Breather induced quantized superfluid vortex filaments and their characterization
BBreather induced quantized superfluid vortex filaments and their characterization
Hao Li , , Chong Liu , , ∗ Wei Zhao , , Zhan-Ying Yang , , † and Wen-Li Yang , , School of Physics, Northwest University, Xi’an 710069, China Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710069, China Institute of Modern Physics, Northwest University, Xi’an 710069, China and Institute of Photonics and Photon-technology, Northwest University, Xi’an 710069, China (Dated: May 21, 2019)We study and characterize the breather-induced quantized superfluid vortex filaments whichcorrespond to the Kuznetsov-Ma breather and super-regular breather excitations developing fromlocalised perturbations. Such vortex filaments, emerging from an otherwise perturbed helicalvortex, exhibit intriguing loop structures corresponding to the large amplitude of breathers dueto the dual action of bending and twisting of the vortex. The loop induced by Kuznetsov-Mabreather emerges periodically as time increases, while the loop structure triggered by super-regularbreather— the loop pair —exhibits striking symmetry breaking due to the broken reflection symmetryof the group velocities of super-regular breather. In particular, we identify explicitly the generationconditions of these loop excitations by introducing a physical quantity—the integral of the relativequadratic curvature—which corresponds to the effective energy of breathers. Although the natureof nonlinearity, it is demonstrated that this physical quantity shows a linear correlation with theloop size. These results will deepen our understanding of breather-induced vortex filaments and behelpful for controllable ring-like excitations on the vortices.
I. INTRODUCTION
Quantum fluid [1, 2] has recently been the subject ofextensive investigations that contains vortex generation,interaction, and reconnection of vortex lines influencedby vortices [3, 4]. The motion of quantum fluid ismost succinctly illustrated by vortex filament due thatit consists of vorticity of infinite strength concentratedalong the filament and gives an intuitive geometricinterpretation of the evolution of the vorticity field.In the case of ideal inviscid fluid, the motion of thefluid elements is constrained by Biot-Savart law whichprovides valuable information of vortex tangles [5, 6].In particular, a variety of excitations are generated andevolve along the vortex filament due to the self-inducedvelocity [7–11]. Such excitations are physically importantsince they turn out to be the main degrees of freedomremaining in a superfluid in the ultra low temperatureregime. Therefore, investigation of vortex structures ofdifferent fundamental excitations is both relevant andnecessary.The first prototype of such excitations is the so-called‘Kelvin wave’ [12]. The latter, which is originated fromthe small deformations of vortex lines, plays an importantrole in the decay of turbulence energy [13]. However, oneshould note that Kelvin waves are low amplitude linearexcitations of a straight vortex. In contrast, there are alsosome larger amplitude excitations propagating along thefilament. Such large-amplitude excitations are inducedby nonlinearity, i.e., ‘nonlinear excitations’.There exists an interesting link between nonlinear ∗ Electronic address: [email protected] † Electronic address: [email protected] excitations and vortex dynamics that has attractedconsiderable attention recently. It is presently knownas ‘Hasimoto transformation’ [7] that allows us map themotion of vortex filament onto a scalar cubic nonlinearSchr¨odinger equation (NLSE) of the self-focusing typeand the resulting loop structure of bright soliton on avortex filament is demonstrated [7]. In addition to theclassical solitons, the NLSE possesses rich ‘breathing’excitations on a plane wave background which are knownas ‘breathers’ [14–16]. Such breathers are stronglyassociated with the modulation instability (MI) [14–18], where its nonlinear stage has been regarded asthe prototype of rogue wave events [19]. Surprisingly,although breather has been one of the center subjects innonlinear physics and its observation has been realizedwidely in many nonlinear systems [15, 20–23], thelink between one special type of breather—Akhmedievbreather [24] (as well as its multiple counterpart) andvortex filaments in a quantized superfluid has beenrevealed only recently [8, 9]. In fact, the resultingnew loop structure of Akhmediev breather, which differsfrom that of bright solitons [7], provides significantcontributions to our understanding of quantum fluidand superfluid turbulence. This is therefore aninterdisciplinary research—in the case of quantizedsuperfluid vortex filaments, MI, and breathers—thatneeds more explorations.However, the Akhmediev breather is merely the exactdescription for the MI emerging from a special purelyperiodic perturbations [24]. There is another type ofbreathers describing the MI developing from localisedperturbations that has not been studied in a quantizedsuperfluid. This includes the Kuznetsov-Ma breather [25]admitting localised single-peak perturbation and super-regular breather [16] supporting localised multi-peakperturbation. Indeed, it is recently demonstrated that a r X i v : . [ n li n . PS ] M a y Kuznetsov-Ma breather describes not only the MI in thesmall amplitude regime but also the interference betweenbright soliton and plane wave in the large amplituderegime [26]; while the super-regular breather admitsthe MI growth rate that coincides with the absolutedifference of group velocities of the breather [27, 28].Given that these breathers are qualitatively different,two questions of fundamental importance now arise: Howabout the loop excitations triggered by these breathers?Is there a physical quantity to identify explicitly all thesebreather-induced loop excitations?In this paper, we study the quantized superfluid vortexfilaments induced by Kuznetsov-Ma breather and super-regular breather admitting localised perturbations. Suchvortex filaments exhibit striking loop structures due tothe dual action of bending and twisting of the vortex.Remarkably, an intriguing loop structure triggeredby super-regular breather—the loop pair—exhibitsspontaneous symmetry breaking, due to the brokenreflection symmetry of the group velocities of super-regular breather. In particular, we identify explicitlythese loop excitations by introducing the integral ofthe relative quadratic curvature, which corresponds tothe effective energy of breathers. Although the natureof nonlinearity, it is demonstrated that this physicalquantity shows a linear correlation with the loop size.
