The direct scattering study of the parametrically driven nonlinear Schrödinger equation

The development of inverse scattering theory [1] – [5] led to the realization that the scattering data can be used as nonlinear modes to analyse solitons in nonintegrable systems. Kaup [6] pioneered the adiabatic analysis, a method that determines the effect of small perturbations on solitons. The method was generalized to the perturbed inverse scattering method [7, 8] so that the effects of radiation could be taken into account. These methods led to insight into the effect of many perturbations, including linear and nonlinear dissipative and forcing terms. The restriction of these methods is that they can only be applied to known solutions of the unperturbed equation. One way to overcome this difficulty is to calculate the direct scattering data numerically. We refer to this method as the direct scattering study.

The direct scattering study was introduced by Overman and co-workers [9] who developed a numerical scheme to calculate the scattering data for periodic potentials. They applied this scheme to investigate chaotic solitons that arise in the damped-driven sine-Gordon equation. Subsequently Ablowitz and co-workers [10, 11] applied this scheme to analyse chaotic solutions that were observed in the integrable NLS equation for periodic potentials. Their study led to the identification of homoclinic crossings, the result of numerical errors, as the source chaos. Bofetta and Osborne [39] developed a numerical scheme for the calculation of the scattering data for decaying potentials defined on the real line. Their scheme was used to study soliton generation in the integrable NLS equation from Gaussian initial potentials [12], chirped potentials [13] and disordered optical fields propagating in nonlinear bulk waveguides [14]. It was also applied to study solitons in many nonintegrable NLS equations where the inclusion of various effects were studied, including linear damping [18, 19], periodic amplifiers [15], higher-order dissipation combined with linear gain [16], quintic nonlinearities [20] and many more [17] – [24].

In this paper we illustrate the ability of the direct scattering study to analyse soliton dynamics in two different ways. Firstly the identification of different soliton structures can lead to insight into the dynamical structure and spatial features of solitons in nonlinear equations. Secondly the ability to calculate energy-related measures of radiation allows one to analyse the effect of radiation in the dynamics. For this purpose we consider a model that exhibits a rich diversity of soliton solution, namely the parametrically driven nonlinear Schrödinger (PDNLS) equation, given by

Here $h>0$ corresponds to the driving strength, while $\gamma>0$ represents the damping coefficient. The PDNLS equation is an important model for resonant phenomena in hydrodynamics [25] – [27] nonlinear optics [28] – [30] and many more (see [31, 32] and the references therein).

The PDNLS equation admits a trivial zero solution. [34] The zero solution is stable whenever the driving strength $h\leq h_{c}$, where

When the driving strength exceeds the damping coefficient $\gamma\leq h$, the PDNLS equation has two additional time-independent soliton solutions [34]

where $A^{\pm}=\left(1\pm\sqrt{h^{2}-\gamma^{2}}\right)^{1/2},$ $\theta^{+}=\frac{1}{2}\mbox{arcsin}\left(\gamma/h\right)$ and $\theta^{-}=\theta^{+}-\frac{\pi}{2}.$ The solution $\psi^{-}$ is unstable for all choices of damping and driving strengths $\gamma$ and $h$ respectively. The $\psi^{+}$ solution has a stable region. For the large damping regime $\gamma\geq 0.378$, the $\psi^{+}$ solution is stable whenever $\gamma\leq h\leq h_{c}$. In the smaller damping regime $\gamma<0.378$ the $\psi^{+}$ solution destabilizes due to a Hopf bifurcation. In this regime the stability criteria for $\psi^{+}$ is given by $\gamma\leq h<h_{H},$ where $h_{H}<h_{c}$ is the critical driving strength where the Hopf bifurcation occurs. The Hopf bifurcation gives rise to time-dependent soliton solutions. We therefore only consider the small damping regime.

Bondila and co-workers [33] generated the time-dependent soliton attractors through direct simulation. They also reported two other attractors in the unstable region of $\psi^{+}$, namely the zero attractor and spatio-temporal chaotic attractors. These results are summarized in Figure 1. They also characterized the temporal structure of the soliton attractors, reporting that the majority of soliton attractors have a simple 1-period mode of oscillation. The only exception is in a narrow band near the zero attractor region where period-doubling bifurcations lead to temporally chaotic solitons. Barashenkov and co-workers [34] showed that these chaotic soliton attractors are caused by homoclinic instabilities.

