Topological Vector Potentials Underlying One-dimensional Nonlinear Waves
TTopological Vector Potentials Underlying One-dimensional Nonlinear Waves
L.-C. Zhao , , , L.-Z. Meng , Y.-H. Qin , Z.-Y. Yang , , , and J. Liu , ∗ School of Physics, Northwest University, Xi’an 710127, China Peng Huanwu Center for Fundamental Theory, 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: February 23, 2021)We reveal intrinsic topological vector potentials underlying the nonlinear waves governed by one-dimensional nonlinear Schr¨odinger equations by investigating the Berry connection of the linearizedBogoliubov-de-Gennes (BdG) equations in an extended complex coordinate space. Surprisingly, wefind that the density zeros of these nonlinear waves exactly correspond to the degenerate points of theBdG energy spectra and can constitute monopole fields with a quantized magnetic flux of elementary π . Such a vector potential consisting of paired monopoles with opposite charges can completelycapture the essential characteristics of nonlinear wave evolution. As an application, we investigaterogue waves and explain their exotic property of “appearing from nowhere and disappearing withouta trace” by means of a monopole collision mechanism. The maximum amplification ratio andmultiple phase steps of a high-order rogue wave are found to be closely related to the number ofmonopoles. Important implications of the intrinsic topological vector potentials are discussed. Introduction —Real-space topology has been found tobe useful for describing some important physical effects,such as the Aharonov-Bohm effect [1], as well as sometopological structures of vortices [2, 3], skyrmions [4],and knots [5, 6]. Topologies can also emerge in param-eter space, as in the case of the Berry phase and vir-tual monopole theory [7], or in momentum space, as intopological energy band theory [8–10], revealing bizarrevirtual particles and characterizing new forms of matter,including topological insulators [11] and Weyl fermionsemimetals [12], and even facilitating quantum comput-ing [13, 14].Topological issues are mainly discussed in the contextof systems with two or more dimensions. The topologicalproperties associated with one-dimensional (1D) nonlin-ear waves such as solitons, rogue waves and breathersare seldom discussed, even though they exist widely inatomic gases [15], plasma [16], water waves [17], and fer-romagnetic materials [18] and play important roles inboth integrability theory [19] and optical communica-tions [20]. It is usually assumed that the phase vari-ation of wavefunctions is a sign of topological excita-tion since 1D nonlinear excitations such as rogue waves(RWs), dark solitons and breathers always contain abun-dant phase information [21–24]. However, in contrastto the case of 2D or 3D topological structures such asvortices, the topological charge defined according to thephase variations is no longer valid for characterizing thetopology of such 1D nonlinear waves [25, 26].In this paper, we present a theory to reveal intrinsicvector potentials underlying 1D nonlinear waves that cancompletely capture the topological properties hidden in ∗ Electronic address: [email protected] their phase variations. By investigating the Berry con-nection of the linearized Bogoliubov-de-Gennes (BdG)equations in an extended complex coordinate space, wesurprisingly find that the density zeros of these nonlin-ear waves exactly correspond to the degenerate pointsof the BdG energy spectra and can constitute monopolefields with a quantized magnetic flux of elementary π .Taking nonlinear RWs as an example, we obtain analyt-ical expressions for the topological vector potentials andfind that the exotic dynamic evolution of an RW solu-tion can be explained by a topological vector