II. HASIMOTO TRANSFORMATION ANDINVERSE MAP
For the incompressible and inviscid fluid, the Biot-Savart equation is reduced to a simpler local inductionapproximation (LIA) equation [29–31] by taking leadingorder v = (Γ / π ) ln ( R/a ) κ t × n = βκ t × n . (1)Here Γ is a circulation, R is local radius of curvatureand a is the effective vortex core radius. v = d r dt isthe flow velocity vector of the vortex filament, t and n are unit vectors corresponding to the tangent andprincipal normal directions, respectively. κ , as a realfunction of arc length variable s and time t , representsthe curvature distribution of the vortex filament. Thisequation makes us obtain more properties of the statesrelated to the motion of vortex especially in the case ofHasimoto transformation. Assuming that β is constantand making use of the Seret-Frenet equations given in[32], r (cid:48) = t , t (cid:48) = κ n , n (cid:48) = τ b − κ t , b (cid:48) = − τ n , (2)where prime denotes a differential of arc length, b isbinormal vector and τ is the torsion of the vortexfilament, Eq. (1) can be transformed into a 1D scalarcubic NLSE of self-focusing type [7] β − ( iψ t ) = − ψ ss − | ψ | ψ. (3) ψ ( s, t ) is a complex function related with the localinstantaneous geometric parameters curvature κ ( s, t )and torsion τ ( s, t ) in the context of vortices by thetransformation ψ ( s, t ) = κ ( σ, t ) e i (cid:82) s τ ( σ,t ) ds . (4)The NLSE (3) possesses rich ‘breathing’ excitations[14], which provides a path for studying exactly breather-induced quantized superfluid vortex filaments. One canobtain the explicit configuration of these excitations byinverse map (see Appendix A). III. KUZNETSOV-MA BREATHER INDUCEDVORTEX FILAMENTS AND EXACTCHARACTERIZATION
We first consider the Kuznetsov-Ma breather thatexhibits periodic pulsating dynamics along t . Its explicitexpression for Eq. (3) is given by ψ ( s, t ) = (cid:20) − χ cos ( ηβt ) + iη sin ( ηβt ) κ b cosh ( χξ ) − κ cos ( ηβt ) (cid:21) ψ , (5)where χ = (cid:112) b − κ with b being a real constant ( b >κ ), η = bχ , and ξ = s − τ βt . Physically, b describes theoscillation period and amplitude of the Kuznetsov-Mabreather. κ and τ are real constants which denote theamplitude and wave vector of the plane wave background ψ respectively. The latter has the from ψ = κ exp i ( τ s + ωt ) , ω = β κ / − β τ . (6)This plane wave corresponds to a trivial uniform helicalvortex without physical interest ( κ and τ describethe curvature and torsion of a uniform helical vortex,respectively). In contrast, the Kuznetsov-Ma breatherdescribes nontrivial structure of vortex filament thathas not been studied fully. From Eq. (5), one canreadily calculate the explicit expressions of the curvatureand torsion of vortex filament induced by Kuznetsov-Mabreather, which are given respectively by κ = (cid:34)(cid:18) κ + 2 χ cos( ηβt ) n (cid:19) + 4 η sin ( ηβt )( n ) (cid:35) / , (7)and τ = τ (cid:20) κ η sin ( ηβt ) sinh ( χξ ) m + m + m + m (cid:21) , (8)with n = κ cos ( ηβt ) − b cosh ( χξ ) ,m = κ − κ b + 8 b , m = κ cos (2 ηβt ) ,m = 4 κ b (cid:0) κ − b (cid:1) cos ( ηβt ) cosh ( χξ ) ,m = κ b cosh (2 χξ ) . FIG. 1: Temporal evolutions of curvature κ ( ξ, t ) (a) andtorsion τ ( ξ, t ) (c) corresponding to a Kuznetsov-Ma breather,see Eqs. (7) and (8). (b) and (d) are variations of κ ( ξ, t )and τ ( ξ, t ) at different times. The parameters are: κ = 1, τ = 0 . b = 1 .
