Dynamics of vortex-antivortex pairs and rarefaction pulses in liquid light

TL;DR

This paper numerically investigates the behavior of vortex-antivortex pairs and rarefaction pulses in liquid light modeled by the cubic-quintic nonlinear Schrödinger equation, revealing complex collision dynamics and instabilities.

Contribution

It demonstrates the generation and diverse collision behaviors of vortex-antivortex pairs and rarefaction pulses in liquid light modeled by the cubic-quintic nonlinear Schrödinger equation.

Findings

Vortex-antivortex pairs can be generated through bright soliton collisions.

Various collision outcomes include vortex exchange, inelastic scattering, and pulse merging.

The dynamics resemble liquid-like behavior with complex interaction phenomena.

Abstract

We present a numerical study of the cubic-quintic nonlinear Schr\"odinger equation in two transverse dimensions, relevant for the propagation of light in certain exotic media. A well known feature of the model is the existence of flat-top bright solitons of fixed intensity, whose dynamics resembles the physics of a liquid. They support traveling wave solutions, consisting of rarefaction pulses and vortex-antivortex pairs. In this work, we demonstrate how the vortex-antivortex pairs can be generated in bright soliton collisions displaying destructive interference followed by a snake instability. We then discuss the collisional dynamics of the dark excitations for different initial conditions. We describe a number of distinct phenomena including vortex exchange modes, quasielastic flyby scattering, soliton-like crossing, fully inelastic collisions and rarefaction pulse merging.

Equations

$$i{\partial }_z\psi =-({\partial }_x^2+{\partial }_y^2)\psi -(|\psi {|}^2-|\psi {|}^4)\psi .$$

$$\psi {|}_{z=0}$$

$$iU{\partial }_{\eta }\Psi =({\partial }_{\eta }^2+{\partial }_y^2)\Psi +\left(|\Psi {|}^2-|\Psi {|}^4-\frac{3}{16}\right)\Psi$$

$$\mathbf{j}=\frac{1}{i}({\psi }^{\ast }\nabla \psi -\psi \nabla {\psi }^{\ast })$$

$$p$$

$$E$$

Full text

I Introduction

The synergy between competing nonlinearities in the Schrödinger equation can give rise to very interesting dynamics PhysRevLett.102.203903 ; setzpfandt2009competing , including, for instance, solitons PhysRevA.83.053838 ; Laudyn:15 and phase transitions PhysRevLett.116.163902 ; 0295-5075-98-4-44003 . In this paper, we provide novel insights on the (focusing) cubic- (defocusing) quintic model, which has been thoroughly studied in the context of nonlinear optics mihalache1988exact ; pushkarov1996bright ; dimitrevski1998analysis , where it was shown that large power solitons have neat similarities with regular liquids, thereby motivating the term “liquid light” michinel2002liquid . The same equation has been applied in other frameworks too, see e.g, josserand1997coalescence ; muryshev2002dynamics ; khaykovich2006deviation ; carretero2008nonlinear ; davydova2003two .

The cubic-quintic equation is an appropriate model for the propagation of light in certain optical materials, see for instance smektala2000non and references in caplan2012existence . It has also been used as an approximation to the process of filamentation piekara1974analysis ; centurion2005dynamics ; novoa2010filamentation . Recent experimental advances reinforce the significance of new theoretical studies. Despite damping, (limited) soliton propagation has been observed in carbon disulfide falcao2013robust . Furthermore, the droplet-like behavior of cubic-quintic propagation has been demonstrated in atomic gases at low optical powers wu2013cubic ; wu2015solitons , using quantum coherence and interference as proposed in michinel2006turning ; alexandrescu2009liquidlike . Other setups in which the fifth order nonlinearity can be enhanced through quantum effects comprise Rydberg atoms bai2016enhanced and quantum dots peng2014tunneling ; tian2015giant . Confinement and guiding of light in a (defocusing) cubic- (focusing) quintic has also been reported reyna2016guiding .

