Figure 1 shows snapshots of the evolution of the density ${\left|u(x,t)\right|}^{2}$ of the numerical solution of the problem (1), (2), (4), for $\nu =\sigma =1$, Γ = 0.5, and Ω = 1, for the algebraically decaying initial condition (2). We observe that the initial datum evolves towards a localised waveform, which is strongly reminiscent of a PRW. The numerical solution is plotted by the continuous (blue) curve, against the dashed (red) curve depicting the evolution of the PRW profile (5), ${u}_{\text{PS}}(x,t-2.415;0.84)$. The maximum amplitude of the event is attained at ${t}^{\ast}=2.415$. The time *t*^{∗} is used to define the time translation of the analytical PRW solution of the integrable NLS. The power of its background *P*_{0} = 0.84 is numerically detected, so that the amplitude of the analytical PRW coincides with the maximum amplitude of the numerical event. Note that for the above set of parameters, we found that *K*_{0} = 1.54 and Λ = 1.19.

For $t\in [1.3,2.5]$, the centred localised waveform exhibits an algebraic in time growth/decay rate, close to that of the PRW soliton; notably, both the time-growing and then time-decaying centred profiles appear to manifest a locking to a PRW-type mode, as it is depicted in the left panel (a) of the top row of Figure 2, showing the time evolution of the density of the centre ${\left|u(0,t)\right|}^{2}$, for $t\in [0,3]$. The middle panel (b) of the top row of Figure 2 shows a detail of the spatial profile of the maximum event at ${t}^{\ast}=2.415$, close to the right of the two symmetric minima of the exact PRW ${u}_{\text{PS}}(x,t-2.415;0.84)$. This detail illustrates that the emerged extreme event preserves the algebraic spatial decay of the PRW soliton.

Figure 2: (Colour online) Parameters: *ν* = 1, *σ* = 1, Γ = 0.5, Ω = 1, *L* = 100. Top left panel (a): evolution of the density of the centre, ${\left|u(0,t)\right|}^{2}$, for the initial condition (2), against the evolution of the density of the centre of the PRW (5), ${u}_{\text{PS}}(x,t-2.415;0.84)$, with *K*_{0} = 1.54 and Λ = 1.19. Top middle panel (b): a detail of the spatial profile of the maximum event at ${t}^{\ast}=2.415$, close to the right of the two symmetric minima of the exact PRW ${u}_{\text{PS}}(x,t-2.415;0.84)$. Top right panel (c): Another view of the numerical density, at time ${t}^{\ast}=2.415$, where the extreme event attains its maximum amplitude, for $x\in [-100,100]$. The PRW-type structure is formed on a finite background. Bottom panels: Contour plots of the spatiotemporal evolution of the density, for the above evolution, for $t\in [0,10]$ (d), for $t\in [10,20]$ (e), and $t\in [20,30]$ (f).

The top right panel (c) of Figure 2 depicts a rescaled, extended view of the maximum extreme event, plotted for $x\in [-L,L]$. It reveals a remarkable feature of the dynamics: the extreme event occurs on the top of a finite background of amplitude ${\left|h\right|}^{2}\sim 1.19$, which is formed at the early stages of the evolution of the vanishing initial condition; on the one hand, we observed that the core of the PRW-solitonic structure is proximal to the analytical PRW of the integrable NLS with *P*_{0} = 0.84 (as numerically detected by the fitting argument of [50] discussed above), and on the other hand, it does not tend to the background of unit amplitude as the PRW of the integrable NLS when $\nu =\sigma =1$. This fact that the amplitude of the background of the PRW-type event differs from *P*_{0} = 0.84 or *P*_{0} = 1 suggests that it may be determined by the driver, as it will be analysed below. Note that the same effects were observed for increased values of *L*.

The bottom row of Figure 2 depicts contour plots of the spatiotemporal evolution of the density ${\left|u(x,t)\right|}^{2}$. The bottom left panel (d) portrays the evolution for $t\in [0,10]$. The PRW-type soliton discussed above is the first extreme event (FE) corresponding to the spot marked by the arrow. After its formation, the sustaining finite background exhibits MI dynamics, characterised by the emergence of large-amplitude localised modes. A first interesting effect is that the solution preserves at the early stage of the evolution the even spatial symmetry $u(x,t)=u(-x,t)$. The even symmetry breaking due to the presence of the driver occurs for *t* ≳ 14, as shown in the contour plot of the bottom middle panel (e), which portrays the dynamics for $t\in [10,20]$. A second interesting effect is that the later stages of MI are manifested by spatial energy localisation (initiated prior to the even symmetry breaking) and the formation of extreme amplitude solitary modes. The survived solitary modes are dominating in the dynamics, as shown in the right contour plot (f) (which shows the relevant evolution for $t\in [20,30]$), and their amplitude is increasing.