2, and β = 4 π . The variations of curvature and torsion of Kuznetsov-Ma breather on the ( ξ , t ) plane (note here that ξ = s − τ βt denotes the moving frame on the group velocity),with initial conditions b = 1 . κ = 1, τ = 0 .
05, areshown in Fig. 1(a) and (c). As expected, the curvatureof Kuznetsov-Ma breather, starting from a localised non-periodic (single-peak) perturbation, evolves graduallyinto its maximum at t = 0 [see the profiles in Fig. 1(b)].The curvature then exhibits periodic oscillation with theperiod 2 π/ ( ηβ ) as t increases [see Fig. 1(a)].A notable feature is that the torsion, as a functionof arc length s and time t , exhibits singular behavioras t → π/ ( ηβ ). Figure 1(d) clearly indicates that thephase becomes ill defined at the point κ = 0 near t = 0,which leads to the severe twisting of the vortex filament.This is not surprising since the Kuznetsov-Ma breatheradmits a π phase shift at the valleys. This phase shiftresults in the singular behavior of the torsion. Note thatthe singular does not make the Hasimoto transformationill-defined due to the π phase shift of nonlinear waves.Figure 2 shows the corresponding vortex configurationof Kuznetsov-Ma breather within one growth-decaycycle. One can see clearly from the figure that thevortex filament, emerging from an otherwise perturbedhelical vortex at t = − .
314 [see Fig. 2 (a)], exhibits astriking loop structure at t = 0 due to the dual action ofbending and twisting of the vortex. This loop structuredisappears gradually as t increases. At t = 0 . t increases due to the featureof the Kuznetsov-Ma breather. One should note thatwhen κ →
0, the loop of Kuznetsov-Ma breather reducesto the classical loop structure of bright solitons [7]; theperiodic recurrence of the loop is gone. The Kuznetsov-Ma breather can transform into thePeregrine rogue wave [34] with double localization in thelimit of b → κ . The latter is also the limiting case ofthe Akhmediev breathers. All these breathers can induceloop-structure excitations on vortex filaments, as shownabove and in Ref. [8, 9].One then wonder how to identify explicitly these vortexfilaments induced by breathers, since each kind of vortexfilament has a similar loop structure corresponding to themaximum curvature. This is the question of fundamentalimportance that has not been answered before. To dothis, we introduce the following physical quantity— theintegral of the relative quadratic curvature of the form∆ K = (cid:90) ∞−∞ (cid:2) κ ( s, t ) − κ ( s, t ) (cid:3) ds. (9)Eq. (9) corresponds to the effective energy of breathersin optics [35, 36]. Namely, it coincides with the energyof breathers against plane wave, i.e., (cid:82) ∞−∞ (cid:0) ψ − ψ (cid:1) ds .For a quantum condensate fluid, Eq. (9) stands forthe effective atom numbers [1]. Generally, this is aquantity of physical importance which can be monitoredeffectively for localised nonlinear waves in experiments[1, 37]. Here we highlight that Eq. (9) can be used forcharacterising the breather induced vortex filaments inquantized superfluid.It is interesting to note that for the Kuznetsov-Mabreather (5), one obtains exactly ∆ K = 8 (cid:112) b − κ ,which indicates ∆ K >
0; while for the Peregrine roguewave and the Akhmediev breather, we find that ∆ K = 0(see Appendix B). This is the immanent reason whythe loop structure induced by Kuznetsov-Ma breatherexhibits periodic oscillation as t increases, while the loopstructure triggered by the Peregrine rogue wave and theAkhmediev breather appears only once during the timeevolution. On the other hand, the condition ∆ K = 0indicates that the resulting vortex filament starts froma uniform helical vortex structure. This corresponds tothe case of the Peregrine rogue wave and the Akhmedievbreather. However, the uniform helical vortex structurewill never appear for the vortex filament induced by theKuznetsov-Ma breather.Let us take a closer look at Eq. (9) by considering therelation between ∆ K and the Kuznetsov-Ma breather-induced loop structure. To this end, we define thecharacteristic size of the loop structure, r k , whichdescribes the minimum radius of the structure.Figure 3 shows the relation between ∆ K ( κ , t ) and r ( κ , t ) on logarithmic coordinates with random values of b in the region b ∈ [2 κ , ∞ ]. This parametercondition allows us to study the qualitative link between∆ K ( κ , t ) and r k ( κ , t ) from the vortex filaments inducedby random Kuznetsov-Ma breathers (i.e., a seriesof Kuznetsov-Ma breathers with random period andamplitude).For the fixed t ( t = 0), we show the characteristicsof ln(∆ K ) and ln( r k ) with increasing κ in Fig. 3(a).It is interesting that, despite the random values of b , FIG. 2: Configuration of vortex filaments of Kuznetsov-Ma breathers computed by LIA for parameters given by κ = 1, τ = 0 . b = 1 . β = 4 π at different time (a) t = − . t = − . t = − .