In the cubic-quintic model, there is a one-parameter family of form-preserving traveling dark wave solutions within a critical bright background, which was computed in paredes2014coherent following the numerical methods of chiron2016travelling . For small velocities, it consists of vortex-antivortex pairs of charges $\pm 1$ (we will make a usual abuse of language and refer to “velocity” for what in the optical setup corresponds to the propagation angle with respect to the axis). For larger subsonic velocities, the solutions are rarefaction pulses, namely dark blobs without vorticity. The fainter the pulse is, the faster it moves within the bright background. This family of solutions is similar to the one existing for third order defocusing nonlinearity jones1982motions ; jones1986motions ; berloff2004motions ; bethuel2009travelling . Rarefaction pulses should not be confused with the unstable quiescent bubbles of barashenkov1988soliton ; barashenkov1989stability .

A separate issue is how these dark soliton-like excitations can be generated dynamically. In the context of Bose-Einstein condensates (BEC), they have been generated by phase imprinting proud2016jones . In the framework of superfluids, it was shown that they can appear when the fluid flows past an obstacle josserand1995cavitation , a process that in optics can be mimicked by the nonlinear interaction with an incoherently coupled beam feijoo2014drag and in BECs with a laser beam (see mironov2010structure and references therein).

A remarkable result of paredes2014coherent is that, for the liquid of light, rarefaction pulses can be generated by interference in the collision of two bright solitons of very different sizes and powers. The analogy with bubbles in fluids motivates the usage of the term cavitation for this kind of process. The produced caviton excitation propagates within the large soliton and can exit it becoming a bright soliton again. This bright-dark-bright conversion is familiar in one dimension, see e.g. kim2000soliton ; garralon2013numerical , but it is a distinctive feature of the cubic-quintic equation in two dimensions. This peculiarity facilitates the creation of dark traveling waves in a controlled manner from initial conditions comprising only bright solitons. With three initial bright solitons, two separate traveling waves can be created within the same fluid.

The natural question that we address in the present paper is how these traveling waves interact with each other. It would be really interesting to implement this kind of processes in experimental setups as those described in falcao2013robust ; wu2013cubic ; wu2015solitons . For the case of defocusing cubic nonlinearity, the dynamics of the dark excitations in a nontrivial background was analyzed in smirnov2012dynamics ; mironov2012propagation ; mironov2013scattering and their interaction with a single vortex in smirnov2015scattering .

In section II, we fix notation and review some features of the cubic-quintic model. In section III, we show that vortex-antivortex pairs can be produced by a soliton-soliton collision. Sections IV-VI describe the result of our simulations concerning dark wave interactions. We discuss in turn the collision of two vortex-antivortex pairs, that of a rarefaction pulse with a vortex-antivortex and that of two rarefaction pulses. In section VII we outline our conclusions and make some final remarks. The supplemental material suppl contains animations for all of the examples of dynamical evolution that are presented along the paper and a few extra illustrative cases.

II Solitons and traveling waves

In this section we briefly review well-known results concerning the cubic-quintic model in order to provide the basic ingredients for the following. In the paraxial approximation, the canonical equation governing the wave amplitude $\psi(x,y,z)$ reads:

[TABLE]

A refractive index of the form $n=n_{0}+n_{2}I+n_{4}I^{2}$ has been assumed, where $I\sim|\psi|^{2}$ is the intensity. It is straightforward to check that the equation in terms of physical quantities can be rescaled to the dimensionless variables of Eq. (1) without loss of generality as long as $n_{2}>0$, $n_{4}<0$.

