A crucial and intriguing question in nonlinear dynamics concerns the persistence of dynamical features of integrable systems in the presence of perturbations. In the context of the nonlinear Schrödinger (NLS) partial differential equations, and particularly for the integrable NLS (with a focusing, cubic nonlinearity), one of these features that is receiving tremendous interest concerns the emergence of rogue waves: extreme wave events possessing spatiotemporal localisation and large amplitude, which are mathematically described by its class of rational solutions. For the fundamental representatives of this class, namely, the Peregrine rogue wave (PRW), and the space- or time-periodic solutions as the Akhmediev and Kuznetsov–Ma (KMb) breathers, respectively , , , , their physical relevance is justified by numerous episodes in the ocean , ,  and experimental observations in hydrodynamics , , , nonlinear optics and lasers , , , superfluidity , and plasma physics . Such natural and experimental evidences motivated recent advances on the predictability of extreme wave events based on studies analysing the interactions of energy localisation and strong local nonlinearity , .
In the context of the aforementioned persistence question, the robustness of rational solutions and rogue wave dynamics in the presence of perturbations has been identified for various special cases of extended NLS equations. Representative key works refer to third-order (including modified Hirota  and Dysthe  equations) , ,  as well as fourth- ,  and fifth-order  models. Important extensions to coupled equations and systems include  for parity-time symmetric systems,  for NLS systems with derivative nonlinearities, and  for Manakov systems (physically relevant in the context of Bose-Einstein condensates). In a different perspective, results on the spectral analysis of the PRW as a limiting case of the KMb are presented in . Furthermore, the findings of  suggest that a suitably defined dispersion or nonlinearity management, if applied to a continuous wave (cw) background, may effectively stabilise the supported PRW and KMb waveforms.
In this spirit, and motivated by key works on the linearly forced/driven NLS equations in the context of rogue waves and the robustness of localised waveforms in nearly integrable systems , , , , we consider, instead of the linear forcing, the action of an external, time-periodic driver. The relevance of such a driving term, arising in a number of physical systems (as in the theory of charge-density waves and plasma physics), has been extensively analysed in . Then, continuing along the lines of our recent work , we examine, starting herein numerically, the potential emergence of spatiotemporal algebraically decaying waveforms in the dynamics of the associated, periodically driven NLS model.
However, the approach we will follow regarding the initial conditions differs drastically from investigations exciting PRWs, from initial data defined as interactions between a pulse of small amplitude and a cw , , , . Instead of such initial conditions, we ask for the possibility to excite PRWs from vanishing initial conditions, decaying either at an algebraic or exponential rate.
The answer is positive: the main finding is that extreme waveforms, strongly reminiscent to the PRW, can be excited from the decaying initial data. Their profile and statiotemporal decay rates are close to that of the analytical PRW solution. An important feature of this finding is that these PRW-solitonic structures emerge on the top of a finite background, which is formed at the early stages of the evolution, although the initial condition decays to zero. The birth of the PRW-type waveforms can be potentially understood by a synergy of the modulational instability (MI) of the cw solutions of the model and the persistence of the localisation of the initial condition on the top of the emerged unstable background. In other words, for vanishing initial data, in the presence of the periodic forcing, the system self-induces the effects of the MI mechanism analysed in , , , , for the excitation of PRWs.
The article is structured as follows: In Section 2, we report the results of the numerical simulations and analytical considerations on the stability of spatial homogeneous solutions. We also briefly comment on the dynamics of the linearly damped and forced model, and of the integrable NLS, initiated from the same, decaying initial conditions. In Section 3, we summarise our results with an eye towards future work.
2 Numerical Results
2.1 Brief Description of the Model
In this section, we report the results of direct numerical simulations on the dynamics of the periodically driven, NLS equation
In (1), the parameter ν is the second-order (group velocity) dispersion, and σ is the strength of the nonlinearity. The parameters Γ and Ω correspond to the amplitude and frequency of the driver, respectively. Let us recall that under the change of variable
Equation (1) defines a nonintegrable perturbation of the focusing integrable NLS, which corresponds to the case Γ = 0. It is one of the fundamental partial differential equations exhibiting complex , even spatiotemporal chaotic behaviour , , together with its dissipative counterparts , , . The impact of the breaking of the hyperbolic structure of the integrable NLS, in the emergence of complex dynamics for the damped and forced models, has been rigorously analysed in , , , . For equations incorporating higher-order dissipation, we refer to ,  and references therein.