01, (d) t = 0 .
157 and (e) t = 0 . K ( κ , t ) and r k ( κ , t ) onlogarithmic coordinates (ln ∆ K , ln r k ) (a) as κ varies withfixed t = − .
01; (b) as t varies with fixed κ = 1. The solidlines are precise description of relation between ln ∆ K andln r k as b → ∞ . The values of the parameter b are randomnumbers in the region b ∈ [2 κ , κ ]. ln(∆ K ) decreases linearly as ln( r k ) increases and thecorresponding rates α are exactly consistent at a fixedtime ( α = − κ and τ and variational t , as shownin Fig. 3(b).We then explain the linear relation above exactly.We note that for the Kuznetsov-Ma-breather-inducedloop structure, the minimum loop radius r k is inverselyproportional to the maximum curvature κ m , i.e., r k = 1 κ m . It is given explicitly by Eq. (7) at ξ = 0: r k = (cid:34)(cid:18) κ + 2 χ cos( ηβt ) n (cid:19) + 4 η sin ( ηβt )( n ) (cid:35) − / (10)with n = κ cos ( ηβt ) − b . Here r k is the function of b , κ and t . Thus the accurate description of ∆ K · r k reads∆ K · r k = 8 (cid:112) b − κ (cid:20)(cid:16) κ + χ cos( ηβt ) n (cid:17) + η sin ( ηβt )( n ) (cid:21) / . (11) FIG. 4: Profile of ∆ K · r k , Eq. (11) as b increases. Otherparameters are κ = 1, t = 2 nπ/ ( ηβ ) ( n is an integer) and β = 4 π . Clearly, for the case of Kuznetsov-Ma breather, ∆ K · r k (cid:54) = 0; while for case of the Peregrine rogue wave andAkhmediev breather, ∆ K · r k = 0, since ∆ K = 0.We show the profile of Eq. (11) as b increases in Fig.4. One can see that ∆ K · r k increases monotonously withincreasing b . Remarkably, as b → ∞ , we find ∆ K · r k → b → ∞ ∆ K · r k = 4 , (12)Namely, ln ∆ K = − ln r k + ln 4 , (13)on logarithmic coordinates.We show the linear relation by the solid lines in Fig. 3.Observably, the numerical results are in good agreementwith the analytical relation (12) (solid line). Physically,the Kuznetsov-Ma breather in the region b ∈ [2 κ , ∞ ] canbe approximatively described by the linear interference between a bright soliton and a plane wave [26]. As b →∞ ( b (cid:29) κ ), i.e., the amplitude of the bright soliton ismuch bigger than that of the plane wave, the plane wavecan be neglected. As a result, the effective energy ∆ K is quadruple of the amplitude of the remaining brightsoliton, which directly leads to Eq. (12). FIG. 5: Temporal evolutions of curvature κ ( ξ, t ) (a) and torsion τ ( ξ, t ) (c) corresponding to a super-regular breather, see Eq.(C2) in Appendix C. (b) and (d) are variations of κ ( ξ, t ) and τ ( ξ, t ) at different times. Other parameters are κ = 1, τ = 0 . R = 1 . φ = π/ IV. SUPER-REGULAR BREATHER INDUCEDLOOP PAIR AND SYMMETRY BREAKING
Let us then consider the vortex filament induced bythe super-regular breather. The latter, which recentlyserves as the exact MI scenario excited from localisedmulti-peak perturbations, is formed by the nonlinearsuperposition of two quasi-Akhmediev breathers [16,27, 28, 38, 39]. The exact solution of super-regularbreather with τ = 0 is first provided in Ref. [16].However, the general solution with τ (cid:54) = 0 in the infiniteNLSE is presented recently in Ref. [27]. By usingthe transformation above and the super-regular solutionin Ref. [27], the corresponding properties of vortexfilaments can be achieved effectively. Here we omit thetedious explicit expression but show the important andcompact results. At the first step, the integral of therelative quadratic curvature of super-regular breathercan be obtained explicitly from the exact solution inAppendix C. It reads,∆ K = 16 κ (cid:20) ε cos φ + π sin φ csch (cid:18) π sin φε cos φ (cid:19)(cid:21) , (14)where ε = R − ε (cid:28) R ( >
1) and φ [ ∈ ( − π/ , π/ R (or ε ) and φ are two important parametersthat describe directly the amplitude and period of thesuper-regular breathers. It is therefore crucial to study the property of vortex filaments induced by super-regularbreathers by the choice of parameters R and φ .Just as the case of Kuznetsov-Ma breather, Eq.(14) is also greater than zero, i.e., ∆ K >
0. Thisindicates that the super-regular breather also admitslong-time dynamics which is different from the Peregrinerogue wave and Akhmediev breather. Unlike thecase of Kuznetsov-Ma breather, the evolutions ofcurvature and torsion of super-regular breather exhibitremarkably different characteristics. This stems fromthat the super-regular breather possesses a localisedmulti-peak perturbation rather than a localised single-peak perturbation.Figure 5 shows the variation of curvature and torsioninduced by super-regular breather with the initialparameters κ = 1, R = 1 .