From a spatial point of view, soliton attractors consist of a soliton part and a radiation part. The soliton part consists of a quiescent (non-propagating) soliton that oscillates temporally. The radiation part consists of waves with small amplitudes and widths relative to the soliton part of the solutions. These waves arise symmetrically about the soliton, and propagate away from the soliton. Radiation waves are subject to dissipation associated with the linearized equation, and account for losses in the system. Time-dependent soliton attractors arise due to a balance between energy gained through driving and energy lost through dissipation [32].

The interaction between solitons and radiation plays an important role in the dynamics of the PDNLS equation. Alexeeva and co-workers [40] showed that radiation is responsible for instabilities in the undamped case. Shchesnovich and Barashenkov [37] used inverse scattering theory to construct a reduced system of ODE’s using a single nonlinear mode and a single radiation mode. Their system produced the period-doubling route to chaos, as well as the zero attractor region. However they were unable to reproduce the full complexity of the attractor chart Figure 1. Barashenkov and co-workers [32] used a linearized analysis to study the effect of damping on radiation emissions. Their analysis show that smaller damping strength suppresses the emission of radiation waves. It should be noted that this analysis ignores the effects of interaction between solitons and radiation.

In this paper we use the direct scattering study to gain insight into the role that soliton structures, radiation waves and their interaction play in the dynamics of the PDNLS equation. The paper is structured as follows: In Section 2 we discuss the generation of soliton attractors of the PDNLS equation. We analyse these results by calculating basic characteristics associated with these attractors. We show the difficulties of separating radiation from the soliton by inspection. In Section 3 we apply the direct scattering study to the solitons of the PDNLS equation. We reveal four different types of soliton attractors, based on the soliton content. The results show that large radiation emissions can excite additional nonlinear modes. We also reveal soliton complexes in the form of breather-like structures that may result from large driving strengths. Based on these results we describe the effect of damping and driving strengths on the structure of the soliton and its radiation emissions. In Section 4 we consider the period-doubling route to temporal chaos and its effect on the radiation emissions of the soliton. Here radiation emission measurement is used to describe the effect of period-doubling bifurcations on radiation emissions. We also consider soliton transients within the zero attractor region, and identify the role of solitons and radiation in the annihilation of the soliton transient. In Section 5 we consider the spatio-temporally chaotic region. We identify the formation of multiple solitons as the source of chaos formation, and discuss the role of soliton complex structures and radiation in this process. In Section 6 we conclude with a brief discussion of the results.

To generate the soliton attractors we follow the direct simulation approach of Bondila and co-workers [33]. This is done by integrating the unstable $\psi^{+}$ solution (2) numerically. Numerical errors provide destabilizing perturbations that repulse the orbit, forcing it to a different solution, or attractor, after a sufficiently long interval of integration. One could interpret this as constructing a heteroclinic orbit that approaches $\psi^{+}$ and the attractor as $t\rightarrow-\infty$ and $t\rightarrow\infty$ respectively. We used a fourth-order pseudospectral split-step scheme for integration with an interval length $L=200$, $N=2048$ grid points and a step size $\Delta t=0.005$. Direct simulation is ideal for constructing quasi-periodic and temporally chaotic solutions. It also allows one to calculate soliton transients in regions where no soliton attractors exist.

In this section we use the numerically generated solutions to study the effects of damping and driving on the oscillating soliton attractors. For 1-period soliton attractors, the temporal period $T$ of these attractors depends on both the damping and driving strengths. For a fixed driving strength, an increase in damping strength results in a decrease in the temporal period of the resulting soliton attractor. Likewise, for a fixed damping strength, an increase in driving strength leads to a decrease in the temporal period of the resulting soliton attractor. Mathematically this means that the temporal period of 1-period attractors can be written as a function $T\left(\gamma,h\right)$ that satisfies

Both the soliton and radiation parts of the soliton attractors also depend on the damping and driving strengths. Lets first consider the soliton part of the solution. One of the most important properties of the soliton is its amplitude, defined as

Here $\psi_{\gamma,h}\left(x,t\right)$ corresponds to the soliton attractor that arises for damping and driving strengths $\gamma$ and $h$ respectively. We define the magnitude of temporal oscillation in terms of the soliton amplitude, given by

The magnitude of temporal oscillation reflects the change that the amplitude undergoes during each temporal oscillation. For a fixed damping strength, the magnitude of temporal oscillation increases when the driving strength is increased, i.e.

However, for a fixed driving strength, an increase in the damping strength leads to a decrease in the magnitude of temporal oscillations, i.e.

