Breather-induced quantised superfluid vortex filaments and their characterisation
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Hao Li1,2,3, Chong Liu,1,2,3,6,7, Wei Zhao1,5, Zhan-Ying Yang1,2,3,6, Wen-Li Yang1,2,3,41School of Physics, Northwest University, Xi’an 710127, China 2Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710127, China 3NSFC-SPTP Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China 4Institute of Modern Physics, Northwest University, Xi’an 710127, China 5Institute of Photonics and Photon-technology, Northwest University, Xi’an 710127, China 6State Key Laboratory of Photoelectric Technology and Functional Materials, Xi’an 710127, China
First author contact:7Author to whom any correspondence should be addressed. Received:2020-02-2Revised:2020-03-20Accepted:2020-03-23Online:2020-06-24
Abstract We study and characterise the breather-induced quantised superfluid vortex filaments which correspond to the Kuznetsov-Ma breather and super-regular breather excitations developing from localised perturbations. Such vortex filaments, emerging from an otherwise perturbed helical vortex, exhibit intriguing loop structures corresponding to the large amplitude of breathers due to the dual action of bending and twisting of the vortex. The loop induced by the Kuznetsov-Ma breather emerges periodically as time increases, while the loop structure triggered by the super-regular breather—the loop pair—exhibits striking symmetry breaking due to the broken reflection symmetry of the group velocities of the super-regular breather. In particular, we identify explicitly the generation conditions of these loop excitations by introducing a physical quantity—the integral of the relative quadratic curvature—which corresponds to the effective energy of breathers. Despite the nature of nonlinearity, it is demonstrated that this physical quantity shows a linear correlation with the loop size. These results will deepen our understanding of breather-induced vortex filaments and be helpful for controllable ring-like excitations on the vortices. Keywords：vortex filament;loop structure;local induction approximation;Kuznetsov-Ma breather;super-regular breather
PDF (1208KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Hao Li, Chong Liu, Wei Zhao, Zhan-Ying Yang, Wen-Li Yang. Breather-induced quantised superfluid vortex filaments and their characterisation. Communications in Theoretical Physics, 2020, 72(7): 075802- doi:10.1088/1572-9494/ab8a1c
1. Introduction
Quantum fluid [1, 2] has recently been the subject of extensive investigations that concern vortex generation, interaction, and reconnection of vortex lines influenced by vortices [3, 4]. The motion of quantum fluid is most succinctly illustrated by vortex filament due to the fact that it consists of vorticity of infinite strength concentrated along the filament, and gives an intuitive geometric interpretation of the evolution of the vorticity field. In the case of ideal inviscid fluid, the motion of the fluid elements is constrained by Biot-Savart law, which provides valuable information about vortex tangles [5, 6]. In particular, a variety of excitations are generated and evolve along the vortex filament due to the self-induced velocity [7–11]. Such excitations are physically important since they turn out to be the main degrees of freedom remaining in a superfluid in the ultra low temperature regime. Therefore, investigation of vortex structures of different fundamental excitations is both relevant and necessary.
The first prototype of such excitations is the so-called ‘Kelvin wave’ [12]. The latter, which is originated from the small deformations of vortex lines, plays an important role in the decay of turbulence energy [13]. However, one should note that Kelvin waves are low-amplitude linear excitations of a straight vortex. In contrast, there are also some larger amplitude excitations propagating along the filament. Such large-amplitude excitations are induced by nonlinearity, i.e. ‘nonlinear excitations’.
There exists an interesting link between nonlinear excitations and vortex dynamics that has attracted considerable attention recently. It is presently known as ‘Hasimoto transformation’ [7] that allows us to map the motion of vortex filament onto a scalar cubic nonlinear Schrödinger equation (NLSE) of the self-focusing type, and the resulting loop structure of bright soliton on a vortex filament is demonstrated [7]. In addition to the classical solitons, the NLSE possesses rich ‘breathing’ excitations on a plane wave background, which are known as ‘breathers’ [14–16]. Such breathers are strongly associated with the modulation instability (MI) [14–18], where its nonlinear stage has been regarded as the prototype of rogue wave events [19]. Surprisingly, although the breather has been one of the central subjects in nonlinear physics, and its observation has been realised widely in many nonlinear systems [15, 20–23], the link between one special type of breather—the Akhmediev breather [24] (as well as its multiple counterpart) and vortex filaments in a quantised superfluid has been revealed only recently [8, 9]. In fact, the resulting new loop structure of the Akhmediev breather, which differs from that of bright solitons [7], provides significant contributions to our understanding of quantum fluid and superfluid turbulence. This is therefore an interdisciplinary research—in the case of quantised superfluid vortex filaments, MI, and breathers—that needs more explorations.
However, the Akhmediev breather is merely the exact description for the MI emerging from a special purely periodic perturbation [24]. There is another type of breather, describing the MI developing from localised perturbations, which has not been studied in a quantised superfluid. This includes the Kuznetsov-Ma breather [25] admitting localised single-peak perturbation and super-regular breather [16] supporting localised multi-peak perturbation. Indeed, it is recently demonstrated that the Kuznetsov-Ma breather describes not only the MI in the small amplitude regime, but also the interference between bright soliton and plane wave in the large amplitude regime [26], while the super-regular breather admits the MI growth rate that coincides with the absolute difference of group velocities of the breather [27, 28]. Given that these breathers are qualitatively different, two questions of fundamental importance now arise: How about the loop excitations triggered by these breathers? Is there a physical quantity to identify explicitly all these breather-induced loop excitations?