1, and φ = π/
8. As canbe seen from Fig. 5(a) that the curvature of a super-regular breather triggered from a localised multi-peakperturbation at t = 0 [see Fig. 5(b)] increases graduallydue to the exponential amplification of the MI at thelinear stage. It reaches its maximum at t = 0 .
43 and thensplits into two quasi-Akhmediev breathers propagatingalong different directions during the nonlinear stageof MI. The corresponding torsion also suffers singularbehavior starting from the maximum curvature point t = 0 .
43 [see Fig. 5(c)]. Interestingly, the nonlinearpropagation stage always holds the singular torsion atthe maximum curvature point as t > .
43 [see Fig. 5(d)].Figure 6 displays the corresponding vortex structure
FIG. 6: Configuration of vortex filaments of super-regularbreathers at different time (a) t = 0, (b) t = 0 .
43 and (c) t = 1 .
2. Other parameters are the same as in Fig. 4.FIG. 7: Top view of configuration of vortex filaments of super-regular breathers at a fixed time t = 1 . τ increases. (a) τ = 0 .
01, (b) τ = 0 .
17 and (c) τ = 0 .
24. One can seeclearly the reflection symmetry breaking of the loop pair withnon-zero τ . Other parameters are κ = 1, R = 1 . φ = π/ at t = 0, t = 0 .
43, and t = 1 .
2, respectively. Onecan see clearly that the vortex filament emerges froma perturbed helical vortex at t = 0 and then exhibitsa remarkable loop structure at t = 0 .
43. This is thelinear MI stage that corresponds to one loop excitation.Interestingly, once the vortex filament evolves into thenonlinear stage, the loop structure splits into a loop pair which corresponds to the two quasi-Akhmediev breatherspropagation with different group velocities.In particular, we find that the loop pair induced bysuper-regular breather at the nonlinear stage shows aninteresting reflection symmetry breaking , as shown in Fig.7. We find that this remarkable feature comes fromthe asymmetry of the group velocities of the two quasi-Akhmediev breathers. Indeed, the group velocities ofthe super-regular breather are given by [see Eq. (C4) inAppendix C] V g = 2 βτ + d, V g = 2 βτ − d, (15)where d = βκ ( R +1 ) R − R sin φ . Clearly, due to τ (cid:54) = 0, theabsolute values of this two group velocities are alwaysunequal. Once κ , ε , and φ are fixed, the degree of theasymmetry is proportional to the value of | τ | .Figure 7 shows the corresponding vortex structuresinduced by super-regular breather as τ increases. Wesee that as τ → τ [Figs. 7(b) and 7(c)].It is very interesting to note that, despite the brokenreflection symmetry as τ (cid:54) = 0, the growth rate of modulation instability driven by the super-regularbreather does not depend on τ . Namely, this growthrate is only associated with the absolute difference ofthe group velocities, G = η r | V g − V g | with η r = a ( R − /R ) cos φ , as shown in Ref. [27]. This resultis physically important because that although the super-regular breather induced vortex structures can exhibitdifferent loop pairs with symmetry breaking, the inherentMI property can remain invariable.Finally, we consider the relation between ∆ K andcharacteristic size r s of the super-regular breatherinduced vortex structures. Similar to the case ofKuznetsov-Ma breather, we define characteristic size r s as the minimum radius of the super-regular-breatherinduced vortex structure throughout the whole evolution.Thus, the characteristic size r s , which is also inverselyproportional to the maximum curvature κ ms (i.e., r s =1 /κ ms ), is given by r s = (cid:20) κ + κ (cid:18) ε + 11 + ε (cid:19) cos φ (cid:21) − , (16)where ε = R − ε (cid:28)
1) defined above.Collecting Eq. (14) and Eq. (16), we obtain theexplicit expression of ∆ K · r s by omitting the high-orderterm O ( ε ). It reads∆ K · r s = α s ε, (17)where α s = 16 cos φ/ (1 + 2 cos φ ).In contrast to the case of Kuznetsov-Ma breather, Eq.(11), where only one parameter b can be modulated whenthe plane wave parameters ( κ , τ ) and the structuralparameter β are fixed, Eq. (17) has two free physicalparameters ( ε and φ ). But even so, we highlight thatlinear relations can also hold for the case of super-regularbreather induced vortex structures. FIG. 8: Relations between ∆ K · r s and ε of the vortexfilament induced by super-regular breather as φ varies. Otherparameters are κ = 1. The discrete points are obtained withthe high-order term O ( ε ) considered, while the colored linesretain the first-order term only. FIG. 9: Relations between ∆ K · r s and φ of the vortexfilament induced by super-regular breather as ε varies. Otherparameters are κ = 1. Figure 8 shows the characteristics of ∆ K · r s as ε increases with different values of φ . In particular,we compare the results obtained from the approximateexpression Eq. (17) (the solid lines) and the exactexpression (the dotted lines), respectively. One can seethat for each fixed φ , ∆ K · r s shows a linear relation with ε . The corresponding rate α s decreases in the range of[48 ,
0] as φ increases from 0 to π/ K · r s as φ decreases with different values of ε . For each case withfixed ε , ∆ K · r k increases monotonously with decreasing φ . As φ →
0, one obtains that ∆ K · r k → ε/ φ →
0, the super-regular breather transforms itselfinto two colliding Kuznetsov-Ma breathers, so that thesimilar linear relation can be maintained.