The damping and driving have a similar effect on radiation emissions of the soliton attractors. To measure the radiation we use the radiation emission amplitude defined as

where $R_{\gamma,h}\left(t\right)=\psi_{\gamma,h}^{(m)}\left(x_{0},t\right)-\psi_{\gamma,h}^{(m)}\left(x_{1},t\right),$ $\psi_{\gamma,h}^{(m)}\left(x,t\right)=\mbox{Im}\left\{\psi_{\gamma,h}\left(x,t\right)\right\},$ and $x_{0}\left(t\right)>0$ and $x_{1}\left(t\right)>0$ are the closest points to the origin where $\psi_{\gamma,h}^{(m)}$ has a local maximum and local minimum respectively.The radiation emission amplitude indicates the size of radiation waves that is emitted during each temporal period. For a fixed damping strength, an increase in the driving strength leads to an increase in the radiation emission amplitude, i.e.

For a fixed driving strength, an increase in damping strength leads to a decrease in the radiation emission amplitude, i.e.

It is also interesting to note that the velocity of radiation waves is the same for all damping and driving strengths, provided that the radiation wave is sufficiently separated from the soliton core. These results of the dependence of soliton attractors on the damping and driving strengths are summarized in Table 1.

Another aspect of radiation waves that is influenced by the driving strength is the position where the radiation waves are formed. Radiation waves that form in the tail of the soliton can be easily identified. Moreover, if radiation waves form in the tail and move away from the quiescent soliton, it is reasonable to expect little interaction with the soliton part of the attractor. Conversely if the radiation waves form inside the soliton, it complicates the analysis of both the soliton part and the radiation part of the soliton attractor, as well as the interaction between the two components.

Numerical results show that, for smaller driving strengths, radiation waves are formed on the tail of the soliton. However, for larger driving strengths, radiation waves are formed inside the soliton core. To illustrate this, we consider the imaginary part of the soliton attractors that arises for damping strength $\gamma=0.3$. We use the spatial profile to show the formation and behaviour of the radiation waves. The spatial profile consists of all pairs $\left(x,t\right)$ that are local extremes with respect to $x,$ i.e. pairs that satisfy

Solid lines represent local minima, and dotted lines denote local maxima. In Figure 2 we show the spatial profiles of attractors that arise for a damping strength of $\gamma=0.3$ and for different driving strengths. In Figure 2 (a) we see that the radiation waves form at $x\approx\pm 3.5$. In this case the radiation waves simply move away from the soliton. However, as driving strength increases, the distance between the point of formation and the soliton decreases. This is shown in Figure 2 (b) where the driving strength is $h=0.46$. Here we see that the radiation waves form at $x\approx\pm 1.5$. Moreover, we see that the local maxima move through the soliton. When the driving strength is increased further, the radiation waves form at the origin, the same position as the centre of the soliton. This is shown in Figure 2 (c), where the spatial profile is shown for $h=0.53$. Here we see that both radiation waves are formed at the origin.

The complex formation of radiation waves within the soliton core, associated with larger driving strengths, motivates the use of the direct scattering study. The inverse scattering transform method is an analytical method for solving the focusing NLS equation. The method maps the initial condition onto an auxiliary space where the transformed set, known as the scattering data, evolves in a trivial manner. The evolved scattering data can then be used to reconstruct the solution. An important aspect about the mapping of the potential onto the scattering data is that it divides the potential into a discrete set and a continuous set. The discrete set describes the soliton part of the solution, while the continuous set describes the radiation part of the solution. For soliton attractors of the PDNLS equation the scattering data can be used as a filter to separate the soliton part from the radiation part.

The discrete eigenvalues of the associated Zakharov-Shabat (ZS) eigenvalue problem form part of the discrete scattering data. The ZS eigenvalue problem is defined as

Here $\psi$ is a solution of the PDNLS equation at a fixed point in time and $\zeta$ is an eigenvalue. The discrete eigenvalues of the ZS eigenvalue problem (6) are associated with eigenfunctions $v_{1,2}$ that decay to zero when $\left|x\right|\rightarrow\infty$. We refer to the discrete eigenvalues as the ZS eigenvalues. The set of ZS eigenvalues are referred to as the soliton content, due to its relationship with solitons. In the literature these eigenvalues are also sometimes referred to as nonlinear modes. Their relationship with solitons is briefly discussed in the appendix.