Figure 3 is an attempt to shed light to the structure of the above solitary modes. The left panel (a) depicts the evolution of the density of the centre for $t\in [0,16]$, showing that it undergoes chaotic oscillations. The oscillations in the subinterval [10], [16] correspond to those of the top of the solitary mode, which was depicted in the contour plot of the evolution of the density of Figure 2e. This is first evidence that the solitary mode possesses the structure of a (large amplitude) “chaotic soliton” in the sense of [51], [42], [43], [44] than a breather. A second evidence is illustrated in the middle panel (b), showing a plot of the evolution of the density of *x* = 0.33, i.e. ${\left|u(0.33,t)\right|}^{2}$, for $t\in [28,30]$; it is actually a detail – in this time subinterval – of the evolution of the solitary mode depicted in the contour plot of Figure 2e. The centred mode has slightly slided at *x* = 0.33, and the oscillations of the mode are reminiscent of those presented in [42, Fig. 1, p. 4]. We may conjecture that the system, for the considered example of parameters, is locked to a “chaotic” soliton and not to a large-amplitude breathing mode. Nevertheless, varying the parameters of the driver, the appearance of more breather-like waveforms may not be excluded (at least, at the early stages of the evolution), as shown in Figure 4, depicting the dynamics when Γ = 1 and Ω = 2.7. These breather-like modes may evolve to the aforementioned “chaotic” solitons at later stages of the dynamics.

Figure 3: Parameters: *ν* = 1, *σ* = 1, Γ = 0.5, Ω = 1, *L* = 100. Left panel (a): Temporal evolution of the density of the centre ${\left|u(0,t)\right|}^{2}$, for the initial condition (2). Middle panel (b): A detail of the temporal evolution of the density at *x* = 0.33 (${\left|u(0.33,t)\right|}^{2}$), for $t\in [28,30]$.

Figure 4: Parameters: *ν* = 1, *σ* = 1, Γ = 1, Ω = 2.7, *L* = 100, and initial condition (2). Left panel (a): Contour plot of the spatiotemporal evolution of the density for $t\in [0,20]$. Right panel (b): Temporal evolution of the density of the centre ${\left|u(0,t)\right|}^{2}$, for $t\in [0,14]$.

Figure 5: (Colour online) Parameters: *ν* = 1, *σ* = 1, Γ = 0.5, *L* = 100, and initial condition (2). Left panel (a): Contour plot of the spatiotemporal evolution of the density for $t\in [0,30]$, when Ω = 0.5. Left panel (b): as in panel (a), but for Ω = 1.5.

The numerical results that follow come out from an indicative study on the dependencies of the above dynamics on the parameters of the driver. In Figures 5 and 6, we fixed Γ = 0.5 and $\nu =\sigma =1$ as above, and we varied its frequency Ω. The left panel (a) of Figure 5 depicts the contour plot of the spatiotemporal evolution of the density for Ω = 0.5 and the right panel (b) for Ω = 1.5. Despite some changes in the patterns, the overall picture of the dynamics observed in the case Ω = 1 (large-amplitude solitons, following after the emergence of extreme FE) persists for both examples of $\mathrm{\Omega}=0.5<1$ and $\mathrm{\Omega}=1.5>1$, respectively.

Drastic changes appear for larger values of the driver’s frequency Ω. These changes are illustrated in columns (a) to (c) of Figure 6. In each column, the upper panel shows the temporal evolution of the density of the centre ${\left|u(0,t)\right|}^{2}$ for $t\in [0,30]$, and the bottom panel shows a contour plot of the spatiotemporal evolution of the density for $x\in [-10,10]$ and $t\in [0,30]$. Column (a) depicts the numerical results for Ω = 2, a value that in the present study – for the considered set of parameters – can be viewed as “critical”: The large-amplitude peak of the density of the centre observed in the top panel (a) corresponds to a PRW-type event – the spot of the bottom panel (a). Remarkably, afterwards – in contrast with the previous observations – we see that the large-amplitude, “chaotic solitary” modes disappear; the later stages of the dynamics are manifested by small amplitude chaotic oscillations instead, as depicted in the inset of the top panel (a).

Figure 6: (Colour online) Parameters: *ν* = 1, *σ* = 1, Γ = 0.5, *L* = 100, and initial condition (2). Column (a): The upper panel (a) shows the temporal evolution of the density of the centre ${\left|u(0,t)\right|}^{2}$, for $t\in [0,30]$, when Ω = 2. The bottom panel (a) shows the contour plot of the spatiotemporal evolution of the density for $x\in [-10,10]$ and $t\in [0,30]$, when Ω = 2. Column (b): Same as in column (a), but for Ω = 4. Column (c). Same as in column (a), but for Ω = 6.

Increasing the driver’s frequency to Ω = 4, we observe in column (b) yet another remarkable effect: the disappearance of the PRW-type events. The dynamics are locked to a spatially localised mode whose top is oscillating in time almost periodically with moderate amplitudes. The frequency of the oscillations of the top of such “quasiperiodic” solitary modes seems to be dictated by the frequency of the driver and increasing, as shown in column (c), depicting the relevant evolution for the increased value of Ω = 6. When Ω is further increased, the frequency of the above oscillations is also increasing, suggesting that the dynamics tend to lock to a stationary soliton. This is expected, since in the limit of large Ω, as the period of the oscillations is dictated by the frequency of the driver, it should tend to zero.