In this paper, we study the quantised superfluid vortex filaments induced by Kuznetsov-Ma breather and super-regular breather admitting localised perturbations. Such vortex filaments exhibit striking loop structures due to the dual action of bending and twisting of the vortex. Remarkably, an intriguing loop structure triggered by super-regular breather—the loop pair—exhibits spontaneous symmetry breaking, due to the broken reflection symmetry of the group velocities of super-regular breather. In particular, we identify explicitly these loop excitations by introducing the integral of the relative quadratic curvature, which corresponds to the effective energy of breathers. Although the nature of nonlinearity, it is demonstrated that this physical quantity shows a linear correlation with the loop size.
2. Hasimoto transformation and inverse map
For the incompressible and inviscid fluid, the Biot-Savart equation is reduced to a simpler local induction approximation (LIA) equation [29–31] by taking the leading order $ \begin{eqnarray}{\boldsymbol{v}}=\left({\rm{\Gamma }}/4\pi \right)\mathrm{ln}\left(R/{a}_{0}\right)\kappa {\boldsymbol{t}}\times {\boldsymbol{n}}=\beta \kappa {\boldsymbol{t}}\times {\boldsymbol{n}}\end{eqnarray}$here Γ is a circulation, R is local radius of curvature and a0 is the effective vortex core radius. ${\boldsymbol{v}}={\rm{d}}{\boldsymbol{r}}/{\rm{d}}{t}$ is the velocity vector of the vortex filament, ${\boldsymbol{t}}$ and ${\boldsymbol{n}}$ are unit vectors corresponding to the tangent and principal normal directions, respectively. κ, as a real function of arc length variable s and time t, represents the curvature distribution of the vortex filament. This equation makes us obtain more properties of the states related to the motion of vortex, especially in the case of Hasimoto transformation. Assuming that β is constant and making use of the Serret-Frenet equations given in [32], $ \begin{eqnarray}{\boldsymbol{r}}^{\prime} ={\boldsymbol{t}},\,\,{\boldsymbol{t}}^{\prime} =\kappa {\boldsymbol{n}},\,\,{\boldsymbol{n}}^{\prime} =\tau {\boldsymbol{b}}-\kappa {\boldsymbol{t}},\,\,{\boldsymbol{b}}^{\prime} =-\tau {\boldsymbol{n}},\end{eqnarray}$where prime denotes a differential of arc length, ${\boldsymbol{b}}$ is the binormal vector and τ is the torsion of the vortex filament, equation (1) can be transformed into a 1D scalar cubic NLSE of self-focusing type [7] $ \begin{eqnarray}{\beta }^{-1}\left({\rm{i}}{\psi }_{{t}}\right)=-{\psi }_{{ss}}-\displaystyle \frac{1}{2}| \psi {| }^{2}\psi .\end{eqnarray}$$\psi \left(s,t\right)$ is a complex function related with the local instantaneous geometric parameters curvature $\kappa \left(s,t\right)$ and torsion $\tau \left(s,t\right)$ in the context of vortices by the transformation $ \begin{eqnarray}\psi \left(s,t\right)=\kappa \left(s,t\right){{\rm{e}}}^{{\rm{i}}{\int }_{0}^{s}\tau \left(\sigma ,t\right){\rm{d}}s}.\end{eqnarray}$
The NLSE (3) possesses rich ‘breathing’ excitations [14], which provides a path for studying exactly breather-induced quantised superfluid vortex filaments. We can obtain the explicit configuration of these excitations by inverse map (see appendix A).
3. Kuznetsov-Ma breather-induced vortex filaments and exact characterisation
We first consider the Kuznetsov-Ma breather that exhibits periodic pulsating dynamics along t. Its explicit expression for equation (3) is given by $ \begin{eqnarray}\psi (s,t)=\left[1-2\displaystyle \frac{{\chi }^{2}\cos \left(\eta \beta t\right)+{\rm{i}}\eta \sin \left(\eta \beta {\text{t}}\right)}{{\kappa }_{0}b\,\cosh \left(\chi \xi \right)-{\kappa }_{0}^{2}\,\cos \left(\eta \beta t\right)}\right]{\psi }_{0},\end{eqnarray}$where $\chi =\sqrt{{b}^{2}-{\kappa }_{0}^{2}}$ with b being a real constant (b>κ0), η=bχ, and $\xi =s-2{\tau }_{0}\beta t$. Physically, b describes the oscillation period and amplitude of the Kuznetsov-Ma breather. κ0 and τ0 are real constants which denote the amplitude and wave vector of the plane wave background ψ0 respectively. The latter has the form $ \begin{eqnarray}{\psi }_{0}={\kappa }_{0}\exp \left[{\rm{i}}({\tau }_{0}{s}+\omega {t})\right],\,\omega =\beta \,{\kappa }_{0}^{2}/2-\beta \,{\tau }_{0}^{2}.\end{eqnarray}$This plane wave corresponds to a trivial uniform helical vortex without physical interest (κ0 and τ0 describe the curvature and torsion of a uniform helical vortex, respectively). In contrast, the Kuznetsov-Ma breather describes the nontrivial structure of vortex filament that has not been studied fully. From equation (5), we can readily calculate the explicit expressions of the curvature and torsion of vortex filament induced by the Kuznetsov-Ma breather, which are given respectively by $ \begin{eqnarray}\kappa ={\left[{\left({\kappa }_{0}+\displaystyle \frac{2{\chi }^{2}\cos (\eta \beta t)}{{n}_{1}}\right)}^{2}+\displaystyle \frac{4{\eta }^{2}{\sin }^{2}(\eta \beta t)}{{\left({n}_{1}\right)}^{2}}\right]}^{1/2},\end{eqnarray}$and $ \begin{eqnarray}\tau ={\tau }_{0}\left[1+\displaystyle \frac{4{\kappa }_{0}{\eta }^{2}\sin \left(\eta \beta t\right)\sinh \left(\chi \xi \right)}{{m}_{1}+{m}_{2}+{m}_{3}+{m}_{4}}\right],\end{eqnarray}$with $ \begin{eqnarray*}\begin{array}{rcl} & & {n}_{1}={\kappa }_{0}\cos \left(\eta \beta t\right)-b\cosh \left(\chi \xi \right),\\ & & {m}_{1}={\kappa }_{0}^{4}-7\kappa {0}^{2}{b}^{2}+8{b}^{4},\,{m}_{2}={\kappa }_{0}^{4}\cos \left(2\eta \beta t\right),\\ & & {m}_{3}=4{\kappa }_{0}b\left({\kappa }_{0}^{2}-2{b}^{2}\right)\cos \left(\eta \beta t\right)\cosh \left(\chi \xi \right),\\ & & {m}_{4}={\kappa }_{0}^{2}{b}^{2}\cosh \left(2\chi \xi \right).\end{array}\end{eqnarray*}$
The variations of curvature and torsion of the Kuznetsov-Ma breather on the (ξ, t) plane (note here that $\xi =s-2{\tau }_{0}\beta t$ denotes the moving frame on the group velocity), with initial conditions b=1.2, κ0=1, τ0=0.05, are shown in figures 1(a) and (c). As expected, the curvature of the Kuznetsov-Ma breather, starting from a localised non-periodic (single-peak) perturbation, evolves gradually into its maximum at t=0 (see the profiles in figure 1(b)). The curvature then exhibits periodic oscillation with the period 2π/(ηβ) as t increases (see figure 1(a)).
Figure 1.
New window|Download| PPT slide Figure 1.Temporal evolutions of curvature $\kappa \left(\xi ,t\right)$ (a) and torsion $\tau \left(\xi ,t\right)$ (c) corresponding to a Kuznetsov-Ma breather; see equations (7) and (8). (b) and (d) are variations of $\kappa \left(\xi ,t\right)$ and $\tau \left(\xi ,t\right)$ at different times. The parameters are: ${\kappa }_{0}=1$, τ0=0.05, b=1.2, and β=4π.
A notable feature is that the torsion, as a function of arc length s and time t, exhibits singular behaviour as $t\to 2\pi /(\eta \beta )$. Figure 1(d) clearly indicates that the phase 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 breather admits a π phase shift at the valleys. This phase shift results in the singular behaviour of the torsion. Note that the singular does not make the Hasimoto transformation ill-defined due to the π phase shift of nonlinear waves.
Figure 2 shows the corresponding vortex configuration of the Kuznetsov-Ma breather within one growth-decay cycle. We can see clearly from the figure that the vortex filament, emerging from an otherwise perturbed helical vortex at t=−0.314 (see figure 2(a)), exhibits a striking loop structure at t=0 due to the dual action of bending and twisting of the vortex. This loop structure disappears gradually as t increases. At t=0.314, the vortex filament recovers the initial state. This process will emerge periodically as t increases due to the feature of the Kuznetsov-Ma breather. We should note that when ${\kappa }_{0}\to 0$, the loop of the Kuznetsov-Ma breather reduces to the classical loop structure of bright solitons [7]; the periodic recurrence of the loop is gone.
Figure 2.
New window|Download| PPT slide Figure 2.Configuration of vortex filaments of Kuznetsov-Ma breathers computed by LIA for parameters given by κ0=1, τ0=0.05, b=1.2 and β=4π at different time (a) t=−0.314, (b) t=−0.157, (c) t=−0.01, (d) t=0.157 and (e) t=0.314.
The Kuznetsov-Ma breather can transform into the Peregrine rogue wave [33] with double localisation in the limit of $b\to {\kappa }_{0}$. The latter is also the limiting case of the Akhmediev breathers. All these breathers can induce loop-structure excitations on vortex filaments, as shown above and in [8, 9].
We then wonder how to identify explicitly these vortex filaments induced by breathers, since each kind of vortex filament has a similar loop structure corresponding to the maximum curvature. This is the question of fundamental importance that has not been answered before. To do this, we introduce the following physical quantity—the integral of the relative quadratic curvature of the form $ \begin{eqnarray}{\rm{\Delta }}K={\int }_{-\infty }^{\infty }\left[{\kappa }^{2}\left(s,t\right)-{\kappa }_{0}^{2}\left(s,t\right)\right]{\rm{d}}{s}.\end{eqnarray}$Equation (9) corresponds to the effective energy of breathers in optics [34]. Namely, it coincides with the energy of breathers against plane wave, i.e. ${\int }_{-\infty }^{\infty }\left({\psi }^{2}-{\psi }_{0}^{2}\right){\rm{d}}{s}$. For a quantum condensate fluid, equation (9) stands for the effective atom numbers [1]. Generally, this is a quantity of physical importance which can be monitored effectively for localised nonlinear waves in experiments [1, 35]. Here we highlight that equation (9) can be used for characterising the breather-induced vortex filaments in quantised superfluid.
It is interesting to note that for the Kuznetsov-Ma breather (5), we obtain exactly ${\rm{\Delta }}K=8\sqrt{{b}^{2}-{\kappa }_{0}^{2}}$, which indicates ΔK>0; while for the Peregrine rogue wave and the Akhmediev breather, we find that ΔK=0 (see appendix B). This is the immanent reason why the loop structure induced by the Kuznetsov-Ma breather exhibits periodic oscillation as t increases, while the loop structure triggered by the Peregrine rogue wave and the Akhmediev breather appears only once during the time evolution. On the other hand, the condition ΔK=0 indicates that the resulting vortex filament starts from a uniform helical vortex structure. This corresponds to the case of the Peregrine rogue wave and the Akhmediev breather. However, the uniform helical vortex structure will never appear for the vortex filament induced by the Kuznetsov-Ma breather.
Let us take a closer look at equation (9) by considering the relation between ΔK and the Kuznetsov-Ma breather-induced loop structure. To this end, we define the characteristic size of the loop structure, rk, which describes the minimum radius of the structure.
Figure 3 shows the relation between ΔK(κ0, t) and r(κ0, t) on logarithmic coordinates with random values of b in the region $b\in [2{\kappa }_{0},\infty ]$. This parameter condition allows us to study the qualitative link between ΔK(κ0, t) and rk(κ0, t) from the vortex filaments induced by random Kuznetsov-Ma breathers (i.e. a series of Kuznetsov-Ma breathers with random period and amplitude).
Figure 3.
New window|Download| PPT slide Figure 3.Relations between ${\rm{\Delta }}K({\kappa }_{0},t)$ and rk(κ0, t) on logarithmic coordinates ($\mathrm{ln}{\rm{\Delta }}K$, $\mathrm{ln}{r}_{k}$) (a) as κ0 varies with fixed t=−0.01; (b) as t varies with fixed κ0=1. The solid lines are a precise description of the relation between $\mathrm{ln}{\rm{\Delta }}K$ and $\mathrm{ln}{r}_{k}$ as $b\to \infty $. The values of the parameter b are random numbers in the region b ∈ [2κ0, 30κ0].
For the fixed t (t = 0), we show the characteristics of ln(ΔK) and $\mathrm{ln}({r}_{k})$ with increasing κ0 in figure 3(a). It is interesting that, despite the random values of b, $\mathrm{ln}({\rm{\Delta }}K)$ decreases linearly as $\mathrm{ln}({r}_{k})$ increases and the corresponding rates α are exactly consistent at a fixed time (α=−1). The similar linear relation also holds for the case with fixed κ0 and τ0 and variational t, as shown in figure 3(b).
We then explain the linear relation above exactly. We note that for the Kuznetsov-Ma-breather-induced loop structure, the minimum loop radius rk is inversely proportional to the maximum curvature κm, i.e. $ \begin{eqnarray*}{r}_{k}=\displaystyle \frac{1}{{\kappa }_{m}}.\end{eqnarray*}$It is given explicitly by equation (7) at ξ=0: $ \begin{eqnarray}{r}_{k}={\left[{\left({\kappa }_{0}+\displaystyle \frac{2{\chi }^{2}\cos (\eta \beta t)}{{n}_{2}}\right)}^{2}+\displaystyle \frac{4{\eta }^{2}{\sin }^{2}(\eta \beta t)}{{\left({n}_{2}\right)}^{2}}\right]}^{-1/2}\end{eqnarray}$with ${n}_{2}={\kappa }_{0}\cos \left(\eta \beta t\right)-b$. Here rk is the function of b, κ0 and t. Thus the accurate description of ${\rm{\Delta }}K\cdot {r}_{k}$ reads $ \begin{eqnarray}{\rm{\Delta }}K\cdot {r}_{k}=\displaystyle \frac{8\sqrt{{b}^{2}-{\kappa }_{0}^{2}}}{{\left[{\left({\kappa }_{0}+\tfrac{2{\chi }^{2}\cos (\eta \beta t)}{{n}_{2}}\right)}^{2}+\tfrac{4{\eta }^{2}{\sin }^{2}(\eta \beta t)}{{\left({n}_{2}\right)}^{2}}\right]}^{1/2}}.\end{eqnarray}$Clearly, for the case of the Kuznetsov-Ma breather, ${\rm{\Delta }}K\cdot {r}_{k}\ne 0;$ while for the case of the Peregrine rogue wave and Akhmediev breather, ${\rm{\Delta }}K\cdot {r}_{k}=0$, since ΔK=0.
We show the profile of equation (11) as b increases in figure 4. We can see that ${\rm{\Delta }}K\cdot {r}_{k}$ increases monotonously with increasing b. Remarkably, as $b\to \infty $, we find ${\rm{\Delta }}K\cdot {r}_{k}\to 4$. Indeed, we can readily obtain a simple relation from equation (11) as $b\to \infty $ $ \begin{eqnarray}{\rm{\Delta }}K\cdot {r}_{k}=4.\end{eqnarray}$Namely, $ \begin{eqnarray}\mathrm{ln}{\rm{\Delta }}K=-\mathrm{ln}{r}_{k}+\mathrm{ln}4,\end{eqnarray}$on logarithmic coordinates.
Figure 4.
New window|Download| PPT slide Figure 4.Profile of ${\rm{\Delta }}K\cdot {r}_{k}$, equation (11) as b increases. Other parameters are κ0=1, $t=2n\pi /(\eta \beta )$ (n is an integer) and β=4π.
We show the linear relation by the solid lines in figure 3. Observably, the numerical results are in good agreement with the analytical relation (12) (solid line). Physically, the Kuznetsov-Ma breather in the region $b\in [2{\kappa }_{0},\infty ]$ can be approximately described by the linear interference between a bright soliton and a plane wave [26]. As $b\to \infty $ ($b\gg {\kappa }_{0}$), i.e. the amplitude of the bright soliton is much bigger than that of the plane wave, the plane wave can be neglected. As a result, the effective energy ΔK is quadruple the amplitude of the remaining bright soliton, which directly leads to equation (12).
4. Super-regular breather-induced loop pair and symmetry breaking
Let us then consider the vortex filament induced by the super-regular breather. The latter, which recently serves as the exact MI scenario excited from localised multi-peak perturbations, is formed by the nonlinear superposition of two quasi-Akhmediev breathers [16, 27, 28, 36–38]. The exact solution of super-regular breather with τ0=0 is first provided in [16]. However, the general solution with ${\tau }_{0}\ne 0$ in the infinite NLSE is presented recently in [27]. By using the transformation above and the super-regular solution in [27], the corresponding properties of vortex filaments can be achieved effectively. Here we omit the tedious explicit expression but show the important and compact results. At the first step, the integral of the relative quadratic curvature of super-regular breather can be obtained explicitly from the exact solution in appendix C. It reads $ \begin{eqnarray}{\rm{\Delta }}K=16{\kappa }_{0}\left[\varepsilon \,\cos \phi +\pi \,\sin \phi \,\mathrm{csch}\left(\displaystyle \frac{\pi \,\sin \phi }{\varepsilon \,\cos \phi }\right)\right],\end{eqnarray}$where $\varepsilon =R-1$ and $\varepsilon \ll 1$. R(>1) and $\phi [\in (-\pi /2,\pi /2)]$ are two real parameters that denote respectively the radius and angle in polar coordinates (see appendix C). Physically, R (or ϵ) and φ are two important parameters that describe directly the amplitude and period of the super-regular breathers. It is therefore crucial to study the property of vortex filaments induced by super-regular breathers by the choice of parameters R and φ.
Just as in the case of the Kuznetsov-Ma breather, equation (14) is also greater than zero, i.e. ΔK>0. This indicates that the super-regular breather also admits long-time dynamics which is different from the Peregrine rogue wave and Akhmediev breather. Unlike the case of the Kuznetsov-Ma breather, the evolutions of curvature and torsion of the super-regular breather exhibit remarkably different characteristics. This stems from the fact that the super-regular breather possesses a localised multi-peak perturbation rather than a localised single-peak perturbation.
Figure 5 shows the variation of curvature and torsion induced by the super-regular breather with the initial parameters κ0=1, R=1.1, and φ=π/8. It can be seen from figure 5(a) that the curvature of a super-regular breather triggered from a localised multi-peak perturbation at t=0 (see figure 5(b)) increases gradually due to the exponential amplification of the MI at the linear stage. It reaches its maximum at t=0.43 and then splits into two quasi-Akhmediev breathers propagating along different directions during the nonlinear stage of MI. The corresponding torsion also suffers singular behaviour starting from the maximum curvature point t=0.43 (see figure 5(c)). Interestingly, the nonlinear propagation stage always holds the singular torsion at the maximum curvature point as t>0.43 (see figure 5(d)).
Figure 5.
New window|Download| PPT slide Figure 5.Temporal evolutions of curvature $\kappa \left(\xi ,t\right)$ (a) and torsion $\tau \left(\xi ,t\right)$ (c) corresponding to a super-regular breather, see equation (C2) in appendix C. (b) and (d) are variations of $\kappa \left(\xi ,t\right)$ and $\tau \left(\xi ,t\right)$ at different times. Other parameters are κ0=1, τ0=0.05, R=1.1 and φ=π/8.
Figure 6 displays the corresponding vortex structure at t=0, t=0.43, and t=1.2, respectively. We can see clearly that the vortex filament emerges from a perturbed helical vortex at t=0 and then exhibits a remarkable loop structure at t=0.43. This is the linear MI stage that corresponds to one loop excitation. Interestingly, once the vortex filament evolves into the nonlinear stage, the loop structure splits into a loop pair which corresponds to the two quasi-Akhmediev breathers propagation with different group velocities.
Figure 6.
New window|Download| PPT slide Figure 6.Configuration of vortex filaments of super-regular breathers at different times (a) t=0, (b) t=0.43 and (c) t=1.2. Other parameters are the same as in figure 5.
In particular, we find that the loop pair induced by the super-regular breather at the nonlinear stage shows an interesting reflection symmetry breaking, as shown in figure 7. We find that this remarkable feature comes from the asymmetry of the group velocities of the two quasi-Akhmediev breathers. Indeed, the group velocities of the super-regular breather are given by (see equation (C4) in appendix C) $ \begin{eqnarray}{V}_{g1}=2\beta {\tau }_{0}+d,\,\,{V}_{g2}=2\beta {\tau }_{0}-d,\end{eqnarray}$where $d=\beta {\kappa }_{0}\tfrac{\left({R}^{4}+1\right)}{{R}^{3}-R}\sin \phi $. Clearly, due to ${\tau }_{0}\ne 0$, the absolute values of these two group velocities are always unequal. Once κ0, ϵ, and φ are fixed, the degree of the asymmetry is proportional to the value of $| {\tau }_{0}| $.
Figure 7.
New window|Download| PPT slide Figure 7.Top view of configuration of vortex filaments of super-regular breathers at a fixed time t=1.2 as τ0 increases. (a) τ0=0.01, (b) τ0=0.17 and (c) τ0=0.24. We can see clearly the reflection symmetry breaking of the loop pair with non-zero τ0. Other parameters are κ0=1, R=1.1 and φ=π/8.
Figure 7 shows the corresponding vortex structures induced by super-regular breather as τ0 increases. We see that as ${\tau }_{0}\to 0$ the resulting loop pair exhibits quasi-reflection symmetry (figure 7(a)), while the reflection symmetry of the loop pair breaks greatly with increasing τ0 (figures 7(b) and (c)).
It is very interesting to note that, despite the broken reflection symmetry as ${\tau }_{0}\ne 0$, the growth rate of modulation instability driven by the super-regular breather does not depend on τ0. Namely, this growth rate is only associated with the absolute difference of the group velocities, $G={\eta }_{r}| {V}_{g1}-{V}_{g2}| $ with ${\eta }_{r}=\tfrac{a}{2}\left(R-1/R\right)\cos \phi $, as shown in [27]. This result is physically important because although the super-regular breather-induced vortex structures can exhibit different loop pairs with symmetry breaking, the inherent MI property can remain invariable.
Finally, we consider the relation between ΔK and characteristic size rs of the super-regular breather-induced vortex structures. Similar to the case of the Kuznetsov-Ma breather, we define characteristic size rs as the minimum radius of the super-regular breather-induced vortex structure throughout the whole evolution. Thus, the characteristic size rs, which is also inversely proportional to the maximum curvature κms (i.e. ${r}_{s}=1/{\kappa }_{{ms}}$), is given by $ \begin{eqnarray}{r}_{s}={\left[{\kappa }_{0}+{\kappa }_{0}\left(1+\varepsilon +\displaystyle \frac{1}{1+\varepsilon }\right)\cos \phi \right]}^{-1},\end{eqnarray}$where $\varepsilon =R-1$ is a small value ($\varepsilon \ll 1$) defined above.
Collecting equations (14) and (16), we obtain the explicit expression of ${\rm{\Delta }}K\cdot {r}_{s}$ by omitting the high-order term O(ϵ2). It reads $ \begin{eqnarray}{\rm{\Delta }}K\cdot {r}_{s}={\alpha }_{s}\varepsilon ,\end{eqnarray}$where ${\alpha }_{s}=16\cos \phi /\left(1+2\cos \phi \right)$.
In contrast to the case of the Kuznetsov-Ma breather, equation (11), where only one parameter b can be modulated when the plane wave parameters (κ0, τ0) and the structural parameter β are fixed, equation (17) has two free physical parameters (ϵ and φ). But even so, we highlight that linear relations can also hold for the case of super-regular breather-induced vortex structures.
Figure 8 shows the characteristics of ${\rm{\Delta }}K\cdot {r}_{s}$ as ϵ increases with different values of φ. In particular, we compare the results obtained from the approximate expression equation (17) (the solid lines) and the exact expression (the dotted lines), respectively. We can see that for each fixed φ, ${\rm{\Delta }}K\cdot {r}_{s}$ shows a linear relation with ϵ. The corresponding rate αs decreases in the range of [48,0] as φ increases from 0 to π/2.
Figure 8.
New window|Download| PPT slide Figure 8.Relations between ${\rm{\Delta }}K\cdot {r}_{s}$ and ϵ of the vortex filament induced by the super-regular breather as φ varies. Other parameters are κ0=1. The discrete points are obtained with the high-order term O(ϵ2) considered, while the coloured lines retain the first-order term only.
Figure 9 shows the characteristics of ${\rm{\Delta }}K\cdot {r}_{s}$ as φ decreases with different values of ϵ. For each case with fixed ϵ, ${\rm{\Delta }}K\cdot {r}_{k}$ increases monotonously with decreasing φ. As $\phi \to 0$, we obtain that ${\rm{\Delta }}K\cdot {r}_{k}\to 16\varepsilon /3$. This is similar with the case of the Kuznetsov-Ma breather-induced vortex structures shown in figure 4. Physically, as $\phi \to 0$, the super-regular breather transforms itself into two colliding Kuznetsov-Ma breathers, so that the similar linear relation can be maintained.
Figure 9.
New window|Download| PPT slide Figure 9.Relations between ${\rm{\Delta }}K\cdot {r}_{s}$ and φ of the vortex filament induced by the super-regular breather as ϵ varies. Other parameters are κ0=1.
5. Conclusion
In summary, we have investigated the superfluid vortex filaments induced by the Kuznetsov-Ma breather and super-regular breather, which admit localised perturbations. We have shown that the loop structure induced by the Kuznetsov-Ma breather emerges periodically as time increases, while the loop structure triggered by the super-regular breather—the loop pair—exhibits striking symmetry breaking due to the broken reflection symmetry of the group velocities of the super-regular breather. To characterise these loop excitations, we have introduced the integral of the relative quadratic curvature and demonstrated that this physical quantity shows a linear correlation with the characteristic size of the loop structure.
Note that both of the above vortex structures exist in a non-dissipative system. Actually, at finite temperatures, the vortex filaments are also affected by the mutual friction[11, 39, 40]. It is our future subject to study how the mutual friction affects the dynamics of different vortex structures, and this research will deepen our understanding of quantised superfluid vortices.
Acknowledgments
This work has been supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 11 705 145, 11 875 220, 11 947 301, 11 434 013, and 11 425 522), Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2018JQ1003), and the Major Basic Research Program of Natural Science of Shaanxi Province (Grant Nos. 2017KCT-12, 2017ZDJC-32).
The position vector of the vortex filament
We give the explicit expression of the position vector of the vortex filament by integrating the Frenet-Serret equations given in [32] and the exact expression is formulated explicitly in [41], which should be represented as $ \begin{eqnarray*}\begin{array}{rcl}{\boldsymbol{r}}\left(s,t\right) & = & \left[\begin{array}{c}x\left(s,t\right)\\ y\left(s,t\right)\\ z\left(s,t\right)\end{array}\right]\\ & = & \left[\begin{array}{c}{x}_{0}\left(t\right)+{\sum }_{k=1}^{3}{c}_{k1}\left(t\right){\displaystyle \int }_{0}^{s}{M}_{k}\left(\sigma ,t\right){\rm{d}}\sigma \\ {y}_{0}\left(t\right)+{\sum }_{k=1}^{3}{c}_{k2}\left(t\right){\displaystyle \int }_{0}^{s}{M}_{k}\left(\sigma ,t\right){\rm{d}}\sigma \\ {z}_{0}\left(t\right)+{\sum }_{k=1}^{3}{c}_{k3}\left(t\right){\displaystyle \int }_{0}^{s}{M}_{k}\left(\sigma ,t\right){\rm{d}}\sigma \end{array}\right],\end{array}\end{eqnarray*}$ here, ${x}_{0}\left(t\right)$, ${y}_{0}\left(t\right)$, ${z}_{0}\left(t\right)$ are constants with respect to the initial position of vortex structures. ${M}_{k}\left(k=1,2,3\right)$ are $ \begin{eqnarray*}{M}_{1}=\displaystyle \frac{{\gamma }^{2}+{\alpha }^{2}\cos \lambda }{{\lambda }^{2}},{M}_{2}=\displaystyle \frac{\alpha \sin \lambda }{\lambda },{M}_{3}=\displaystyle \frac{\alpha \gamma \left(1-\cos \lambda \right)}{{\lambda }^{2}},\end{eqnarray*}$ where $\alpha ={\int }_{0}^{s}\kappa \left(\sigma ,t\right){\rm{d}}\sigma $, $\gamma ={\int }_{0}^{s}\tau \left(\sigma ,t\right){\rm{d}}\sigma $, and $\lambda \,=\sqrt{{\left({\int }_{0}^{s}\kappa \left(\sigma ,t\right){\rm{d}}\sigma \right)}^{2}+{\left({\int }_{0}^{s}\tau \left(\sigma ,t\right){\rm{d}}\sigma \right)}^{2}}$.
Explicit expressions of ΔK of the Akhmediev breather and Peregrine rogue wave
ΔK, the integral of the relative quadratic curvature, is expressed explicitly in the form $ \begin{eqnarray}{\rm{\Delta }}K={\int }_{-\infty }^{\infty }\left[{\kappa }^{2}\left(s,t\right)-{\kappa }_{0}^{2}\left(s,t\right)\right]{\rm{d}}{s},\end{eqnarray}$here, $\kappa \left(s,t\right)$ and ${\kappa }_{0}\left(s,t\right)$ represent the curvature distribution of the vortex filament and the curvature of the background uniform helical vortex corresponding to the plane wave ψ0 (6) respectively.
As mentioned in section 3, in addition to the Kuznetsov-Ma breather, the plane wave (6) also admits other breathing waves, including the Akhmediev breather and the Peregrine rogue wave. As a comparison, we show here ΔK for the vortex filaments induced by the Akhmediev breather and Peregrine rogue wave.
We first consider the Akhmediev breather that exhibits the explicit description for the MI emerging from periodic perturbations. Its exact expression is given by $ \begin{eqnarray}{\psi }_{A}(s,t)=\left[1-2\displaystyle \frac{{\chi }_{1}^{2}\cosh \left({\eta }_{1}\beta t\right)+{\rm{i}}{\eta }_{1}\sinh \left({\eta }_{1}\beta {t}\right)}{{\kappa }_{0}^{2}\,\cos \left({\eta }_{1}\beta t\right)-{\kappa }_{0}b\,\cosh \left({\chi }_{1}\xi \right)}\right]{\psi }_{0},\end{eqnarray}$where ${\chi }_{1}=\sqrt{{\kappa }_{0}^{2}-{b}^{2}}$ with b<κ0, η1=bχ1, and $\xi =s\,-2{\tau }_{0}\beta t$. The corresponding exact expression of the curvature is given by $ \begin{eqnarray}{\kappa }_{A}={\left[{\left({\kappa }_{0}-\displaystyle \frac{2{\chi }_{1}^{2}\cosh ({\eta }_{1}\beta t)}{A}\right)}^{2}+\displaystyle \frac{4{\eta }_{1}^{2}{\sinh }^{2}({\eta }_{1}\beta t)}{{A}^{2}}\right]}^{1/2}\end{eqnarray}$with $A={\kappa }_{0}\cosh \left({\eta }_{1}\beta t\right)-b\cos \left({\chi }_{1}\xi \right)$. A substitution of equation (B3) into equation (B1) yields ΔKA=0.
We then consider the Peregrine rogue wave with double localisation. The latter corresponds to the limiting case of equation (B2) as $b\to {\kappa }_{0}$. Its exact expression is given by $ \begin{eqnarray}{\psi }_{P}\left(s,t\right)=\left[1-\displaystyle \frac{4{\rm{i}}{\kappa }_{0}^{2}\beta {t}+4}{1+{\kappa }_{0}^{4}{\beta }^{2}{t}^{2}+{\kappa }_{0}^{2}{\left(s-2\beta {\tau }_{0}t\right)}^{2}}\right]{\psi }_{0},\end{eqnarray}$whose curvature is in the form of $ \begin{eqnarray}{\kappa }_{P}={\kappa }_{0}\sqrt{\displaystyle \frac{16{\kappa }_{0}^{4}{\beta }^{2}{t}^{2}}{{a}^{2}}+{\left(1-\displaystyle \frac{4}{a}\right)}^{2}}\end{eqnarray}$with $a=1+{\kappa }_{0}^{4}{\beta }^{2}{t}^{2}+{\kappa }_{0}^{2}{\left(s-2\beta {\tau }_{0}t\right)}^{2}$. By calculating equation (B1), we demonstrate also that ΔKP=0.
As a result, both the Akhmediev breather and the Peregrine rogue wave share the vanishing ΔK, which indicates that the corresponding vortex filaments start from a uniform helical vortex structure.
Explicit expression of the super-regular breather
The explicit expression of the super-regular breather for equation (3) is given by the Darboux transformation [27], where the spectral parameter λ is parameterised by the Jukowsky transform [16] as follows: $ \begin{eqnarray}\lambda ={\rm{i}}\displaystyle \frac{{\kappa }_{0}}{2}\left({\rm{\Delta }}+\displaystyle \frac{1}{{\rm{\Delta }}}\right)-\displaystyle \frac{{\tau }_{0}}{2},\,{\rm{\Delta }}={R}{{\rm{e}}}^{{\rm{i}}\phi },\end{eqnarray}$here, R and φ define the location of the spectral parameter λ in the polar coordinates. They represent radius and angle respectively in the region R>1 and $\phi \in \left(-\pi /2,\pi /2\right)$. For τ0=0, equation (C1) reduces to the spectral parameter used in [16]. With different values of R and φ, the resulting exact solution can describe different breather dynamics [16]. A more general phase diagram of breathers has been obtained recently in [42]. Here we consider the super-regular breather formed by two quasi-Akhmediev breathers with ${R}_{1}={R}_{2}=R=1+\varepsilon $ ($\varepsilon \ll 1$), φ1=−φ2=φ. Its explicit expression of the solution is in the form: $ \begin{eqnarray}\psi \left(s,t\right)={\psi }_{0}\left[1-4\rho \varrho \displaystyle \frac{\left({\rm{i}}\varrho -\rho \right){{\rm{\Xi }}}_{1}+\left({\rm{i}}\varrho +\rho \right){{\rm{\Xi }}}_{2}}{{\kappa }_{0}\left({\rho }^{2}{{\rm{\Xi }}}_{3}+{\varrho }^{2}{{\rm{\Xi }}}_{4}\right)}\right],\end{eqnarray}$here $ \begin{eqnarray*}\begin{array}{rcl} & & \varrho =\displaystyle \frac{{\kappa }_{0}}{2}\left(R-\displaystyle \frac{1}{R}\right)\sin \phi ,\,\rho =\displaystyle \frac{{\kappa }_{0}}{2}\left(R+\displaystyle \frac{1}{R}\right)\cos \phi \\ & & {{\rm{\Xi }}}_{1}={\varphi }_{21}{\phi }_{11}+{\varphi }_{22}{\phi }_{21},\,\,\,{{\rm{\Xi }}}_{2}={\varphi }_{11}{\phi }_{21}+{\varphi }_{21}{\phi }_{22},\\ & & {{\rm{\Xi }}}_{3}={\varphi }_{11}{\phi }_{22}-{\varphi }_{21}{\phi }_{12}-{\varphi }_{12}{\phi }_{21}+{\varphi }_{22}{\phi }_{11},\\ & & {{\rm{\Xi }}}_{4}=\left({\varphi }_{11}+{\varphi }_{22}\right)\left({\phi }_{11}+{\phi }_{22}\right),\end{array}\end{eqnarray*}$with $ \begin{eqnarray*}\begin{array}{rcl} & & {\phi }_{{jj}}=\cosh \left({{\rm{\Theta }}}_{2}\mp {\rm{i}}\psi \right)-\cos \left({{\rm{\Phi }}}_{2}\mp \phi \right),\\ & & {\varphi }_{{jj}}=\cosh \left({{\rm{\Theta }}}_{1}\mp {\rm{i}}\psi \right)-\cos \left({{\rm{\Phi }}}_{1}\mp \phi \right),\\ & & {\phi }_{j3-j}=\pm i\cosh \left({{\rm{\Theta }}}_{2}\mp {\rm{i}}\phi \right)-\cos \left({{\rm{\Phi }}}_{2}\mp \theta \right),\\ & & {\varphi }_{j3-j}=\pm i\cosh \left({{\rm{\Theta }}}_{1}\mp {\rm{i}}\phi \right)-\cos \left({{\rm{\Phi }}}_{1}\mp \theta \right),\end{array}\end{eqnarray*}$where $\theta =\arctan \left[\left(1-{{iR}}^{2}\right)/\left(1+{R}^{2}\right)\right]$. ${{\rm{\Theta }}}_{j}$ and φj are related with group and phase velocities respectively, which is in the form of $ \begin{eqnarray}{{\rm{\Theta }}}_{j}=2{\eta }_{r}\left(s-{V}_{{gj}}t\right),\,{\phi }_{j}=2{\eta }_{{ij}}\left(s-{V}_{{pj}}t\right),\end{eqnarray}$where $ \begin{eqnarray}\begin{array}{rcl} & & {\eta }_{i1}=-{\eta }_{i1}=\displaystyle \frac{\kappa }{2}\left(R+\displaystyle \frac{1}{R}\right)\sin \phi ,\\ & & {\eta }_{r}=\displaystyle \frac{\kappa }{2}\left(R-\displaystyle \frac{1}{R}\right)\cos \phi ,\\ & & {V}_{p1}=2\beta {\tau }_{0}-{d}_{1},\,{V}_{p2}=2\beta {\tau }_{0}+{d}_{2},\\ & & {V}_{g1}=2\beta {\tau }_{0}+d,\,{V}_{g2}=2\beta {\tau }_{0}-d\end{array}\end{eqnarray}$with ${d}_{1}=\beta {\kappa }_{0}\left(R-\tfrac{1}{R}\right)\tfrac{\cos \left(2\phi \right)}{\sin \phi }$, ${d}_{2}=\beta {\kappa }_{0}\tfrac{\left(R-\tfrac{1}{R}\right)}{\sin \phi }$ and $d\,=\beta {\kappa }_{0}\tfrac{\left({R}^{4}+1\right)}{{R}^{3}-R}\sin \phi $. The initial state of the super-regular breather can be extracted from the above solution at t=0. It reads, as $\varepsilon \to 0$, $ \begin{eqnarray}\psi \left(s,0\right)={\psi }_{0}\left(1-{\rm{i}}\displaystyle \frac{4\varepsilon \cos \phi \cos \left({\kappa }_{0}s\sin \phi \right)}{\cosh \left({\kappa }_{0}\varepsilon s\cos \phi \right)}\right).\end{eqnarray}$Note that for a given plane wave background (6) (i.e. κ0 and τ0 are fixed), R and φ determine the amplitude and period of the breathers. In particular, R and φ effect the profile of the initial state (C5) of super-regular breathers.
The integral of the relative quadratic curvature of the super-regular breather ΔK (see equation (14)) is obtained explicitly from the initial state (C5), since the NLSE shares the same ΔK at different times.
,1,2,3,6,7, Wei Zhao1,5, Zhan-Ying Yang1,2,3,6, Wen-Li Yang1,2,3,41School of Physics, 
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