We calculated the soliton content numerically by using the optimal Floquet exponent of Hill’s method, described in [35]. The results reveal that there are four different types of soliton attractors, each with a unique behaviour pattern with respect to the soliton content. After briefly illustrating the different types of attractors in Sections 3.1 – 3.4, we discuss the general structure of soliton attractors in Section 3.5.

Spatio-temporal chaos forms as a result of exceedingly large driving strengths. When $\psi^{+}$ is integrated in this region a soliton transient emerges. At a critical point in time additional solitons start to form. This moment marks the onset of spatio-temporal chaos. As time evolves beyond this point more solitons are formed, resulting in the solution to spread across the spatial domain. A typical example of the formation of chaos is shown in Figure 13. Here we integrated $\psi^{+}$ for $\gamma=0.2$ and $h=0.82.$ For $t<105$ we see a soliton transient. At $t\approx 108$ we see the formation of two solitons, marking the onset of spatio-temporal chaos. Beyond $t\approx 108$ we see that the formation of more solitons results in the the spreading of the solution across the spatial domain.

To investigate the transition from soliton transients to spatio-temporal chaos we integrate $\psi^{+}$ numerically for damping and driving strengths near the lower boundary of the spatio-temporal chaotic region. We also calculate the soliton content during the transition from soliton transient to spatio-temporal chaos in order to monitor the role of the solitons and radiation in this transition.

Numerical results show that, depending on the damping strength, either lateral waves or radiation tails are responsible for the formation of additional solitons. In particular, for smaller damping strengths $\gamma\leq 0.34$ lateral waves are responsible for the formation of additional solitons, leading to chaos. In this case the lateral waves become solitons. Note that this is in contrast to the Type IV attractors where lateral waves become radiation waves. For larger damping strengths $\gamma>0.34$ the formation of additional solitons is associated with radiation tails, consisting of multiple radiation waves, that couple to form additional solitons. In the latter case the role of the soliton is to “feed” the radiation tail, resulting in the formation of solitons in the tail of the soliton. We refer to the role of the soliton in the smaller and larger damping regime as direct and indirect respectively.

The direct role of the soliton in the formation of spatio-temporal chaos for the smaller damping regime is illustrated in Figure 14 where the results are shown for damping strength $\gamma=0.2$, and driving strength $h=0.802$. In Figure 14 (a) the transition from soliton transient to spatio-temporal chaos is shown. The lower arrow shows the lateral wave that forms during the penultimate temporal oscillation. One can clearly see that this wave dissipates harmlessly, and plays no role in the formation of additional solitons. The upper arrow shows the lateral wave that form during the final temporal oscillation before the formation of multiple solitons. We see that this wave is transformed from lateral wave to soliton at $t\approx 120$. In Figure 14 (b) and (c) we show the imaginary and real parts of the associated ZS eigenvalues respectively. Notice that splitting eigenvalues are observed during the formation of multiple solitons on the interval $115\leq t\leq 120$. Splitting eigenvalues are observed whenever the lateral waves are responsible for the formation of additional solitons. This shows that the formation of lateral waves in the breather-like structures of Type III and Type IV attractors are responsible for the formation of chaos in the smaller damping regime.

The indirect role of the soliton in the formation of spatio-temporal chaos for the larger damping regime is illustrated in Figures 15 and 16, corresponding to damping strengths $\gamma=0.35$ and $\gamma=0.37$ respectively. In Figure 15 (a) we show the solution that arises when $\psi^{+}$ is integrated numerically for $\gamma=0.35$ and $h=1.019$. The arrow shows the lateral wave formed during the penultimate temporal oscillation of the soliton transient. At $t\approx 879$ we see that this wave combines with another radiation wave (formed during the final temporal oscillation of the transient) to form additional solitons. In Figure 15 (b) and (c) we show the imaginary and real parts of the associated ZS eigenvalues respectively. In the former we see that no splitting occurs until the formation of additional solitons at $t\approx 877.$ Figure 15 (c) shows that the soliton content of the soliton transient contains side eigenvalues. In Figure 16 (a) we show the solution that arises when $\psi^{+}$ is integrated numerically for damping and driving strengths $\gamma=0.37$ and $h=1.063$ respectively. The arrow shows the radiation wave that is emitted at $t\approx 2472$, three oscillations before the formation of multiple solitons. Here we see that this radiation wave does not dissipate, but remains in the tail of the soliton. Finally at $t\approx 2477$ this wave combines with other radiation waves to form multiple solitons. In (b) and (c) we show the real and imaginary parts of the associated ZS eigenvalues respectively. In this case we see no splitting eigenvalues. Instead we see the formation of multiple purely imaginary ZS eigenvalues. This is typical of the transition from the soliton transient to chaos when radiation tails are responsible for the formation of multiple solitons.