V. CONCLUSION
In summary, we have investigated the superfluidvortex filaments induced by Kuznetsov-Ma breatherand super-regular breather, which admit localisedperturbations. We have shown that the loop structureinduced by Kuznetsov-Ma breather emerges periodicallyas time increases, while the loop structure triggeredby super-regular breather—the loop pair—exhibitsstriking symmetry breaking due to the broken reflectionsymmetry of the group velocities of super-regularbreather. In particular, we have characterized andidentified explicitly these loop excitations by introducingthe integral of the relative quadratic curvature, whichcorresponds to the effective energy of breathers.Although the nature of nonlinearity, it is demonstratedthat this physical quantity shows a linear correlation withthe loop size.
ACKNOWLEDGEMENTS
This work has been supported by the NationalNatural Science Foundation of China (NSFC) (GrantNos. 11705145, 11875220, 11434013, and 11425522),Natural Science Basic Research Plan in Shaanxi Provinceof China (Grant No. 2018JQ1003), and the MajorBasic Research Program of Natural Science of ShaanxiProvince (Grant Nos. 2017KCT-12, 2017ZDJC-32).
Appendix A: THE POSITION VECTOR OF THEVORTEX FILAMENT
We give the explicit expression of the position vectorof the vortex filament by integrating the Frenet-Serretequations given in [32] and the exact expression isformulated explicitly in Ref. [33], which should berepresented as r ( s, t ) = x ( s, t ) y ( s, t ) z ( s, t ) = x ( t ) + (cid:88) k =1 c k ( t ) (cid:90) s M k ( σ, t ) dσy ( t ) + (cid:88) k =1 c k ( t ) (cid:90) s M k ( σ, t ) dσz ( t ) + (cid:88) k =1 c k ( t ) (cid:90) s M k ( σ, t ) dσ . Here, x ( t ), y ( t ), z ( t ) are constants with respect tothe initial position of vortex structures. M k ( k = 1 , , M = γ + α cos λλ , M = α sin λλ , M = αγ (1 − cos λ ) λ , where α = (cid:82) s κ ( σ, t ) dσ , γ = (cid:82) s τ ( σ, t ) dσ , and λ = (cid:113)(cid:0)(cid:82) s κ ( σ, t ) dσ (cid:1) + (cid:0)(cid:82) s τ ( σ, t ) dσ (cid:1) . Appendix B: EXPLICIT EXPRESSIONs OF ∆ K OF AKHMEDIEV BREATHER ANDPEREGRINE ROGUE WAVE ∆ K , the integral of the relative quadratic curvature , isexpressed explicitly in the form∆ K = (cid:90) ∞−∞ (cid:2) κ ( s, t ) − κ ( s, t ) (cid:3) ds. (B1)Here, κ ( s, t ) and κ ( s, t ) represent the curvaturedistribution of the vortex filament and the curvature ofthe background uniform helical vortex corresponding toplane wave ψ (6) respectively.As mentioned in Sec III, in addition to the Kuznetsov-Ma breather, plane wave (6) also admits other breathingwaves, including the Akhmediev breather and thePeregrine rogue wave. As comparison, we show here ∆ K for the vortex filaments induced by Akhmediev breatherand Peregrine rogue wave.We first consider the Akhmediev breather that exhibitsthe explicit description for the MI emerging from periodicperturbations. Its exact expressions is given by ψ A ( s, t ) = (cid:20) − χ cosh ( η βt ) + iη sinh ( η βt ) κ cos ( η βt ) − κ b cosh ( χ ξ ) (cid:21) ψ , (B2)where χ = (cid:112) κ − b with b < κ , η = bχ , and ξ = s − τ βt . The corresponding exact expression of thecurvature is given by κ A = (cid:34)(cid:18) κ − χ cosh( η βt ) A (cid:19) + 4 η sinh ( η βt ) A (cid:35) / (B3)with A = κ cosh ( η βt ) − b cos ( χ ξ ). A substitution ofEq. (B3) into Eq. (B1) yields ∆ K A = 0.We then consider the Peregrine rogue wave with doublelocalization. The latter corresponds to the limiting caseof Eq. (B2) as b → κ . Its exact expression is given by ψ P ( s, t ) = (cid:34) − iκ βt + 41 + κ β t + κ ( s − βτ t ) (cid:35) ψ , (B4)whose curvature is in the form of κ P = κ (cid:115) κ β t a + (cid:18) − a (cid:19) (B5)with a = 1+ κ β t + κ ( s − βτ t ) . By calculating Eq.