There are stable solitary waves of the form $\psi=e^{i\beta z}f(r)$ with $\lim_{r\to\infty}f(r)=0$ which we laxly call bright solitons, as it is customary in the literature. The numerical study of dimitrevski1998analysis ; michinel2002liquid shows that there are solutions for $0<\beta<\beta_{cr}=\frac{3}{16}$. The power $P=2\pi\int rf(r)^{2}dr$ grows monotonically with $\beta$ in the range $P_{0}<P<\infty$ where $P_{0}$ is the minimal value that leads to self-trapping. For small $\beta$, the function $f(r)$ is bell-shaped. Near the $\beta_{cr}$ eigenvalue cutoff prytula2008eigenvalue , $f(r)$ tends to a flat-top profile. This means that $f\approx\Psi_{cr}=\frac{\sqrt{3}}{2}$ for $r<r_{sol}$ and around the soliton radius $r_{sol}$ there is a quick drop to $f\approx 0$ for $r>r_{sol}$. This limit is the liquid-like phase, in which the soliton resembles a fluid with constant density and fixed surface tension subject to the Young-Laplace equation novoa2009pressure .

We will use these bright solitons to define the initial conditions of simulations in the following sections, by considering:

[TABLE]

where the ${\bf x_{i}}$ are the initial positions of the solitons, ${\bf v_{i}}$ their initial velocities and $\phi_{i}$ their initial phases. Boldface symbols are two-dimensional vectors. The $f_{i}(.)$ are the soliton profiles, where $f_{1}$ is a flat-top soliton, corresponding to the liquid where the dynamics takes place and $f_{2}$, $f_{3}$ are smaller solitons that dynamically generate the dark excitations. In Fig. 1, we plot the profiles of the particular solitons that will be used in all the examples below.

Let us now turn to the dark traveling waves paredes2014coherent . They are form-preserving solutions of Eq. (1) moving at constant speed $U$ in, say, the $x$-direction, embedded in an infinite liquid. Inserting the ansatz $\psi(x,y,z)=e^{i\,\beta_{cr}\,z}\Psi(\eta,y)$ jones1982motions . where $\eta=x-Uz$, we can write:

[TABLE]

subject to the boundary condition $\lim_{\eta^{2}+y^{2}\to\infty}\Psi=\Psi_{cr}=\frac{\sqrt{3}}{2}$. There is a family of solutions parameterized by $0<U<\frac{\sqrt{3}}{2}$. For small $U$ they are vortex-antivortex pairs, with $|\psi|^{2}=0$ at the phase singularities. When $U$ grows, the vortex and antivortex merge into a rarefaction pulse, whose $|\psi|^{2}$ is nowhere vanishing. It is important to remark that the transition is completely smooth and, roughly, one can think of the rarefaction pulse as a bound state of vortex and antivortex. In fact, under nontrivial dynamical evolution both kinds of eigenstates can transform into each other mironov2012propagation ; smirnov2015scattering .

An interesting quantity is the current density which, in the hydrodynamical picture, represents the flow of the fluid.

[TABLE]

The ${\bf j}$ is essential to understand how the dark excitation modifies the medium around it and therefore to understand the interaction between traveling waves. In figure 2, we depict this quantity for three examples of traveling waves.

Momentum and energy are conserved quantities defined by:

[TABLE]

Within the family of solutions, one can check that $U=\partial E/\partial p$ and three virial identities are satisfied jones1982motions ; jones1986motions ; paredes2014coherent .

The analyisis in the coming sections results from the numerical integration of Eq. (1) with initial conditions (2). The computations are done using a standard split-step beam propagation method agrawal2007nonlinear . The evolution associated to the non-derivative terms is computed with a fourth order Runge-Kutta method. The plotted figures are built using grids of 800$\times$600 points. We have checked convergence of the method by comparing results with different grids in $(x,y)$ and steps in $z$.

III Coherent generation of vortex-antivortex pairs

In paredes2014coherent , it was shown that a rarefaction pulse can appear when two coherent bright solitons meet with appropriate relative velocity and phase. Roughly speaking, destructive interference generates a void at the collision point which can acquire the necessary velocity thanks to the incoming momentum. Although, definitely, an exact solution of (3) is not realized in the dynamical process, the resulting robust dark excitation can indeed be identified with a traveling wave solution. This fact was checked in paredes2014coherent by comparing the dispersion relations. Even if the size of the medium (the large soliton) is not infinite, it can support the traveling wave if it is much larger than the dark structure.