potentialreconnection mechanism, analogous to magnetic field re-connection in astrophysics [27]. Virtual magnetic monopole and topological vector po-tential —We choose one of the simplest models, i.e., thescalar nonlinear Schr¨odinger equation (NLSE) i ∂ψ∂t = − ∂ ψ∂x + g | ψ | ψ , to demonstrate our theory. The 1DNLSE describes the dynamics of nonlinear waves in opti-cal fibers [28], water wave tanks [29], and plasma systems[30]. It has a nonlinear wave solution ψ ( x, t ), which canbe a dark soliton (for g > g < ψ = ψ ( x, t ) + f ( t ) exp[ ikx ] + f ∗ ( t ) exp[ − ikx ], we obtain a r X i v : . [ n li n . PS ] F e b a Hamiltonian for BdG excitation [32]: H = (cid:32) k + 2 g | ψ ( x, t ) | gψ ( x, t ) − gψ ( x, t ) ∗ − k − g | ψ ( x, t ) | (cid:33) . (1)Because the Hamiltonian is non-Hermitian, theleft eigenvector of each branch is no longer equalto the transposed conjugate of its right eigenvector[33]. We obtain instantaneous eigenvalues of E ± = ± (cid:113) k + 3 g | ψ | + 2 k g | ψ | . The instantaneous eigen-vectors | V R ± (cid:105) and (cid:104) V L ± | are given by H | V R ± (cid:105) = E ± | V R ± (cid:105) and (cid:104) V L ± | H = E ± (cid:104) V L ± | , and they take the followingforms: | V R ± (cid:105) = (cid:18) k + 2 g | ψ | + E ± − gψ ∗ (cid:19) , (2) (cid:104) V L ± | = (cid:16) − gψ ∗ − k − g | ψ | + E ± (cid:17) . (3)Here, we can see that (cid:104) V L ± | V R ∓ (cid:105) = 0.We consider the mode of k = 0, and the energy de-generacy corresponds to density zeros. The BdG spectraare usually not degenerate in real space. Nevertheless,new phenomena would remain hidden if one were to re-strict one’s attention to real physical parameters [34].Two famous examples are the Lee-Yang zeros [35] andFisher zeros [36] reported for imaginary magnetic fieldsand imaginary temperatures, respectively. Inspired bythese, we extend the real coordinate variable x to theimaginary variable z = x + yi to explore the density zerosand underlying monopole topological potentials. Thus,we introduce the parameters C = ( x, y ).The Berry connection is then calculated in the param-eter space: A ± = i (cid:104) V L ± ( C ) |∇ C | V R ± ( C ) (cid:105)(cid:104) V L ± ( C ) | V R ± ( C ) (cid:105) (4)= i (cid:104) V L ± | ∂ z | V R ± (cid:105)(cid:104) V L ± | V R ± (cid:105) ( e x + i e y ) . (5)We note that there are always two branches of BdG ex-citations with positive and negative energies, correspond-ing to a pair of quasiparticles with opposite energies [37].The effective vector potential should be the average overthe two branches, which is A eff = A + + A − F ( z )( e x + i e y ) , (6)with F ( z ) = − (cid:82) ∂ t | ψ | dz | ψ | . To obtain Eq. (6), we haveused the fluid conservation law ∂ t | ψ | + ∂ x ( ψ | ∂ x φ ) = 0and ignored the purely imaginary term of i∂ x | ψ | / | ψ | .Here, | ψ | and φ are the amplitude and phase of the wave-function, respectively.The existence of the density zero points ( | ψ ( z N ) | =0) implies that A eff might have N singularities on thecomplex plane (denoted by z N = x N + iy N ). According to the Cauchy integral formula, a meromorphic functioncan be expressed in terms of these singularities [38], thatis, A eff = (cid:80) N Res [ F ( z N )] z − z N ( e x + i e y ), where Res [ F ( z N )] =Ω / πi is the residue.The real part of the above Berry connection constitutesa 2D vector potential in the following explicit form: A = Re[ A eff ] = (cid:88) N Ω[(x − x N ) e y − (y − y N ) e x ]2 π [(x − x N ) + (y − y N ) ] . (7)Here, A takes the form of a monopole’s topological vectorpotential in a 2D case [39, 40]: the corresponding mag-netic field will be zero everywhere except at those singu-lar points, that is, B = e z (cid:80) N Ω δ [ r − r N ]. Interestingly,we find that the magnetic flux is Ω = ± π , correspondingto a monopole with a charge of ± /
