Stable long-distance propagation and on-off switching of colliding soliton sequences with dissipative interaction
We study propagation and on-off switching of two colliding soliton sequences in the presence of second-order dispersion, Kerr nonlinearity, linear loss, cubic gain, and quintic loss. Employing a Lotka-Volterra (LV) model for dynamics of soliton amplitudes along with simulations with two perturbed coupled nonlinear Schrödinger (NLS) equations, we show that stable long-distance propagation can be achieved for a wide range of the gain-loss coefficients, including values that are outside of the perturbative regime. Furthermore, we demonstrate robust on-off and off-on switching of one of the sequences by an abrupt change in the ratio of cubic gain and quintic loss coefficients, and extend the results to pulse sequences with periodically alternating phases. Our study significantly strengthens the recently found relation between collision dynamics of sequences of NLS solitons and population dynamics in LV models, and indicates that the relation might be further extended to solitary waves of the cubic-quintic Ginzburg-Landau equation.
pacs:42.65.Tg, 05.45.Yv, 42.65.Sf
The cubic nonlinear Schrödinger (NLS) equation, which describes wave propagation in the presence of second-order dispersion and cubic (Kerr) nonlinearity, is one of the most widely researched nonlinear wave models in physics. It was successfully employed to describe water wave dynamics Zakharov84 (), nonlinear waves in plasma Horton96 (), Bose-Einstein condensates (BECs) Dalfovo99 (), and pulse propagation in optical waveguides Hasegawa95 (). The fundamental NLS solitons are the most ubiquitous solutions of the cubic NLS equation due to their stability and to the fact that a generic wave pattern typically evolves into a sequence of fundamental solitons in the presence of anomalous dispersion and Kerr nonlinearity. In the absence of additional physical processes (perturbations), the fundamental solitons propagate without any change in their amplitude, group velocity, and shape. However, the presence of perturbations usually breaks this ideal picture, by inducing changes in the solitons’ amplitude, group velocity, and shape. Dissipative perturbations due to linear and nonlinear gain or loss are of particular interest, since they are very common in soliton systems. In optical waveguides, for example, nonlinear loss or gain arise due to multiphoton absorption or emission, respectively Boyd2008 (). Moreover, nonlinear loss and gain play an important role in many phenomena described by the complex Ginzburg-Landau (GL) equation Kramer2002 (), such as convection and pattern formation in fluids and mode-locked lasers.
Despite its success, the single NLS equation is limited to describing a scalar physical field and is also unsuitable for handling generic broadband nonlinear wave systems. If the physical field is a vector, or if the spectra of the waves in a broadband wave system are concentrated about multiple widely separated wavelengths, the single NLS equation should be replaced by a system of coupled-NLS equations. Indeed, in recent years, coupled-NLS models have been employed in studies of a wide range of phenomena in fluid dynamics Fluids (); Onorato2006 (), nonlinear optics Hasegawa95 (); NLoptics (), multiphase BECs BECs (), and plasma physics plasma (). Coupled-NLS models are particularly useful in describing broadband wave systems, in which the waves are organized in sequences (trains) moving with very different group velocities, such as in crossing seas Onorato2006 (), or in broadband optical waveguide transmission Hasegawa95 (). Due to the large group velocity differences in these systems, collisions between pulses from different sequences are very frequent, and thus play a major role in the dynamics.
In the current paper, we study the dynamics of two colliding sequences of fundamental NLS solitons in the presence of dissipative perturbations due to linear loss, cubic gain, and quintic loss (i.e., a GL gain-loss profile). In the absence of perturbations, the solitons’ amplitudes and group velocities do not change in the collisions. However, the presence of dissipative interaction due to cubic gain and quintic loss induces additional amplitude shifts during the collisions, whose magnitude depends on the initial amplitudes of the two colliding solitons. Consequently, amplitude dynamics of solitons in the two sequences become nonlinearly coupled, and important questions arise regarding the nature of this dynamics. In three recent studies we showed that amplitude dynamics of sequences of colliding solitons in the presence of dissipative perturbations might be described by Lotka-Volterra (LV) models for species, where the exact form of the LV model depends on the nature of the perturbations NP2010a (); PNC2010 (); PC2012b (). This potential relation between coupled-NLS equations and LV systems is of great interest, due to the central role of these seemingly unrelated models in the natural sciences. While the importance of coupled-NLS equations was explained in the previous paragraph, LV models are widely used in environmental sciences to describe population dynamics Lotka25 (); Volterra28 (); LV_ecology (), in chemistry, to study dynamics of chemical reactions LV_chem (), in economics, to explain interaction between different technologies LV_economics (), and in neural networks, to compute neuron firing rates LV_nueral (). Due to the importance of coupled-NLS and LV models, it is essential to further enhance and extend the understanding of the relation between them. Moreover, a better understanding of this relation can be used for stabilizing the propagation, for controlling and tuning of soliton amplitudes and group velocities, and even for broadband switching.