Next, keeping the driver’s frequency fixed to Ω = 1, a similar dynamical phenomenology to the one presented in Figure 2 emerged for the reduced forcing amplitude Γ = 0.25. The dynamics for this example are summarised in Figure 7. In this case, the FE is found to be close to the PRW-soliton ${u}_{\text{PS}}(x,t-3.33;1.06)$ of the integrable limit, with *K*_{0} = 0.97 and Λ = 0.94. The presentation is the same as in Figure 2. The top left panel (a) shows the time evolution of the density of the centre, where its time growth and time decay is still close to the PRW for $t\in [2,3.5]$. The top middle panel (b) illustrates that the numerical solution, when the FE attains its maximum density at ${t}^{\ast}=3.33$, yet captures the profile of the PRW around its symmetric minima, even closer than the case of Γ = 0.5. The whole profile of the FE at the time of its maximum density is depicted in the top right panel (c). As a result of the decreased forcing amplitude, the amplitude of the finite background supporting the PRW event is also decreased, i.e. ${\left|h\right|}^{2}\sim 0.34$. Accordingly, the emerging localised modes possess reduced amplitude. Additionally, we observe a delay in the emergence of the extreme amplitude solitary modes, as shown in the panels (d) to (f) of the bottom row, portraying contour plots of the spatiotemporal evolution of the density, for $t\in [0,60]$.

Figure 7: (Colour Online) Dynamics of the initial condition (2), for forcing amplitude Γ = 0.25 and the rest of parameters fixed as in Fig. 2. (a) Evolution of the density of the center, ${\left|u(0,t)\right|}^{2}$, for the initial condition (2), against the evolution of the density of the center of the PRW (5), ${u}_{\text{PS}}(x,t-3.33;1.06)$, with *K*_{0} = 0.97 and Λ = 0.94. (b) A detail of the spatial profile of the maximum event at *t*^{∗} = 3.33, close to the right of the two symmetric minima of the exact PRW ${u}_{\text{PS}}(x,t-3.33;1.06)$. (c) Another view of the numerical density, at time *t*^{∗} = 3.33, where the extreme event attains its maximum amplitude, for *x* ∈ [−20, 20]. The PRW-type structure is still formed on a finite background. Bottom panels: Contour plots of the spatiotemporal evolution of the density, for the above evolution, for $t\in [0,20]$ (d), for $t\in [20,40]$ (e), and $t\in [40,60]$ (f).

Proceeding to a progressive decrease of Γ, we observe a suppression of the extreme wave dynamics (similar to the case of increasing Ω). These suppression effects are illustrated in Figure 8, where the presentation follows that of Figure 6: for Γ = 0.1, we observe in column (a) that first the large-amplitude solitary structures disappear while the emergence of rogue waves still persists. This feature is shown by the large-amplitude peaks of the density of the centre shown in the top panel (a), which correspond to the localised spots of the contour plot portrayed in the bottom panel (a). Further suppression occurs for Γ = 0.05 as shown in column (b) (manifested by the decrease of amplitude of the FE), while for Γ = 0.01, the dynamics seems again to tend to lock to a stationary soliton.

Figure 8: (Colour online) Parameters: *ν* = 1, *σ* = 1, Ω = 1, *L* = 100, and initial condition (2). Column (a): The upper panel (a) shows the temporal evolution of the density of the centre ${\left|u(0,t)\right|}^{2}$, for $t\in [0,30]$, when Γ = 0.1. The bottom panel (a) shows the contour plot of the spatiotemporal evolution of the density for $x\in [-10,10]$ and $t\in [0,30]$, when Γ = 0.1. Column (b): Same as in column (a), but for Γ = 0.05. Column (c): Same as in column (a), but for Γ = 0.01.

A complete study of the bifurcations in the full parameter $(\mathrm{\Gamma},\mathrm{\Omega})$-parametric space, apart from being essential, might be a formidable task (as it may involve the nontrivial analysis of resonances given in [39], [40], [41]) and is beyond the scope of the present work. However, we may already conjecture on the dependencies of the exhibited dynamics on the amplitude Γ and the frequency Ω of the driver. For instance, for fixed *ν*, *σ*, *L*, we may identify thresholds *Γ*_{thresh} and *Ω*_{thresh} such that for suitably fixed Ω (or Γ), if $\mathrm{\Gamma}>{\mathrm{\Gamma}}_{\text{thresh}}$ (or $\mathrm{\Omega}<{\mathrm{\Omega}}_{\text{thresh}}$), extreme wave dynamics emerge. The above numerical studies provided the following examples for the thresholds: we found that for $\nu =\sigma =1$ and *L* = 100, when Ω = 1, ${\mathrm{\Gamma}}_{\text{thresh}}<0.05$, and when Γ = 0.5, then ${\mathrm{\Omega}}_{\text{thresh}}>2$. Furthermore, in the suppression regimes, in the limit of small Γ (for fixed Ω) or in the limit of large Ω (for fixed Γ), the dynamics tend to lock to a stationary soliton of the integrable NLS. This is expected, as in the limit of small Γ, the system approximates the integrable limit.

The above observations will be further underlined by the comments on the behaviour of the integrable limit Γ = 0, for the same type of vanishing conditions.

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