2, for all known 1Dnonlinear waves [41].In particular, the line integral of the vector potentialalong the real axis can predict the phase variations ofsuch a nonlinear wave. From Eq. (6), we can also findthat A eff [ x, y ] = ∂ z φ ( e x + i e y ). By reducing the problemto the real axis, we obtain φ ( x ) = (cid:82) A ( x, y = 0) · e x dx .It is interesting to compare the above results with theAharonov-Bohm effect [1], which predicts a topologicalphase when an electron moves on a close path around asolenoid. A 1D nonlinear wave moving on the real axiscannot see the magnetic fields scattered on the complexplane; however, it will acquire a phase due to the presenceof the vector potential. The evolution of such a nonlin-ear wave can be understood from the transformed equa-tion based on the topological vector potential, i∂ t ˜ ψ =( (ˆ p x + A ( x,y =0) · e x ) + ( g | ˜ ψ | + ∂ (cid:82) A ( x,y =0) · e x dx∂t ) ˜ ψ , witha transformation ˜ ψ = ψ exp [ − i (cid:82) A ( x, y = 0) · e x dx ].The effective magnetic field and electric field can be de-rived as B = ∇ × A and E = − ∂ A ∂t − ∇ ∂ (cid:82) A ( x,y =0) · e x dx∂t ,respectively [42]. In this sense, the phase variations ofthese nonlinear waves can be viewed as a 1D counter-part to the Aharonov-Bohm phase. Usually, the topo-logical vector potential exhibits time dependence, whichprovides an alternative way to understand the dynamicsof such nonlinear waves. Topological vector potentials for rogue waves —Theabove NLSE with g = − ψ = [1 − it )4 t +4 x +1 ] e it . The temporal evolution of the RW ampli-tude depicted in Fig. 1 (a) shows that the wave densityremains almost constant until t = −
5, after which sud-den growth occurs. At t = 0, the amplitude amplifica-tion ratio (defined as the peak amplitude divided by thebackground amplitude) reaches its maximum value of 3.Subsequently, the RW quickly decays, and the wave den-sity recovers to be nearly constant. The amplitude peakis located at x = 0, and on either side of the peak, thereare two valleys at x = ± a ( a = (cid:113) ). Interestingly,phase jumps accompany the rise in amplitude. In Fig. 1(b), we see that there is a π phase jump corresponding FIG. 1: (a) Amplitude distribution, (b) phase distribution,and (c-f) evolution of the topological vector potential for afundamental rogue wave (FRW). The monopoles with positiveand negative charges (i.e., ± ) are indicated by (cid:74) and (cid:78) ,respectively. to each amplitude valley [23, 24], and the direction of thephase jump at x = − a suddenly inverts to − π slightlyafter the moment when the maximum amplitude peakemerges.The characteristic of “appearing from nowhere and dis-appearing without a trace” of RWs [44, 45] is believed tobe the cause of many ocean disasters and therefore hasattracted much attention [46, 47]. Modulational insta-bility can well explain the rapid growth of RWs [45–49];however, it fails to explain the decay process. Moreover,analytical studies have indicated that the amplitude am-plification ratios for RWs of different orders are subjectto certain limits [44, 50], but no physical mechanism forthese ceiling values has been discovered. The π phasejumps and abrupt inversion discussed above are also notfully understood [23]. Here, we attempt to elucidatethese issues with the help of our developed topologicalvector potential theory.According to Eq. (6), the vector potential under-lying the FRW takes an explicit form of A eff = tz t +8 t (4 z +5)+(3 − z ) ( e x + i e y ) = F ( z )( e x + i e y ). Ithas four singularities, i.e., z , = ± ( a + ib ) and z , = ± ( a − ib ), where a = √ t √ t + √ t +64 t +9 − and b = √ t + √ t +64 t +9 − √ . Thus, we have A eff [ x, y ] = (cid:88) N =1 ,..., Res [ F ( z N )] z − z N ( e x + i e y ) , (8)where the residue is Res [ F ( z N )] = lim z → z N ( z − z N ) F ( z ) = ± i . According to Eq. (7), the vector potential A iscomposed of two pairs of monopoles, and in each pair, thetwo monopoles have charges of 1 / t = 0 with speeds of db/dt = (cid:112) / da/dt = 0, and then bounce backafter exchanging their charges (see Fig. 1 (e-f)). As t →± a → ± a and b →