In order to give further motivation for the current study and to clarify the importance of its results, we provide a brief critical description of our three earlier studies of the problem. In Ref. NP2010a (), we studied the propagation of colliding soliton sequences in the presence of delayed Raman response. We showed that amplitude dynamics of soliton sequences is described by an -dimensional predator-prey model, but the results were not checked by numerical simulations with the corresponding coupled-NLS model. Furthermore, the equilibrium states of the predator-prey model were found to be centers, i.e., the equilibria were not asymptotically stable. In order to resolve the problem of asymptotic stability in the LV model, we turned to investigate collision dynamics of soliton sequences in the presence of weak linear gain and cubic loss PNC2010 (). We showed that in this case dynamics of soliton amplitudes might be approximately described by a LV model for competing species with quadratic interaction terms. Stability analysis of the equilibrium states of the model yielded conditions on the physical parameters for stable propagation of two sequences with equal soliton amplitudes. Numerical simulations with the full coupled-NLS model confirmed stability at short-to-intermediate distances, but uncovered an instability at longer distances PNC2010 (). The latter finding was attributed to weak (second-order) radiative instability due to the impact of linear gain on the two sequences. In order to overcome this destabilizing effect, we turned to study a different setup, where the soliton sequences propagate in the presence of weak linear loss, cubic gain, and quintic loss PC2012b (). We showed that in this setup, the LV model for amplitude dynamics contains quadratic and quartic interaction terms. The presence of linear loss was expected to suppress the weak radiative instability and by this to enable stable long-distance propagation. However, numerical solution of the corresponding perturbed coupled-NLS model still showed instability at intermediate-to-long propagation distances PC2012b (). Thus, the important problem of further stabilization of the propagation remained unresolved. Moreover, due to the instability at intermediate distances, all the previous studies were unable to explore the possibility of achieving stable on-off and off-on switching, i.e., the turning on and off of transmission of one sequence by a fast change of one or more of the physical parameters. In addition, the earlier studies were limited in scope to small values of the dissipative coefficients and to uniform initial distribution of soliton amplitudes and phases in each sequence.
In the current paper we address these important deficiencies of the earlier studies. Considering propagation of two colliding soliton sequences in the presence of a GL gain-loss profile, we observe stable long-distance transmission over a wide range of the gain-loss coefficients, including values that are outside of the perturbative regime. The stable transmission is enabled by the presence of linear loss, a sufficiently large initial inter-soliton separation, and the absence of significant initial position shift between the two soliton sequences. Moreover, we show that robust on-off and off-on switching of one soliton sequence can be realized by an abrupt change in the ratio of the cubic gain and quintic loss coefficients. In addition, we extend the results to soliton sequences with periodically alternating phases. Our findings significantly enhance the surprising relation between propagation of colliding NLS soliton sequences and population dynamics in LV models. The results also indicate that the relation might be further extended to sequences of solitons of the cubic-quintic GL equation.
The rest of the paper is organized as follows. In Sec. II, we present the coupled-NLS model for the propagation along with the reduced LV model for amplitude dynamics. We also discuss the predictions of the latter model for stability of propagation. In Sec. III, we present the results of numerical simulations with the coupled-NLS model for long distance transmission, as well as for on-off and off-on switching. In addition, we compare the simulation results with the predictions of the LV model. Our conclusions are presented in Sec. IV.
Ii Coupled-NLS and Lotka-Volterra models
We consider propagation of sequences of soliton pulses in the presence of second-order dispersion, Kerr nonlinearity, and a GL gain-loss profile, consisting of linear loss or gain, cubic gain, and quintic loss. We denote the physical field of the th sequence by . In the context of propagation of light through optical waveguides, for example, is proportional to the envelope of the electric field of the th sequence. In the absence of gain and loss, the propagation of the th soliton-sequence is described by the cubic NLS equation
where we adopt the waveguide optics notation, in which is propagation distance, and is time. The fundamental soliton solution of the NLS equation with group velocity is , where , , and , , and are the soliton amplitude, position, and phase, respectively.
We focus attention on the dynamics of two sequences of fundamental NLS solitons propagating with group velocities , where . Assuming a large group velocity difference , the solitons undergo a large number of fast inter-sequence collisions. We now take into account the effects of linear gain-loss, cubic gain, and quintic loss as well as inter-sequence interaction due to Kerr nonlinearity. Thus, the propagation is described by the following system of perturbed coupled-NLS equations:
where , , is the linear gain-loss coefficient for the th sequence, and and are the cubic gain and quintic loss coefficients, respectively. The term in Eq. (2) describes inter-sequence interaction due to Kerr nonlinearity, while , , and correspond to intra-sequence effects due to linear gain-loss, cubic gain and quintic loss, respectively. In addition, the term represents inter-sequence effects due to cubic gain, while and describe dissipative inter-sequence interaction due to quintic loss.