The numerical experiments will simulate the dynamics of (1), excited by either the quadratically decaying initial condition
or the exponentially decaying
which resembles the profile of a bright-soliton solution of the integrable NLS. The model will be supplemented with periodic boundary conditions
The system was integrated by using a pseudospectral method for the spatial discretisation and an adaptive step Runge–Kutta method for the time stepping. Concerning the implementation of the periodic boundary conditions, for cases (2) and (3) of the initial data, let us recall the following: These conditions are strictly satisfied only asymptotically, as L → ∞. For a finite length L, the initial profiles, as well as their spatial derivatives, have jumps across the end points of the interval
where the parameters
2.3 Results of the Direct Numerical Simulations
Figure 1 shows snapshots of the evolution of the density
The top right panel (c) of Figure 2 depicts a rescaled, extended view of the maximum extreme event, plotted for
The bottom row of Figure 2 depicts contour plots of the spatiotemporal evolution of the density
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
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
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
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
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.
A complete study of the bifurcations in the full parameter
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.
2.4 Effects from Continuous Wave Solutions
The existence and stability properties of cw solutions of the forced NLS (1) should have an important role on the birth of the transient PRW-type dynamics. In what follows, we fix for simplicity
There exist for (1), under the dispersion relation
For Γ > 0, the case we are restricted herein, (7) has only real solutions for h: Indeed, let
From the second equation, we have that either B = 0, or
for small ε > 0, which we insert into (1). By using the dispersion relation (7) and linearising the system, i.e. neglecting terms of order ε2 and higher, we derive the equation this small influence satisfies:
To examine MI, we may assume that the perturbation u1 is harmonic, i.e.
where k and ω denote the wavenumber and the frequency of the perturbation, respectively. Next, substitution of the expression (10) for u1 in the linearised (9) yields the following algebraic system for c1 and c2:
Seeking for nontrivial solutions c1 and c2 of the above system, we require the relevant determinant to be zero; this way, we obtain the following dispersion relation:
The perturbation (8) suggests that solutions (6) are modulationally unstable if ω is complex, i.e. when the right-hand side of (11) is negative. Solving the equation
in terms of k, we find the following solutions:
The roots (12) define the instability bands of the cw solutions (6). There is not a loss of generality to be restricted for k > 0. Then, the instability bands are defined as follows:
, if .
, if .
Modulational instability has been proved an essential mechanism for the emergence of rogue waves , , . In the light of the MI analysis recalled above, let us reconsider the dynamics presented in the example of Figures 1 and 2. When Ω = 1 and Γ = 0.5, (7) has one real root h1 = 1.19. Thus, the corresponding cw solution (6), with
In this regard, we may conjecture that the background sustaining the PRW-type waveform shown in the snapshots of Figure 1 (and Fig. 2c) is self-induced as the system tends transiently to lock its cw solution in the presence of the forcing, e.g. the solution of amplitude
We should remark that the above arguments are further supported by the detailed analysis of , on the adiabatic excitation and control of N-band solutions (N-phase waves) for the forced NLS. Particularly relevant is the analysis on the excitation of the spatially homogeneous (0-band) solutions (6) from zero initial conditions, which are continuously synchronised with the driver (despite the variation of the driver’s frequency); in our case, the vanishing initial conditions define perturbations of the zero background (being modulationally stable in the case of the integrable limit Γ = 0).
2.5 Comment on the Dynamics of the Damped Counterpart
The effects of linear loss, solely influencing the evolution of Peregrine solitons in the 1D-focusing NLS, have been analysed via nonlinear spectral analysis in ; the unforced, damped NLS equation is physically significant in hydrodynamics and nonlinear optics , , . Numerical and experimental studies  confirmed the observation of higher-order MI dynamics in water waves.
Here, we illustrate that dynamical behaviour of (1) discussed in the previous paragraphs seems to be robust for small damping strengths in the presence of the periodic forcing. For instance, this robustness was identified for the linearly damped counterpart of (1):
Yet, this model is of particular interest in various physical contexts, as in plasma physics ,  (governing the dynamics of a collisional plasma driven by an external rf field). The dynamics of (13) are captured by a finite dimensional global attractor. For its existence and analyticity properties, we refer to , .
Figure 9 depicts the results of the numerical study for ν = 2, σ = 1, damping strength γ = 0.02, and Γ = 0.5. This time, we have used the sech-profiled initial condition (3), for
2.6 Comment on the Dynamics of the Integrable NLS Limit
It is important to note that the dynamics exhibited by the integrable NLS assuming the initial condition (2) or (3) totally differs from those of the forced (and damped) counterpart discussed in the previous paragraphs, although well understood. Figure 10 summarises the results of the numerical study of (1), for Γ = 0.
The left column (a) depicts numerical results for the evolution of the initial condition (2), when
The middle column (b) depicts the dynamics of the initial condition (2), for ν = 2, σ = 1. The initial condition disperses, as shown in the snapshot of the density for t = 20 of the upper panel (b), and the contour plot of its spatiotemporal evolution, shown in the bottom panel (b).