In this section, we show that a similar process can result in the formation of a vortex-antivortex pair. In fact, the difference with paredes2014coherent is simply that the incomnig soliton has to be larger. What happens is that during a collision in phase opposition, an elongated dark region is created. It cannot be stable because there are no rarefaction pulse solutions of similar size. Consequently, it evolves and decays through a snake instability giving rise to the separate vortex and antivortex, which move forward together with a given velocity $U$. Since the resulting configuration is not exactly equal to the stationary solution, the dark regions can change, reconnect and split again. However the vortex-antivortex profile becomes apparent after long enough propagation in $z$. An example is depicted in figure 3. Obviously, the third soliton of (2) is not included in the initial condition.

In figure 4, we expose the phase structure of the wavefunction of the example at a particular propagation distance $z$. The plots prove that the two dark spots of figure 3 correspond indeed to a vortex-antivortex pair.

Concerning the reconversion into a bright soliton paredes2014coherent , we notice that it can take place when the excitation reaches the boundary of the liquid of light as a single dark pulse. On the other hand, when it does so as a vortex-antivortex pair, two waves propagating in opposite directions along the edge of the large soliton get excited suppl .

It must be emphasized that the generation of vortex and antivortex is only one of the possible qualitative outcomes that emerge depending on the relative velocity and phase. As in paredes2014coherent , the droplets can simply coalesce into one. The collision can also result in rarefaction pulses of different energies and speeds. For low velocities, part of the energy can bounce back evolving into a smaller bright soliton. In all cases, surface and bulk sound waves are excited during the process. If the collision is very violent, the large soliton can be severely distorted, ceasing to be a liquid-like approximately homogeneous medium.

We close this section by noting that there are vortex solutions of the cubic-quintic equation (1) of the form $\psi=e^{i\beta z}e^{il\theta}f(r)$ with $\lim_{r\to\infty}f(r)=0$, where $l$ is the topological charge and $\theta$ is the polar angle. Their profiles and stability have been studied in quiroga1997stable ; towers2001stability ; berezhiani2001dynamics ; malomed2002stability ; michinel2004square and their collisional dynamics in paz2004collisional . We remark that the vortices that we are studying in this paper as solutions of Eq. (3) are different objects: they live within the vorticity-less liquid of light and they only exist in pairs and moving with a finite velocity.

IV Collisions of vortex-antivortex pairs

Let us start illustrating the interactions by computing the head-on encounter of two vortex pairs created as described in section III. A typical example is displayed in Fig. 5. The result is an exchange in which the vortex of each pair recombines with the antivortex of the other one (the exchange of a single vortex with a vortex-antivortex pair was described in smirnov2015scattering with cubic nonlinear potential.) The solitary waves come out perpendicular to the incoming direction. This can be understood in terms of the flow lines of Eq. (4), considering that, during the approach, each pair generates a smooth inhomogeneity in the background in which the other one propagates mironov2013scattering . For instance, the antivortex on the top right (see the panels (c) and (d) of Fig. 5) feels the flow lines generated by the phase structure of the vortex on the top left (see Fig. 4) and is pushed upwards. Conversely, the vortex in the bottom right turns downwards because of the antivortex in the bottom left. Since these bends tend to associate again vortex and antivortex, the propagation can continue after the exchange. In Fig. 5, we have considered slightly different phases for the initial solitons in order to show that a perfect symmetry is not needed for this process.

Similar exchanges can happen for collisions at angles. Figure 6 depicts an example where the incoming excitations are perpendicular to each other. In this case, the vortex moving downwards and the antivortex moving leftwards attract each other and coalesce into a dark blob which can be considered an excited version of a rarefaction pulse. It comes out heading the top right of the plot and is finally reconverted into a (highly excited) bright soliton when it reaches the edge of the medium. The remaining vortex and antivortex eventually couple to each other and continue to propagate towards the bottom left. Notice that the velocity of this pair is much lower than that of the aforementioned rarefaction pulse, as expected from the stationary solutions characterized in section II.