In Ref. PC2012b (), we showed that under certain assumptions, amplitude dynamics of solitons in the two sequences can be approximately described by a LV model for two species with quadratic and quartic interaction terms. The derivation of the LV model was based on the following assumptions. (1) The temporal separation between adjacent solitons in each sequence is a large constant: . In addition, the amplitudes are equal for all solitons from the same sequence, but are not necessarily equal for solitons from different sequences. (2) The pulses circulate in a closed loop, e.g., in an optical waveguide ring. (3) As , the pulses in each sequence are temporally well-separated. As a result, intra-sequence interaction is exponentially small and is neglected. (4) High-order effects due to collision-induced frequency shift and emission of radiation are also neglected.
Since the soliton sequences are periodic, the amplitudes of all pulses in a given sequence undergo the same dynamic evolution. Taking into account collision-induced and single-pulse amplitude changes, we obtain the following equation for the rate of change of the amplitude of the solitons in the th sequence PC2012b ():
where . Note that Eq. (3) can be described as a LV model for two species with quadratic cooperation terms and quartic competition terms.
The results described in the previous paragraph indicate that there is a relation between collision-induced dynamics of soliton sequences in nonlinear waveguides with a GL gain-loss profile and dynamics of population size of species with nonlinear competition and cooperation. Alternatively, one may speak about a connection between the perturbed coupled-NLS model (2) and the LV model (3). This relation along with knowledge about properties of LV models can be used to develop ways for controlling the dynamics of the colliding solitons. A straightforward way for achieving this goal is by tuning of the linear gain-loss coefficients . In optical waveguide systems, for example, this can be realized by adjusting the linear amplifier gain.
As a particular example, we consider the important case where the values of the coefficients are chosen such that the LV model (3) has a steady state with equal amplitudes for both sequences. This case is of special interest, since in optical waveguide systems it is usually easier to control the transmission when the amplitudes of all pulses are equal. Requiring that is a steady state, we obtain , where and . Substituting this relation into Eq. (3), we arrive at the following LV model for amplitude dynamics:
We note that and are equilibrium points of Eq. (4) for any positive values of , , and .
Let us discuss the predictions of the LV model (4) regarding stability of propagation of the two soliton sequences. For concreteness, we choose , and require that the equilibrium points at and are stable nodes (sinks). The stability requirement on , which is important for suppression of weak instability due to radiation emission, leads to for , i.e., the solitons are subject to net linear loss. The above stability requirements yield the following conditions on and PC2012b (): for , and for .
Iii Numerical simulations with the coupled-NLS model
As explained above, the LV model (4) provides an approximate description for collision-induced dynamics of circulating soliton sequences. The propagation is fully described by the coupled-NLS system (2) with periodic boundary conditions. A key question about the collision-induced dynamics concerns the possibility to suppress the instability observed in numerical simulations in Ref. PC2012b (). Here we address this important question. More specifically, we show that the instability can be effectively mitigated by using a larger temporal separation value T, compared with the one used in PC2012b (), and by eliminating the initial inter-sequence position-shift, which was introduced in PC2012b () to avoid incomplete collisions. Thus, the numerical simulations setup in the current paper consists of the coupled-NLS model (2) with periodic boundary conditions and an initial condition in the form of two periodic sequences of overlapping solitons with amplitudes and zero phase:
where . The temporal separation is taken as , compared with that was used in Ref. PC2012b (). Four different values of are used: , , , and , compared with that was considered in Ref. PC2012b (). Note that a value of is outside of the parameter range, where the perturbative approach leading to the LV model (3) is expected to be valid. The other physical parameters in the simulations are taken as , , , , and .
The -dependence of obtained by numerical solution of Eq. (2) with initial amplitudes and is shown in Fig. 1(a) for , , and , and in Fig. 1(b) for . The predictions of the LV model (4) are also presented. The agreement between the coupled-NLS simulations and the predictions of the LV model are excellent. Moreover, in all four cases tends to the equilibrium value of and the approach to is faster as increases. Similar results are obtained for and for other initial amplitude values. Note that the distances over which stable propagation is observed are larger by a factor of 6.67 compared with the the distances in Ref. PNC2010 (), and by a factor of 2 or more compared with the distances in Ref. PC2012b (). The fact that the predictions of the LV model hold even for values as large as 0.5, i.e., outside of the perturbative regime, is quite surprising. Furthermore, as shown in the inset of Fig. 1(b), the shape of the two soliton trains is retained during the propagation despite of the magnitude of and the large number of collisions.