(B1), we demonstrate also that ∆ K P = 0.As a result, both the Akhmediev breather and thePeregrine rogue wave share the vanishing ∆ K , whichindicates that the corresponding vortex filaments startfrom a uniform helical vortex structure. Appendix C: EXPLICIT EXPRESSION OFSUPER-REGULAR BREATHER
The explicit expression of super-regular breather forEq. (3) is given by the Darboux transformation [27],where the spectral parameter λ is parameterized by theJukowsky transform [16] as follows: λ = i κ (cid:18) ∆ + 1∆ (cid:19) − τ , ∆ = Re iφ . (C1)Here, R and φ define the location of the spectralparameter λ in the polar coordinates. They representradius and angle respectively in the region R > φ ∈ ( − π/ , π/ τ = 0, Eq (C1) reduces to the spectralparameter used in Ref. [16]. With different values of R and φ , the resulting exact solution can describe different breather dynamics [16]. A more general phase diagram ofbreathers has been obtained recently in Ref. [40]. Herewe consider the super-regular breather formed by twoquasi-Akhmediev breathers with R = R = R = 1 + ε ( ε (cid:28) φ = − φ = φ . Its explicit expression of thesolution is in the form: ψ ( s, t ) = ψ (cid:20) − ρ(cid:37) ( i(cid:37) − ρ ) Ξ + ( i(cid:37) + ρ ) Ξ κ ( ρ Ξ + (cid:37) Ξ ) (cid:21) . (C2)Here (cid:37) = κ (cid:18) R − R (cid:19) sin φ, ρ = κ (cid:18) R + 1 R (cid:19) cos φ Ξ = ϕ φ + ϕ φ , Ξ = ϕ φ + ϕ φ , Ξ = ϕ φ − ϕ φ − ϕ φ + ϕ φ , Ξ = ( ϕ + ϕ ) ( φ + φ ) , with φ jj = cosh (Θ ∓ iψ ) − cos (Φ ∓ φ ) ,ϕ jj = cosh (Θ ∓ iψ ) − cos (Φ ∓ φ ) ,φ j − j = ± i cosh (Θ ∓ iφ ) − cos (Φ ∓ θ ) ,ϕ j − j = ± i cosh (Θ ∓ iφ ) − cos (Φ ∓ θ ) , where θ = arctan (cid:2)(cid:0) − iR (cid:1) / (cid:0) R (cid:1)(cid:3) . Θ j and φ j are related with group and phase velocities respectively,which is in the form ofΘ j = 2 η r ( s − V gj t ) , φ j = 2 η ij ( s − V pj t ) (C3)where η i = − η i = κ (cid:18) R + 1 R (cid:19) sin φ,η r = κ (cid:18) R − R (cid:19) cos φ,V p = 2 βτ − d , V p = 2 βτ + d ,V g = 2 βτ + d, V g = 2 βτ − d. (C4)with d = βκ (cid:0) R − R (cid:1) cos(2 φ )sin φ , d = βκ ( R − R ) sin φ and d = βκ ( R +1 ) R − R sin φ . The initial state of the super-regularbreather can be extracted from the above solution at t =0. It reads, as ε → ψ ( s,
0) = ψ (cid:18) − i ε cos φ cos ( κ s sin φ )cosh ( κ εs cos φ ) (cid:19) . (C5)Note that for a given plane wave background (6) (i.e., κ and τ are fixed), R and φ determine the amplitudeand period of breathers. In particular, R and φ effect theprofile of the initial state (C5) of super-regular breathers.The integral of the relative quadratic curvature ofsuper-regular breather ∆ K [see Eq. (14)] is obtainedexplicitly from the initial state (C5), since the NLSEshares the same ∆ K at different times. [1] C. F. Barenghi and N. G. Parker, A primer on quantumfluids (Springer, 2016).