0. The vector potential on thereal axis then takes the following form:lim t →± A = lim b → { − π [( x ± a ) e y + b e x ]2 π [( x ± a ) + b ]+ π [( x ± a ) e y − b e x ]2 π [( x ± a ) + b ]+ π [( x ∓ a ) e y + b e x ]2 π [( x ∓ a ) + b ]+ − π [( x ∓ a ) e y − b e x ]2 π [( x ∓ a ) + b ] } = ( − πδ [ x ± a ] + πδ [ x ∓ a ]) e x . (9)The line integral of the above vector potential can ex-plain the π phase jumps shown in Fig. 1 (b). The phasegradient determines the density flow, and the change inthe phase distribution can provide an understanding ofthe growth and decay of RWs [23]. The collision of themonopoles leads to a sudden rise in the wave amplitude.The exchange of the monopole charges after collision canwell explain the striking phase reversal that induces theRW’s rapid decay.Higher-order RWs admit higher amplitude peaks, moredensity valleys (or humps) and multiple phase steps, asshown in Fig. 2 [23, 44, 50, 51]. For a second-order RW,the maximum amplitude amplification ratio is 5. Thereare five phase steps distributed symmetrically with re-spect to the x = 0 axis, each of which is associated witha phase jump of ± π (see Fig. 2 (a)). The vector field iscomposed of six pairs of monopoles, as shown in Fig. 2(b). They can be divided into two classes: the four pairsthat are closer to the x-axis collide with each other onthe x-axis, leading to four π phase jumps, whereas theother two pairs (upper and lower pairs) collide on theimaginary axis, mainly contributing to a sudden rise inthe wave amplitude. For a third-order RW, the maxi-mum amplitude amplification ratio is 7, and there areseven phase steps distributed symmetrically with respectto the x = 0 axis (see Fig. 2 (c)). The vector field is com-posed of twelve pairs of monopoles, as shown in Fig. 2 (d).The paired monopoles collide and merge when the RW FIG. 2: (a) Amplitude and phase distributions for a second-order rogue wave (RW) at the moment when the wave ampli-tude reaches its highest peak ( t = 0). (b) Monopole vectorpotential at the moment slightly before the monopole colli-sions ( t = − . t = 0). (d) Monopole vector poten-tial at the moment before the monopole collisions ( t = − reaches its highest peak. Among them, six pairs collideand merge on the real axis, and the other six pairs collideon the imaginary axis or in other locations on the com-plex plane. The monopole pairs that do not collide onthe real axis do not contribute to multiple phase jumps,but they do influence the peak amplitude and the energytransfer involving RWs. Energy transfer and topological vector potential re-connection —The energy of an RW can be written as (cid:82) + ∞−∞ [ | ∂ x ψ | − ( | ψ | − ] dx , where the first term isthe kinetic energy E k and the second term is the inter-action energy E int . For an FRW, the time-dependentkinetic energy is E k = π (4 t +1) / . Because the total en-ergy is conserved, the amplitude amplification of the RWcorresponds to the energy transfer process from interac-tion energy to kinetic energy, as indicated in Fig. 3 (a).Quantitatively, the total energy transfer can be evaluatedas the time integral of the kinetic energy ( (cid:82) + ∞−∞ E k dt ).We have numerically calculated the integrals for RWswhose orders are up to 10 and have found that they areequal to the sum of the absolute magnetic flux of themonopoles (see Fig. 3 (b)). From the vector field per-spective, we know that monopole