The observation of long-distance propagation of the two soliton sequences opens the way for studying broadband on-off and off-on switching, i.e., the turning on and off of one of the propagating pulse sequences. The two switching scenarios are based on bifurcations of the steady state of the LV model. More specifically, in on-off switching, the value of the parameter is abruptly increased at the switching distance from to , where , such that the steady state becomes unstable, while another steady state at is stable. As a result, soliton amplitudes in sequences 2 and 1 tend to 1 for , but tend to 0 and , respectively, for . This means that transmission of sequence 2 is effectively turned off at . Off-on switching is realized in a similar manner, by decreasing the value of at the switching distance from to such that the steady state becomes stable. Consequently, for , the amplitudes of solitons in sequences 1 and 2 tend to and 0, respectively, but for , both amplitudes tend to 1. Thus, in this case transmission of sequence 2 is turned on at .
In order to check if these switching scenarios can be realized with sequences of colliding solitons, we numerically solve Eq. (2) with initial pulse patterns of the form (5), where , , and . Since , the critical -value is . As an example, the switching distance is taken as . In on-off switching, , and is increased from to at . In off-on switching, we use and for , and and for . The results of the simulations for are shown in Fig. 2 (a) and (b) for on-off and off-on switching, respectively. Also shown is the prediction of the LV model (4). The overall agreement between the coupled-NLS simulations and the predictions of the LV model is very good for both switching scenarios. The only exception is for small values of in on-off switching, where the LV model underestimates the value obtained with the coupled-NLS model. This is due to the fact that in this regime, the linear loss term is comparable to or larger than the Kerr nonlinearity term, leading to the breakdown of the perturbative description. Despite of this fact, the amplitudes of the solitons in sequence 2 do tend to 0, and thus, the on-off switching is fully realized. Based on the good agreement between coupled-NLS simulations and predictions of the LV model we conclude that on-off and off-on switching can be realized for the two sequences of colliding solitons. It should be noted that stable off-on switching can be achieved only when is larger than some threshold value . For the parameter values used in our simulations, . Thus, in order to implement off-on switching with the current setup of two soliton sequences, the value of the decision level distinguishing between 0 and 1 states should be set larger than .
All the propagation setups discussed so far are limited, in the sense that all amplitudes and phases within each soliton sequence are equal. It is therefore important to extend the results to more general setups. Possible extension can be achieved by launching soliton sequences with periodically alternating phases. Indeed, the LV model (4) is independent of the soliton phases, and as a result, amplitude dynamics for sequences with periodically alternating phases is expected to be the same as the dynamics for soliton sequences with a uniform phase pattern. On the other hand, it is known that intra-sequence soliton interaction strongly depends on the relative phase difference between the solitons. The interplay between this interaction and other effects, such as loss or radiation emission might induce instabilities, which would lead to the breakdown of the LV model description. For this reason, it is important to check whether the predictions of the LV model (4) do apply for soliton sequences with periodically alternating phases. For this purpose, we numerically solve Eq. (2) with pulse sequences consisting of two solitons each with initial amplitudes (sequence 1) and (sequence 2). Two sets of initial phases are used: and [set(a)]; and and [set(b)]. The other physical parameter values are taken as , , and . The results of the numerical simulations for are shown in Fig. 3 along with the predictions of the LV model. As can be seen, for both sets of initial phases the agreement between full-scale simulations and the predictions of the LV model is excellent over the entire propagation distance. Furthermore, the soliton amplitudes tend to the equilibrium value , i.e., the propagation is indeed linearly stable. Thus, our numerical simulations demonstrate that the predictions of the LV model (4) can be applied to generic setups with periodic phase patterns.
In summary, we investigated long-distance propagation as well as on-off and off-on switching of two colliding soliton sequences in the presence of second-order dispersion, Kerr nonlinearity, and a GL gain-loss profile. Using coupled-NLS simulations and a reduced LV model, we showed that stable long-distance propagation is possible for a wide range of the gain-loss coefficients, including values that are outside of the perturbative regime. The stable propagation is enabled by the presence of linear loss, the choice of a sufficiently large initial inter-pulse separation within each sequence, and the absence of significant initial time-delay between the two sequences. Furthermore, we found that robust on-off and off-on switching of one of the propagating sequences can be achieved by an abrupt change in the ratio of cubic gain and quintic loss coefficients. Additionally, we demonstrated that the results can be extended to pulse sequences with periodically alternating phases. Our study significantly enhances the recently found relation between collision-induced dynamics of sequences of NLS solitons and evolution of population sizes in LV models. Moreover, it indicates that the relation might be further extended to sequences of solitary waves of the cubic-quintic GL equation, at least in some regions of parameter space. Finally, our results might find useful applications in nonlinear waveguides with saturable absorption for stable energy equalization and broadband switching.
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