The simulation of Fig. 6 is also interesting because it shows other generic features of the dynamics, which can be better appreciated in the animation presented in the supplemental material suppl . In particular, we must emphasize that the evolution of the dark excitations is not elastic, in the sense that some energy is radiated away in the form of sound waves. Moreover, faint rarefaction pulses, of small energy regarding Eq. (5), can be generated. These radiation processes take place during collisions and also during the relaxation of the coherently generated dark bubbles towards their stationary vortex pair form.

We also remark that, for encounters like that of Fig. 6, small changes in the initial conditions can determine how the dark regions combine and greatly affect the outcoming pulses. For instance, if we just change ${\bf v_{3}}$ from (-0.2,0) to (-0.21,0), therefore breaking the symmetry, between both incoming vortex pairs, the one moving horizontally arrives first. Instead of performing a exchange with the other vortex, it merges with the antivortex, creating an elongated void of net vorticity -1. This snake-like structure starts rotating and eventually decays emitting a rarefaction pulse. We present this evolution in suppl . Thus, the encounter gives rise to a vortex-antivortex pair and a rarefaction pulse, just as in Fig. 6, but their resulting propagation directions are rather different. This simulation also shows that, when there is an eventual dark-bright reconversion, the outgoing dark soliton does not necessarily come out with the same propagation direction as the dark blob which generates it.

We close this section by considering a head-on encounter in which the vortices of each pair meet each other (instead of heading an antivortex as in Fig. 5). This can be accomplished by slightly shifting the $y$-position of the bright solitons defined in the initial conditions. An example is depicted in Fig. 7.

This evolution can be qualitatively understood noting that the vortices repel each other and therefore are slowed down while the antivortices continue advancing. This induces a rotation of the whole dark structure, which eventually breaks down resulting in two separate pulses which come out at an angle, different from the incoming one. This is a kind of pseudo-elastic collision. Notice, however, that the scattered pulses cannot be neatly considered vortex-antivortex as the incoming ones. Vortex pairs and rarefaction pulses can be cleanly defined for stationary situations but in dynamical evolutions like the present one, the separation between both is not obvious and they can even transform into each other, as noticed in smirnov2015scattering in a different but somewhat related scenario.

V Collision of a rarefaction pulse with a vortex-antivortex pair

We now consider the encounter of a rarefaction pulse with a vortex-antivortex pair. An illustrative case is sketched in Fig. 8. In the example, the dark regions moving in opposite directions pass near each other but do not experience a direct contact. They keep their distinct identities during the whole evolution and therefore this process is very similar to a elastic scattering. The pulses continue their propagation away from each other and therefore we call this a flyby mode, following smirnov2015scattering . In the figure, it can be appreciated that the propagation of the rarefaction pulse is rotated by a small angle when both waves meet (the horizontal dashed line has been included in the plots to guide the eye). Again, this is due to the flow lines defined in Eq. (4) and represented in Fig. 2, whose structure explains why the caviton turns upwards. The vortex-antivortex pair is also affected by the encounter, by since its energy and momentum (5) is quite larger that that of the rarefaction pulse, it is much harder to appreciate the diversion. Notice that this flyby mode is only relevant for a narrow window of the scattering impact parameter. If the caviton pulse moves far from the dipolar structure, the phase gradients are tiny and their effect is negligible. On the other hand, if both waves are too near, the dark regions recombine giving rise to more complicated evolutions, as we show in the next example.