[2] I. Carusotto and C. Ciuti, Rev. Mod. Phys. , 299(2013).[3] S. Zuccher, M. Caliari, A. Baggaley, and C. Barenghi,Phys. Fluids , 125108 (2012).[4] S. K. Nemirovskii, Phys. Rep. , 85 (2013).[5] P. G. Saffman, Vortex Dynamics (Cambridge UniversityPress, Cambridge, 1992).[6] W. Gilpin, V. N. Prakash, M. Prakash, Nat. Phys. ,380 (2017).[7] H. Hasimoto, J. Fluid Mech , 477 (1972).[8] H. Salman, Phys. Rev. Lett , 165301 (2013).[9] H. Salman, J. Phys.: Conf. Ser. , 012005 (2014).[10] B. K. Shivamoggi, Phys. Rev. B , 012506 (2011).[11] R. A. Van Gorder, Phys. Rev. E , 052208 (2016); Phys.Rev. E 91, 053201 (2015); R. Shah, R. A. Van Gorder,Phys. Rev. E 93, 032218 (2016).[12] W.T. Kelvin, Vibrations of a columnar vortex. Phil. Mag.Ser. 5, 155-168 (1880).[13] E. Kozik and B. Svistunov, Phys. Rev. Lett. 92, 035301(2004).[14] N. Akhmediev and A. Ankiewicz, Solitons: NolinearPulses and Beams (Chapman and Hall, London, 1997).[15] J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, Nat.Photonics , 755 (2014).[16] V. E. Zakharov and A. A. Gelash, Phys. Rev. Lett. , 054101 (2013); A. A. Gelash and V. E. Zakharov,Nonlinearity , R1 (2014); B. Kibler, A. Chabchoub, A.Gelash, N. Akhmediev, and V. E. Zakharov, Phys. Rev.X , 041026 (2015).[17] J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N.Akhmediev, Opt. Express , 21497 (2009).[18] L. C. Zhao and L. M. Ling, J. Opt. Soc. , 850 (2016).[19] N. Akhmediev, J. Soto-Crespo, and A. Ankiewicz, Phys.Lett. A , 2137 (2009); N. Akhmediev, J. Soto-Crespo,and A. Ankiewicz, Phys. Rev. A , 043818 (2009); N.Akhmediev, A. Ankiewicz, and M. Taki, Phys. Lett. A , 675 (2009).[20] M. Onorato, S. Residori, U. Bortolozzo, A. Montina, andF. T. Arecchi, Phys. Rep. , 47 (2013). [21] N. Akhmediev, et al , J. Optics , 063001 (2016).[22] M. Onorato, S. Residori, and F. Baronio, Rogue andShock Waves in Nonlinear Dispersive Media (Springer,2016).[23] S. Wabnitz,
Nonlinear Guided Wave Optics: a testbed forextreme waves (IOP Publ., Bristol, UK, 2017).[24] N. Akhmediev and V. I. Korneev, Theor. Math. Phys. , 1089 (1986); N. Akhmediev, Nature , 267 (2001).[25] E. Kuznetsov, Sov. Phys. Dokl. , 507 (1977); Y. C.Ma, Stud. Appl. Math. , 43 (1979).[26] L. C. Zhao, L. M. Ling, and Z. Y. Yang, Phys. Rev. E , 022218 (2018).[27] C. Liu, Z. Y. Yang, and W. L. Yang, Chaos, , 083110(2018).[28] Y. Ren, C. Liu, Z. Y. Yang, and W. L. Yang, Phys. Rev.E , 062223 (2018).[29] R. J. Arms, and F. R. Hama, The Physics of fluids , 553(1965).[30] L. S. Da Rios, Rend. Circ. Mat. Palermo , 117 (1906).[31] R. Betchov, J. Fluid Mech , 471 (1965).[32] L. M. Pismen, Vortices in Nonlinear Fields (Clarendon,Oxford, 1999).[33] R. Shah,
Rogue waves on a vortex filament (Oxford,2015).[34] D. H. Peregrine, J. Aust. Math. Soc. Ser. B, Appl. Math. , 16 (1983).[35] N. Akhmediev and A. Ankiewicz, (Eds.), Dissipativesolitons , Lect. Notes Phys. 661 (Springer, BerlinHeidelberg 2005).[36] N. Akhmediev and A. Ankiewicz, (Eds.),
DissipativeSolitons: From Optics to Biology and Medicine , Lect.Notes Phys. 751 (Springer, Berlin Heidelberg 2008).[37] P. Grelu and N. Akhmediev, Nat. Photonics 6, 84 (2012).[38] C. Liu, Y. Ren, Z. Y. Yang, and W. L. Yang, Chaos ,083120 (2017).[39] Y. Ren, X. Wang, C. Liu, Z. Y. Yang, and W. L.Yang, Commun. Nonlinear. Sci. Numer. Simulat.63