classical correspondence [41, 42].We then can calculate the vector potential by deter-mining the singularity locations in the complex plane,i.e., Z N : x N = 0 , y N = ± y + N T, ( N = 0 , ± , ... ) (3)with y = arccos ( √ − v ) √ − v and T = π √ − v is the period.The vector potential is shown in Fig. 2 (a). The cor-responding magnetic filed B = ∇ × A can be obtainedaccordingly, which is shown in Fig. 2 (b). The corre-sponding phase jump for dark soliton is shown in Fig. 2(c). We see that the positive and negative magnetic fluxemerge in pairs and locate periodically on the imaginaryaxis. Both the distance between the singularities and theperiod depends explicitly on the velocity of soliton. Forinstance, in the limit of v →
0, the magnetic fields at ± y tends to merge each other at origin. With increasing themoving velocity to sound speed (here c s = 1), they tendto merge with other two magnetic flux at y = ∓ y ± T ,respectively. In this limit, the dark soliton degeneratesinto a plane wave.We now calculate the merge process when v → π jump of the soliton from the vector potentialperspective. When v → | y | →
0. In this limit,lim v → A = lim | y |→ [ π ( x e y + | y | e x )2 π ( x + | y | ) + − π ( x e y − | y | e x )2 π ( x + | y | ) ]= πδ [ x ] e x . (4)Note that other singularities will merge each other in theprocess, and then ∆ φ ds = (cid:82) + ∞−∞ A x dx = π , indicatingthat the π jump arises from the quantized magnetic fluxcorresponding to a monopole.The phase jump for the anti-dark soliton [19] is also ofinterest. It is zero for the static solution and tend to be − π/ v → Topological properties of vector dark solitons — Wenow extend our discussions to a more complicated three-component coupled NLS system. With using the devel-oped Darbox transformation [43, 44] and after lengthydeductions, we obtain a kind of dark-bright-bright soli-ton (DBBS)[45], for which there is a dark soliton witha double-valley density profile in one component, andbright soliton density profiles in the other two compo-nents. Interestingly, the phase jump for this dark solitoncomponent is in the regime [0 , π ]. The double-valleystructure might be symmetric or asymmetric. Corre-sponding magnetic fields and the dark soliton phase φ ds are calculated and plotted in Fig. 3, respectively. Forthe symmetric one, the singular points scattered on twoseparate lines with x = x and x = x with a periodof π along y axis. The phase variation induced by thepoint-like magnetic fields periodically scattered on twoseparate lines is found to be ∆ = ∆ = 0 . π , sum ofthem gives total phase jump of the dark soliton. Whilefor the asymmetric one, the singularities locate on fiveseparate lines at x j ( j = 1 , , , , Types ∆ φ ds T g M χ DS [1] [0 , π ] π √ c s − v S × R /Z Z -4DS [19] [ π , π ]A-DS [19] [ − π , , π ]DS [46] [0 , π ] π √ c s − v DS [20] π πv DBS [35] [0 , π ] π DBBS [38] [0 , π ] π DS [17, 18] [0 , π ] – – – –MS [37] [ π , π ][ − π ,
0] – – – –DBBS1 [0 , π ] π S × R /Z ...Z -8DBBS2 [0 , π ] 2 π S × R /Z ...Z -20DBBBS [0 , π ] π S × R /Z ...Z -12TABLE I: ∆ φ ds the interval of phase jump; T is the periodin y axis where c s is the sound speed; g refers to the numberof the singularities in one period; M is topological manifoldspace; Z N represents singular points; χ = 0 − g is the Eulercharacteristic number. The solutions of DBBS1, DBBS2 andDBBBS are obtained from present work[45]. For MS [37], thephase jump regimes [ π , π ] and [ − π ,
0] correspond to dark andanti-dark soliton components, respectively. ‘–’ means corre-sponding quantities can not be calculated due to the absenceof exact explicit expressions. y axis turns to be 2 π . We also calculate the phase varia-tions ∆ j ( j = 1 , , , ,
5) corresponding to the magneticfields on the five separate lines and show the results inthe subgraphs of Fig. 3 (b). The phase profiles in Fig. 3demonstrate multi-steps structures due to the compli-cated topological potentials.
Topological classification of the dark solitons — We col-lect all known dark soliton solutions and calculate theircorresponding vector potential fields and the Euler char-acteristic of the topological manifolds. The results arepresented in Table I. We find that that all previous darksoliton solutions correspond to Euler index of −
4, whilethe complicated vector solitons obtained in the presentwork have the higher topological number up to − π independent of the soliton velocity [20].However, the singularities of vector potential locate at ± i πv + iN T with T = 4 π/v , whose distribution obvi-ously depends on the velocity. Nevertheless, after a sim-ple scaling transformation, an equivalent topological vec-tor potential can be derived in a velocity independentform, i.e., ¯ A = (cid:80) N ± [ x e y − ( y − ¯ y N ) e x ]2[ x +( y − ¯ y N ) ] with the singularitiesof ¯ y N = ± π + 2 πN , whose line integral will gives a defi-nite π/ Conclusion — We show that the dark solitons servingas a kind of simple nonlinear excitations can demon-strate very interesting topological properties. The phasejump can be viewed as the 1D counterpart to the famousAharonov-Bohm phase, where the solenoid that mightcarry an arbitrary flux is replaced by a monopole witha quantized magnetic flux of elementary π . Underlyingvector fields demonstrate the topology of Wess-Zuminoterm. Our investigations have resolved a long-standingpuzzle on the topological origin of dark solitons and pro- vides a possibility to investigate topological vector po-tential via the generation of dark solitons that are con-trollable in current BEC, optic and microcavity polaritoncondensates experiments. [1] Y.S. Kivshar and B. Luther-Davies, Dark optical soli-tons: physics and applications, Phys. Rep. , 81(1998).[2] Y.S. Kivshar and G.P. Agrawal, Optical Solitons: FromFibers to Photonic Crystals (Academic, NewYork, 2003).[3] P.G. Kevrekidis, D.J. Frantzeskakis, and R. Carretero-Gonz´alez,
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