The initial conditons in Fig. 9 resemble those of Fig. 8, but the initial $y$-displacement of the bright solitons is slightly smaller, yielding a smaller impact parameter for the collision of the dark waves. In this case, the dark regions associated to the antivortex and the rarefaction pulse come into contact and merge, initially giving rise to a large blob of vorticity -1. Since the vortex-antivortex pair has the larger momentum and energy, the subsequent evolution can be roughly described as an absorption of the rarefaction pulse by the pair, which becomes highly excited, but continues its propagation rigthwards. This structure slowly relaxes towards the stationary vortex-antivortex solution by the emission of sound waves and faint rarefaction pulses suppl . In suppl , we also present a simulation in which the vortex pair and the caviton approach each other with zero impact parameter. Roughly, the dynamics can be understood in terms of the previous discussion: when the dark regions touch each other, the rarefaction pulse is swallowed by the vortex-antivortex which, albeit excited, continues its propagation. We have checked that this kind of qualitative behavior is quite generic, regardless of the incoming angles and velocities.

VI Collisions of rarefaction pulses

Let us now discuss the case of two interacting rarefaction pulses. First of all, we notice the existence of flyby modes, similar to those described in the previous section, when the impact parameter is not too large but enough to avoid direct contact.

It is also worth commenting on the dynamics of head-on collisions. The most common result is illustrated in Fig. 10. When the pulses meet, a larger dark blob is created with, possibly, a bright spot inside (see panel (d) of Fig. 10.) Then, two rarefaction pulses appear again and continue their propagation. During the encounter, part of the energy is radiated away and, therefore, the pulses after the collision are slightly fainter and faster. Thus, in this respect, the rarefaction pulses behave as dark quasi-solitons. We remark that this happens for symmetric encounters as the one of the figure or asymmetric ones with pulses of different energies. As expected, when the cavitons reach the edge of the large soliton, they can be reconverted in bright solitons again. In fact, the simulation of Fig. 10 can be interpreted as a bright-dark-bright-dark-bright transformation of the propagating excitation suppl .

Curiously, the picture changes completely if the initial conditions are properly fine tuned. Figure 11 depicts an example in which the rarefaction pulses annihilate each other and their energy is radiated in the form of a circular sound wave. Visibly, the behavior of the rarefaction pulses in this case totally differs from that of form preserving solitons. As a matter of fact, the seemingly antagonistic character of Figs. 10 and 11 can be continuously connected, by noticing that in all head-on encounters the outgoing energy is shared by a bulk wave and two rarefaction pulses. In Fig. 10, most of the energy goes to the latter whereas in Fig. 11 it is mostly acquired by the former, while other initial conditions lead to intermediate possibilities.

Finally, we comment on the encounter of rarefaction pulses at angles. Figure 12 illustrates this case by considering a perpendicular concurrence. As in the previous cases, the dark regions combine producing a dark blob, which is larger than the incoming ones. However, in this case this blob can survive and, in a loose sense, propagate in the direction required by momentum conservation. Thus, the simulation of Fig. 12 can be neatly portrayed as the merging of two rarefaction pulses into a more energetic one. Similarly to all of the presented examples, part of the energy is radiated away during the process.

We close the section by noticing that there is a second typical qualitative behavior, which we show in the last animation of suppl . What happens there is that the dark blob splits giving rise again to two rarefaction pulses (we emphasize that, even if in suppl it may seem that the dark pulses coming out of the collision propagate almost in parallel, they are not vortex and antivortex). Roughly, this last possibility can be thought of as another example of quasi-elastic scattering or as a bounce of the pulses against each other.

VII Summary and outlook

In this work, we have numerically analyzed Eq. (1), reporting on a number of novel qualitative phenomena for the cubic-quintic model in 1+2 dimensions.

The interplay of diffraction with focusing and defocusing nonlinear effects endows the cubic-quintic nonlinear Schrödinger equation with an extremely rich phenomenology. In particular, there are dark traveling waves and bright solitons, which for large powers become liquid-like. Noticeably, the dark and bright stationary and stable solitary waves can transform into each other during evolution. In particular, a bright soliton can excite a rarefaction pulse when it meets a bright soliton of larger power paredes2014coherent . We have shown that a vortex-antivortex pair can be generated in a similar way. The process, however, is not as clean as in the previous case. The incoming soliton has to be larger and gives rise to a more pronounced distortion of the flat-top soliton. Moreover, the vortex and antivortex are not generated directly, but only as the end result of a snake instability of an initial dark blob. Thus, the radiation of part of the excess energy is essential in approaching the stationary vortex-antivortex solution. When a strong enough rarefaction pulse reaches the border of the liquid of light, it typically generates an outgoing bright soliton. On the other hand, the vortex-antivortex pair excites a couple of surface waves propagating in opposite directions.

The possibility of creating the dark states by interference and nonlinear evolution has allowed us to propose numerical experiments concerning their scattering with initial conditions which only include bright solitons, see Eq. (2). We have made a qualitative analysis of the encounters between vortex-antivortex pairs and rarefaction pulses. In brief, our results can be summarized as follows:

• •

If the vortex of a pair meets an antivortex of another pair and viceversa, they tend to get exchanged resulting in two new pairs with different propagation directions.

• •

If the impact parameter of a collision is large enough and the dark regions do not touch each other, there are elastic flyby modes and the propagation direction of each wave is altered because of the flow lines associated to the opposite structure.

• •

When a vortex or a rarefaction pulse touches a vortex-antivortex pair, an excited dark blob is created. It propagates for a while and eventually decays approaching the stationary states. The end result is strongly dependent on initial conditions.

• •

Rarefaction pulses which collide head-on typically cross each other, losing some energy by radiating sound waves. In particular situations, the radiation can take most of the energy. If the pulses collide at an angle, they can merge into a larger rarefaction pulse or scatter quasielastically.

This list does not exhaust the possibilities but it certainly provides a qualitative description for most of the collisions between dark traveling waves. It is tempting to interpret the traveling waves as quasiparticles and to try to understand collisions in terms of their energy-momentum conservation, Eq. (5). Implicitly, this has been our point of view when using the words “elastic” and “inelastic”. Notice that $p$ and $E$ as a whole are conserved in a collision. Nevertheless, if we only take into account the dark traveling waves, the conservation breaks down, as it obvious from figure 10. The main reason is that sound waves take a sizable fraction of energy and momentum in many processes. Moreover, as we have already emphasized, the dark waves typically appear in excited form and therefore the velocity-momentum and dispersion relations that can be deduced from the stationary solutions only apply approximately. Excited dark states have complicated dynamics and cannot always be easily identified with their stationary counterparts. Thus, the quasiparticle interpretation is illustrative but it should be clear that it is just a qualitative rough description.

Our results open some interesting possibilities. First of all, it would be nice to realize the described phenomena in optical setups along the lines of falcao2013robust ; wu2013cubic ; wu2015solitons . It would also be desirable to study similar effects in two dimensions for the cubic defocusing nonlinearity, since it is relevant for Bose-Einstein experiments like proud2016jones , see also verma2016snake and references therein. Moreover, it would be worth considering the three dimensional cubic-quintic case, which supports top-flat stable spatiotemporal solitons desyatnikov2000three ; jovanoski2001light and vortices desyatnikov2000three ; mihalache2002stable . Their collsional dynamics has been analyzed in hong2008energy ; adhikari2016elastic but the dynamics of dark traveling waves has not been described yet. Using the cubic defocusing Schrödinger equation, interesting dynamical analysis of the interplay of rarefaction pulses, vortex rings and vortex lines in 1+3 dimensions have been presented in the context of Bose-Einstein condensates berloff2002evolution ; berloff2004interactions ; komineas2005collisions and superfluids caplan2014scattering . It would be desirable to make contact with these analyses in the cubic-quintic case. Finally, we remark that our setup has partial similarities with other physical systems as, e.g., the scattering by impurities in superfluids as recently modeled in pshenichnyuk2016inelastic . It could be worth exploring analogies between different frameworks.

Acknowledgements.

We thank David Nóvoa and José Ramón Salgueiro for useful comments. This work is supported by grants FIS2014-58117-P from Ministerio de Economía y Competitividad and grants GPC2015/019 and EM2013/002 from Xunta de Galicia. The work of D. F. is supported by the FPU Ph.D. program