collisions induce theconversion of interaction energy into kinetic energy. Thisprocess is analogous to the magnetic field reconnectionprocess identified in astrophysics, in which magnetic fieldenergy is transformed into the kinetic energy of a plasma[27].We have also investigated the relation between thepeak amplitude of an RW and the number of monopoles FIG. 3: (a) Temporal evolution of the kinetic energy and in-teraction energy of an FRW. (b) Energy transfer vs. totalmagnetic flux. Here, the energy transfer refers to the time in-tegral of the kinetic energy, and the total magnetic flux repre-sents the sum of the absolute magnetic fluxes of all monopolesfor a high-order RW. and found that they are closely related. According toour discussion above, an FRW admits 4 monopoles. Be-cause the n th-order RW solution is a nonlinear superpo-sition of n ( n +1)2 FRWs [50], it will contain N = 2 n ( n + 1)monopoles on the complex plane. Among them, thereare 4 n monopoles colliding on the real axis that are re-sponsible for the multiple phase steps, whereas the other2 n ( n −
1) monopoles will collide in other locations on thecomplex plane. On the other hand, the square of themaximum amplitude amplification ratio P for an n th-order RW can be calculated to be (2 n + 1) [44, 52]. Wethus obtain the following explicit relation: P = 2 N + 1.Based on this observation, we can predict that the am-plitude amplification ratio of a high-order RW will besubject to a certain limit because an RW contains onlya limited number of monopoles due to the finite numberof valleys in its geometric configuration [50, 51]. Conclusion and discussion —Theoretically, we revealan intrinsic topological vector potential that is composedof paired monopoles hidden in the phase variations of 1Dnonlinear waves by investigating the Berry connectionof BdG equations in an extended complex coordinatespace, which is in contrast to the topological genus ona Riemann surface of complex parameters [53–56]. Asan application, we demonstrate that an n th-order RWsolution contains n ( n + 1) pairs of monopoles with op-posite charges and that the collision of these monopolesand the reconnection of the corresponding vector fieldlead to energy conversion from interaction energy to ki-netic energy and are responsible for the exotic propertyof “appearing from nowhere and disappearing without atrace” of RWs. Our theory is also applicable to other1D nonlinear models [57–62] and thus may inspire newresearch in directions related to the topological fluid fieldrepresentation of 1D nonlinear systems. It can be furtherextended to study the semiclassical dynamics of BdG ex-citations in Bose-Einstein condensation under the actionof intrinsic topological potentials [63]. Acknowledgments
This work was supported by theNational Natural Science Foundation of China (Con-tract No. 12022513, 11775176, 12047502 and 11775030),NSAF No. U1930403, and the Major Basic Research Pro- gram of Natural Science of Shaanxi Province (Grant No.2018KJXX-094). We thank Prof. C. H. Lee for noticingthe connection to Lee-Yang zeros. [1] Y. Aharonov and D. Bohm, Significance of Electromag-netic Potentials in the Quantum Theory, Phys. Rev. ,485 (1959).[2] M. M. Salomaa and G. E. Volovik, Quantized vortices insuperfluid He, Rev. Mod. Phys. , 533 (1987).[3] L. M. Pismen, Vortices in Nonlinear fields (Oxford,Clarendon, 1999).[4] A. Fert, N. Reyren, V. Cros, Magnetic skyrmions: ad-vances in physics and potential applications, Nat. Rev.Mater. , 17031 (2017).[5] M. Atiyah, The geometry and physics of knots (Cam-bridge University Press, 1990).[6] D. S. Hall, M. W. Ray, K. Tiurev, E. Ruokokoski, A.H.Gheorghe, and M. M¨ott¨onen, Tying quantum knots, Na-ture Phys. , 478 (2016).[7] M.V. 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