Finally, column (c) depicts the dynamics of the initial condition (3), for
It should be also highlighted that for the generalised focusing Hamiltonian NLS
(δ = 1 corresponds to the integrable NLS), rogue waves can be still excited by spatially decaying initial conditions as it was found in . However, the observed extreme waves therein (excited by generic Gaussian wave packets as their width is varied) are decaying to zero. This is a vast difference of the results of , with those presented in the present article.
Summarising, comparing the dynamics of the integrable NLS (Γ = 0) with those of the forced (Γ > 0) (1) [and damped (γ > 0) (13)], it is clear that the birth of extreme events for the latter, initiated by vanishing initial conditions, is far from any integrable limit approximation, , , and further justifies the potential existence of thresholds for the driver’s amplitude and frequency, with the properties described at the end of Section 2.3.
In this work, direct numerical simulations revealed the excitation of Peregrine-type solitonic waveforms, from vanishing initial conditions (possessing an algebraic or exponential spatial decaying rate), for the periodically driven NLS equation. The PRW-type waveforms emerge as first events of the evolution, on the top of a self-induced finite background. This dynamical behaviour can be understood in terms of the existence and modulation instability of the cw solutions of the model and the preservation of the spatial localisation of the initial condition at the early stages of the evolution. Revisiting the dynamics of the corresponding conservative NLS for the same type of initial conditions, it was shown that the above dynamics should be considered as far from approximations from the integrable limit. We also commented that this behaviour may persist in the linearly damped and forced counterpart, at least under the presence of small damping strengths. Importantly, it appears that the emergence of the Peregrine soliton excited by decaying initial conditions as a universal, coherent structure in the dynamics of the 1D – integrable NLS  – as studied therein, in its semiclassical limit scenario ,  – can be robust in the presence of forcing and damping. Notably, for the persistence of semiclassical type dynamics in the presence of a spatiotemporally localised driver (pending on the spatial/temporal scales of the latter and the magnitude of the damping strength), we refer to our recent work .
Future directions include further investigations on forced and damped NLS models, which may be considered in
This research is co-financed by Greece and the European Union (European Social Fund – ESF) through the Operational Programme (Human Resources Development, Education and Lifelong Learning 2014–2020) in the context of the project “Localized and quasiperiodic solutions for partial differential equations: Dynamical paths from mathematical ecology to nonlinear physics” (MIS 5004244).
E. A. Kuznetsov, Sov. Phys.-Dokl. 22, 507 (1977).
E. Pelinovsky and C. Kharif (Eds .), Extreme Ocean Waves, Springer, New York 2008.
C. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue Waves in the Ocean, Springer, New York 2009.
A. R. Osborne, Nonlinear Ocean Waves and the Inverse Scattering Transform, Academic Press, Amsterdam 2010.
M. Onorato, S. Residori, and F. Baronio, Rogue and Shock Waves in Nonlinear Dispersive Media, Springer-Verlag, Heidelberg 2016.
A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, Phys. Rev. Lett. 106, 1 (2011).
A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, Phys. Rev. X 2, 1 (2012).
C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, Phys. Rev. Lett. 108, 1 (2012).
A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, Phys. Rev. Lett. 101, 1 (2008).
H. Bailung, S. K. Sharma, and Y. Nakamura, Phys. Rev. Lett. 107, 1 (2011).
W. Cousins and T. Sapsis, Phys. Rev. E 91, 1 (2015).
A. Calini and C. M. Schober, in: Extreme Ocean Waves (Eds. E. Pelinovsky and C. Kharif), Springer, New York 2008, p. 31.
W. P. Zhong, M. Belic’, and B. A. Malomed, Phys. Rev. E 92, 053201, 1 (2015).
J. Cuevas Maraver, P. G. Kevrekidis, D. J. Frantzeskakis, N. I. Karachalios, M. Haragus, et al., Phys. Rev. E 96, 012202 (2017).
E. Shlizerman and V. Rom-Kedar, Phys. Rev. Lett. 102, 03390 (2009).
D. Cai, D. W. McLaughlin and K. T. R. McLaughlin, The Nonlinear Schrödinger Equation as Both a PDE and a Dynamical System, Handbook of dynamical systems, vol. 2, North-Holland, Amsterdam 2002, p. 599.
E. Shlizerman and V. Rom-Kedar, Chaos 15, 013107, 1 (2005).
E. G. Charalampidis, J. Cuevas-Maraver, D. J. Frantzeskakis, and P. G. Kevrekidis, Rom. Rep. Phys. 70, 504 (2018).
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes 10 (American Mathematical Society, 2003).
M. Bertola and A. Tovbis, Comm. Pure Appl. Math Comm. Pure Appl. