The dispersion relation of a dark soliton

The energy-velocity relation of a dark soliton is usually derived by its exact solution, which has been used to explain the kinetic motion of the dark soliton widely in many-body physical systems. We perform a variational method to re-derive the dispersion relation, with the consideration that the number of particles of the dark soliton could be conserved. The re-derived dispersion relation is completely different from that given by the exact dark soliton solution. The validity of these two dispersion relations is tested by observing the motion of the dark soliton when we drive impurity atoms that coupled with the soliton. The results suggest that the dispersion relation given by the exact solution usually works better than the one with particle number conservation. This motivates us to reveal that density waves (carrying particle transport) are generated during the acceleration process of a dark soliton, in addition to the previously known sound waves (only carrying energy transport). We further show that the density wave emissions of dark solitons can be inhibited by increasing the impurity atom number, which is trapped by the dark soliton through nonlinear coupling. The discussion is meaningful for investigating and understanding the kinetic motion of dark solitons in many different circumstances.


Introduction
The dispersion relation plays a very important role in condensed matter physics. For example, it can be used to define the inertial (or effective) mass, which captures the response of a particle or quasi-particle to an applied force [1]. This has been used to explain the well-known Bloch oscillation [2,3]. Recently, it is further shown that the Bloch oscillation can be used to measure the dispersion relation in an ultra-cold gas [4]. On the other hand, engineering atomic dispersion relations can be used to find new matter phases [5][6][7] and new solitons [8][9][10][11][12] in Bose-Einstein condensates. The dispersion relations are usually calculated based on an effective linear approximation. There are many different particle-like collective excitations in many-body physical systems, such as bright solitons, dark solitons, and complex solitons. Their excitation energy and effective momentum can also be defined [13], which can be used to calculate the dispersion relation of these nonlinear quasi-particles [14,15].
A dark soliton is a spatially localized density notch on top of a finite background, accompanied by a phase step through the notch [16]. In order to discuss the effective mass and related kinetic dynamics of dark solitons [17][18][19][20][21][22][23][24][25][26][27][28][29][30], it is essential to characterize the dispersion relation of a dark soliton. We note that the usual dark soliton energy is calculated based on the exact dark soliton solution (EDSS) [19][20][21][22][23][24]. The acceleration process of dark soliton has been shown to generate sound waves [31][32][33][34][35]. Note that sound waves only transport energy, and do not carry any particle transport. If the acceleration process only generates sound waves, this means that the particle number of dark soliton is conserved. But we find that the particle number is not conserved for EDSS. We try to re-derive the dispersion relation of the dark soliton by performing a variational method, assuming that the particle number is conserved during the acceleration process. This is meaningful for understanding the dispersion relation and acceleration dynamics of dark solitons in a comprehensive and thorough way.
We investigate the dispersion relations of the dark soliton given by EDSS and the variational method. The two dispersion relations both predict negative mass of dark soliton. But the two dispersion relations predict different trajectories of dark soliton, based on the local density approximation method. We suggest that it is convenient to check which one is reasonable by drawing the moving curves on a soliton position vs. velocity plane. With the aid of soliton-impurity coupling systems, it is possible to drive the dark soliton by adding force to the impurity atoms, without affecting its background wholly. The motion of the dark soliton indicates that its inertial mass is indeed negative (moving against the direction of the force). The dispersion relations given by EDSS work better for predicting dark soliton's motion, when the impurity atoms are much less than the particle number of the dark soliton. This character motivates us to uncover that density waves are generated during the acceleration process of the dark soliton, except the previously known sound waves. Our further numerical simulations indicate that more impurity atoms can inhibit the density wave emission during the acceleration process of dark soliton. This character well explains why the spin soliton with positive-negative mass transition can be well described by the variational method rather than the exact spin soliton solution [36,37]. This paper is organized as follows. In section 2, we first review the well-known dark soliton solution, and derive the dispersion relation of the dark soliton by a variational method while keeping dark soliton's particle number conserved. Next, we present the inertial mass of the dark soliton, which clearly shows the differences between our variational results and the those given by EDSS. Possibilities to test the dispersion relation of the dark soliton are discussed, with the aid of soliton-impurity coupled systems. Then in section 3, we further suggest that density waves are generated during the acceleration process. The density wave emissions can be inhibited by increasing the number of impurity atoms trapped in the dark soliton. Finally, our conclusions and discussions are given in section 4.

Dispersion relation of a dark soliton
We follow the usual mean-field approach to describe the dynamics of a scalar Bose-Einstein condensate, defining ψ as the condensate wave function. Fixing by means of re-scaling the atomic mass and Planck's constant to be 1, the dynamical equation for a quasi-one-dimensional condensate system with repulsive interactions can be written as The EDSS can be written as follows where v denotes the moving velocity of the dark soliton on a uniform density background. We would like to review the dispersion relation given by the EDSS, and then derive the dispersion relation of the dark soliton by the variational method, keeping the number of particles of the dark soliton conserved.

Two forms of the dispersion relation
The excitation energy of a dark soliton can be written as which was suggested by Kivshar and Yang [17]. It can be derived from the free energy of a grand canonical ensemble while preserving the chemical potential [20,21], or from the fundamental definition of excitation energy while keeping particle number conserved [38]. Based on this, the canonical momentum of the dark soliton can be derived as [22] P c =ˆv Then we can obtain the dispersion relation for the dark soliton from EDSS. The energy-velocity relation or dispersion relation given by EDSS has been widely used to explain and predict the motion of a dark soliton in external potentials [18][19][20][21][22][23]. The experimental observation of the dark soliton's motion was realized in a cigar-shaped condensate system [39][40][41][42]. The dark soliton corresponds to a local particle loss, one can define its the particle number of a dark soliton as where x c denotes soliton's center position, and the integral interval L should be comparable to the soliton scale. We can obtain N s = 2 √ 1 − v 2 analytically with L → ∞, when the system admits one ideal EDSS with no other waves. In fact, we can choose L ≈ 10 to calculate the dark soliton's particle number numerically, due to the localized nature of dark solitons. One can see that the particle number N s depends on its moving speed. The acceleration process of dark soliton has been shown to generate sound waves [31][32][33][34][35]. If there only exists sound waves, the particle number of the dark soliton should be conserved. Moreover, the acceleration process of the dark soliton usually generates sound waves, and undergoes deformation, i.e. it will have varying amplitude, width, and velocity due to acceleration. These characteristics make the soliton profile form usually be different from the EDSS's profile [36,37]. Therefore we try to re-derive the dispersion relation of the dark soliton by performing a variational method, assuming that N s is conserved during the acceleration process.
With the dark soliton's particle number conserved [i.e. in this case, the particle number of a dark soliton , we obtain the relations between the soliton energy and the moving velocity by the variational method as (see appendix for details) where p(t) varies in [−1, 1] and v 0 denotes the initial velocity of the dark soliton. The relations between the soliton energy and its moving velocity are shown in figure 1(a). We choose three different initial velocities v 0 = 0 (red dashed line), v 0 = 0.5 (green dotted line), v 0 = 0.8 (purple dash-dotted line), and the ones given by EDSS (black solid line). For low initial moving speed (smaller than half sound speed), the variational results agree well with those based on EDSS for moving speed smaller than 80% sound speed, but they deviate a bit largely for other regime speeds. These features mean that the two dispersion forms predict almost indistinguishable kinetic motion for dark solitons with velocities much lower than the sound speed [19][20][21][22][23][24]. However, their deviation becomes larger for dark soliton with larger initial moving speed. For example, the variational results deviates greatly from EDSS results for almost all moving speed, when the initial moving speed is v 0 = 0.8. From the results for v 0 = 0.8 in figure 1, we see that the soliton energy for the static dark soliton, obtained by accelerating an initial soliton with v 0 = 0.8, is much larger than the static dark soliton energy given by EDSS. This can be well understood by investigating their density and phase profiles. The dark soliton solution derived by variational results can predict the density and phase profiles for arbitrary velocity v, with given initial velocity v 0 . For example, we consider a case where v = 0 is decelerated from v 0 = 0.8. The expression for the static dark soliton can be given as ψ s = tanh{[x − b(t)]/0.6}e iθ0(t) , through solving the p(t) and w(t) by the variational method. In contrast, the EDSS gives ψ s = tanh[x − b(t)]e −it for the static dark soliton. The width of the static dark soliton decelerated from v 0 = 0.8 is much smaller than that the one given by EDSS. This brings the soliton energy difference for v 0 = 0.8 in figure 1. The canonical momentum of the dark soliton can be calculated from P c =´v 0 dE s /(v dv) dv. Then, we can obtain the dispersion relation of the dark soliton with different initial velocities in figure 1(b). We can see that the dispersion relations given by our variational method deviate more obviously from those given by EDSS, for dark soliton with larger initial moving speed.

Inertial mass character of a dark soliton
The concept of inertial mass is widely used in condensed matter physics: it captures the response of a quasi-particle in an interacting system to an applied force, encapsulating the emergent Newton's equations of quasi-particle dynamics [1]. The inertial mass of a soliton can also be defined [13,36,37,43]. The inertial mass [15,43] of a dark soliton can be calculated by The mass character is demonstrated in figure 2 for dark solitons with different initial velocities, compared with the mass character given by EDSS. The inertial mass of the dark soliton indeed admits negative value [24][25][26][27], which is directly supported by the following numerical simulation of the dark soliton's motion.
Recently, the positive mass and negative mass of the soliton have been discussed in a spin-orbit coupled Bose-Einstein condensate (BEC) [12] and an optical-lattice potential [44], which is induced by the effective mass of single atom (single-particle band dispersion). The negative mass of the dark soliton is mainly induced by the nonlinear dispersion relation rather than the single-particle band dispersion [45,46]. This is supported by the fact that the acceleration of a dark soliton is indeed inverse to the direction of the force (see figures 3(a)-(c)). These striking inertial mass properties of the dark soliton could inspire more discussions on the dispersion relation of quasi-particle nonlinear waves. But our variational prediction is quite different from those based on EDSS, even for the dark soliton with low initial moving speed. It is naturally expected that the moving trajectory of the dark soliton should demonstrate the inertial mass differences, based on the Newton's equation.

Possibilities to test the dispersion relations
Then, one crucial problem is to test which theoretical result is more reasonable. Inspired by that the position-space Bloch oscillation can be used to measure the dispersion relation in an ultra-cold gas [4], we intend to drive a dark soliton by a weak force and investigate its motion, which can directly distinguish between the two dispersion relations. But it is impossible to apply a force to the dark soliton directly without affecting its non-zero density background. Individual impurities interacting with a quantum system was suggested to have numerous applications in probing, quantum state engineering, or quantum simulation [47]. By introducing few impurity atoms that are coupled to the dark soliton, we can add a weak force to impurity atoms to drive a dark soliton through the coupling nonlinear interaction. This is somewhat similar to our previous work [36,37], but the atoms in the bright soliton component must be very few [48]. The dynamics of this soliton-impurity system we considered can be described by the following equations [49][50][51], where ψ m and ψ s denote wave functions of the impurity atoms and the dark soliton component respectively. To further lower the effects of impurity atoms on the total energy of the coupled systems, we let the intra-species interaction strength between impurity atoms be zero (g 3 = 0). The interactions between atoms in condensate is repulsive for dark soliton excitation (g 1 = 1), and the interactions between impurity atoms and condensate atoms are repulsive (g 2 = 1), which provides coupling between them. An external potential −Fx is applied to impurity atoms, and can be used to drive the dark soliton without obvious effects on the dark soliton's density background [36]. We choose a very weak force to avoid perturbing the background, and mainly focus on the soliton energy for dark solitons undergoing deformations induced by acceleration.
Considering that impurity atoms are trapped in the ground state [49] in the trapping potential formed by a dark soliton, we introduce the initial condition for the following numerical simulations [48], where p 0 = √ 1 − v 2 0 and w 0 = 1/p 0 denote the initial depth and width of the dark soliton. ε is a small parameter (|ε| 2 ≪ 1) indicating that the impurity atoms are very few. The excitation energy for the above soliton-impurity system can be written as Here, it should be noted that when the external potential is slowly varied on the soliton scale, the potential energy´∞ −∞ −Fx|ψ m | 2 dx = −Fx c´∞ −∞ |ψ m | 2 dx = −2Fx c ε 2 w 0 , where x c is the soliton center position. Since the impurity atoms are very few, the terms representing the nonlinear coupling energy and the kinetic energy of the impurity component are small compared to the others. Therefore, they can be ignored safely, i.e. E sε = E s + (−2Fx c )ε 2 w 0 . It should be emphasized that the nonlinear coupling between the impurity and the condensate plays a very important role for the dark soliton's acceleration, although its energy value is much smaller than the dark soliton energy. From the energy conservation law, we have with the initial position of the soliton is at x c = 0, under the local density approximation [21,22].  3(a)), the variational result (red dashed lines) agrees well with the one (black lines) based on EDSS for v < 0.5, and they both cover the numerical simulation curve (blue triangles). The deviations become more pronounced when the dark soliton is accelerated to higher speeds. It can be seen that the trajectory given by the EDSS's dispersion relation agrees better with the numerical simulation than that of the variational results, for the velocity v > 0.5.
As discussed above, the dispersion relations given by our variational method and EDSS distinguish from each other more clearly for large initial moving speed cases. From figure 3(b) (v 0 = −0.5) and (c) (v 0 = −0.7), we can see that the deviations between the EDSS results (black lines) and the variational results (red dashed lines) are indeed more obvious. The existence of impurity atoms with ε = 0.3 will induce some obvious initial velocity shift [52], since the initial states equations (11) and (12) are not as proper as in the case with v 0 = 0 with the identical nonlinear parameter settings. The numerical results are also closer to the predictions of EDSS, and deviate more greatly from those of the variational results with dark soliton's particle number conserved. This means that the particle number of the dark soliton N s is not conserved during the acceleration (deceleration) process. We expect that the dark soliton suffers density wave emissions during the acceleration process, in addition to the well-known sound waves [31][32][33][34][35].

Density wave emissions and inhibition
We emphasize that the particle number of the whole system is conserved. If we choose a very large value of L to calculate the dark soliton's particle number N s , its variation will be almost invisible. In order to detect the emitted density waves, we numerically calculate N s with taking L = 15, which is a bit larger than the soliton scale. The corresponding results are shown in the panel (b) of figure 3. One can see that the particle number of the dark soliton indeed changes during the acceleration process. Therefore the acceleration of a dark soliton not only generates sound wave emissions, but also induces density wave emissions. The density waves carry the particle transport, and sound waves in general carry only energy without carrying any particle transport.
Then, an intuitive question is how to inhibit the density wave emissions. Our previous study on a spin soliton suggests that when the dark soliton is coupled with a bright soliton that fully fills its notch, density wave emissions can be inhibited [36]. Based on this, we suggest that density wave emissions can be inhibited by increasing ε, since when ε → 1 the coupled soliton-impurity system tends to be a spin soliton. However, if we still keep the parameter settings and give the initial state by the EDSS. Abrupt shock waves will be generated in the early stage of evolution, and then change the profile of the dark soliton, since ε is no longer a small parameter. Therefore, we develop the variational method to prevent the initial emission and ensure that the dark soliton's profile remains unchanged, by changing the intra-component interaction strength g 1 according to ε while p 0 , w 0 and g 2 are fixed [53].
This provides possibilities to make numerical simulations with the initial static states where the dark soliton does not change but ε increases. The constant force is also applied to the ψ m component. To detect the density wave emissions during this process, we calculate the particle number of dark soliton N s in different cases (as shown in figure 4(a)), the dark solitons show periodic decrease and increase in N s . The minimum value of N s increases with the ε. This suggests that the dark soliton's density wave emission is inhibited with increasing ε. This character can be understood by that the existence of non-negligible impurity atoms provide an effective potential being helpful to keep the dark soliton's profile. Since the particle loss of the dark soliton is getting much lower with larger ε, the dispersion relation obtained from EDSS is no longer applicable, and dispersion relation given by the variational method (while keeping the particle number conserved) works better. Particularly, these results indicate that the coupled solitons start to oscillate with negative-positive mass transition [36,37]. During the oscillation process, N s periodically decreases and increases. The decrease indicates the density waves emission. One could intuitively think that the increase means the emitted density waves are re-absorbed by the dark soliton. But our calculations on particle number of local finite scale suggest that the emitted density waves cannot be re-absorbed by the dark soliton. The increase is caused by the re-exciting soliton's wave profile from the soliton excitation energy variation. In order to show the inhibition of density wave more clearly, we calculate the change in particle number of the dark soliton ∆N s , which is defined as the difference between the first maximum and the first minimum value (shown in figure 4(b)). It can be seen that ∆N s decreases as ε increases. Namely, the density wave emissions are inhibited by increasing the impurity atom number. On account of the dominant bright soliton, some dispersive waves are generated in the early stage of evolution when ε > 1, resulting in minor increases both in figures 4(a) and (b). It seems that the numerical results deviate from our predictions (denoted by red triangles) when the ε is very small. These can be attributed to that the dark solitons can be accelerated to large velocities and become much wider and shallower in these cases. Hence, the impurity atom diffuses and decouples with the dark soliton, and we can no longer accelerate the dark soliton.
The effect of the intra-species interaction strength between impurity atoms should be included, i.e. g 3 ̸ = 0, when more and more impurity atoms are trapped by the dark soliton. Here we take ε = 1 (black line with dots in figure 4(a)) as an example, to show its effect on density wave emission. We show the change of N s in three cases: no interaction (g 3 = 0), attractive (g 3 = −0.6), and repulsive (g 3 = 0.6) in figure 4(c). The results show that the attractive interaction between impurity atoms has only a small impact, but the repulsive interaction significantly intensifies the diffusion of the soliton within a short time evolution. Due to the diffusion of the soliton, we cannot precisely find the soliton center and thus the N s in the later half of evolution becomes a bit irregular. Moreover, there are some initial dispersive wave generation, due to that initial states are not admitted well by the system, when g 3 is directly varied with other parameters fixed. The oscillation behavior of dark soliton with trapping more impurity atoms should be discussed systemically with taking g 3 ̸ = 0 and re-deriving proper initial states. We will study this project in the near future.

Conclusion and discussion
We systemically investigate the dispersion relations of dark solitons given by EDSS and a variational method. With the aid of soliton-impurity coupled systems, we numerically study the motion of a dark soliton, when we drive the impurity atoms. The trajectories on a soliton position vs. velocity plane indicate that the dispersion relations given by EDSS work better for predicting the motion of the dark soliton, when the impurity atoms are much less than the particle number of the dark soliton. Density wave emissions during the acceleration process are predicted theoretically, except the previously known sound waves. Furthermore, we demonstrate that more impurity atoms can inhibit the density wave emissions during the acceleration process of a dark soliton. In these cases, the dispersion relation derived from the variational method will work better.
It has been shown that the dispersion relation of the dark soliton is in sharp contrast to that of the well-known scalar bright soliton [19][20][21]54]. The phase and density dip characteristics of localized waves on a non-zero density background could induce some unusual kinetic behaviors. For example, the effective mass of dark-bright solitons could admit some striking properties, through nonlinear coupling between bright soliton and dark soliton [36,37,[55][56][57]. This could motivate further studies to discuss the dispersion relations of other nonlinear excitations with nonzero density background, such as vector solitons [55][56][57][58][59][60][61], dark soliton ring in two-dimensional cases [61], two dark soliton stripes [62], and even vortex [63][64][65].

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Acknowledgment
This work is supported by the National Natural Science Foundation of China (Contract Nos. 12022513, 12235007), and the Major Basic Research Program of Natural Science of Shaanxi Province (Grant No. 2018KJXX-094).

Appendix. The Lagrangian variational results for the particle conserved dark soliton
For the Gross-Pitaevskii equation equation (1), one can consider a trial wavefunction based on the exact dark soliton solution, where p(t), w(t) and b(t) denote the depth, width and center position of the dark soliton, respectively. Then the Lagrangian can be written as: The factor (1 − 1 |ψs| 2 ) is introduced for the dark soliton state, or it is impossible to integrate the term i 2 (ψ * s ∂ t ψ s − ψ s ∂ t ψ * s ). This problem was firstly solved by Kivshar and Królikowski in [18]. Substituting the trial wavefunction into the Lagrangian, and after taking the particularly elaborate integrals, we obtain that where v(t) = b ′ (t) = d dt b(t), etc. It is now straightforward to apply the Euler-Lagrangian equation d dt ( ∂L(t) ∂α ′ ) = ∂L(t) ∂α , where α = b(t), p(t), w(t), and θ 0 (t). Then two nontrivial relations can be derived as follows: where N s is the particle number of the dark soliton. The initial conditions are v 0 = v(t = 0) = d b(t = 0)/dt, and p 0 = p(t = 0) = √ 1 − v 2 0 , which can be used to obtain the initial soliton width w 0 = w(t = 0) = 1/ √ 1 − v 2 0 . From the conservation of the norm of dark soliton component, we have From these expressions, one can obtain the soliton energy and velocity, The dispersion relation can be obtained through the parameter p(t).