We conclude that the soliton part of the soliton transient plays a direct role in the formation of spatio-temporal chaos for the damping regimes $\gamma\leq 0.34$. The splitting eigenvalues show that the soliton breaks up to form lateral waves. The lateral waves are sufficiently large so that the effect of driving dominates the dissipative effects. The result is that the lateral waves become solitons that lead to the formation of spatio-temporal chaos. In contrast, in the large damping regime $0.34<\gamma\leq 0.372$ the soliton part of the soliton transient plays a more indirect role in the sense that its role is restricted to the emission of radiation waves. In this setting the strong driving strength weakens the effect of dissipation, so that the lifetime of radiation waves increase. These radiation waves interact, resulting in the formation of multiple solitons.

It should be noted that these results tie in well with the soliton structure chart Figure 7, where the spatio-temporal chaotic attractor region is bounded below by Type IV attractors for $0.24<\gamma<0.328$. We also know from the transient chart Figure 12 that the spatio-temporal chaotic attractor region is bounded below by Type IV transients for $\gamma<0.24$. These results can be interpreted as follows: Chaos in the regime $\gamma<0.328$ where the spatio-temporally chaotic region is bounded below by a Type IV attractor/transient region is caused by lateral waves. Chaos in the regime $\gamma>0.351$ where the spatio-temporally chaotic region is bounded below by Type I or Type II attractors indicate that, at increased driving strengths, radiation tails are responsible for the formation of spatio-temporal chaos. Chaos in the regime $0.328\leq\gamma\leq 0.351$ where the spatio-temporally chaotic region is bounded below by Type III attractors are formed either by lateral waves ($\gamma\leq 0.34$) or radiation wave trains ($\gamma>0.34$).

In this paper we used the direct scattering study to investigate the dynamics of the parametrically driven nonlinear Schrödinger equation. We used the scattering data in two ways, namely soliton identification and radiation emissions measurement. The former were used to identify four different types of time-dependent soliton attractors, each with a unique soliton structure. The results show that the soliton attractors associated with moderate driving strengths (near the Hopf bifurcation boundary) consist of only a single nonlinear mode. The increase of radiation emissions associated with increased driving strengths lead to the temporary excitation of additional nonlinear modes in the form of side eigenvalues. In the intermediate damping regime large driving strengths lead to the formation of breather-like structures, characterised by the formation of a lateral wave on each side of the soliton. Larger driving strengths lead to the destruction of the soliton structure on a periodic basis, resulting in the loss of nonlinear modes. This is accompanied by the periodic formation of additional nonlinear modes.

Radiation emission measurements were used to investigate the transition from period-doubling bifurcations to temporal chaos to destabilization of soliton attractors associated with increased driving strengths. The calculation of radiation emissions show that the rate of radiation emission, associated with increased driving strengths, increases faster when period-doubling bifurcations occur. This suggests that period-doubling may act as a catalyst for the destabilization of the soliton attractor that results in the zero attractor region.

We also used the direct scattering study to analyse soliton transients. For the zero attractor region we showed that soliton transients have the same structure as soliton attractors. The results show that larger radiation emissions are required to destroy soliton transients for larger driving strengths. This is evident from the restabilization that occurs for the damping regime $0.25<\gamma<0.27.$ A study of the soliton transients that arise for the spatio-temporally chaotic region showed that the formation of breather-like structures, associated with Type III and Type IV attractors, play an essential role in the formation of chaos for the damping regime $\gamma\leq 0.34$. For the larger damping regime $\gamma>0.34$ chaos forms as a result of coupled radiation waves in the radiation tail.

CO would like to thank the National Institute of Theoretical Physics (NITheP) and the University of Cape Town’s postgraduate funding office and PPI fund for financial assistance. He would also like to thank the University of Cape Town’s ICTS High Performance Computing Team for support with the computer simulations.

Here we consider the relationship between the soliton content and the associated soliton. This is done by reconstructing soliton solutions for three different pairs of ZS eigenvalues namely 1) a single ZS eigenvalue, 2) two purely imaginary ZS eigenvalues and 3) two complex ZS eigenvalues symmetric about the imaginary axis. For each case we use the reconstructed solitons to identify the behaviour associated with the soliton content of the four types of attractors reported in Section 3.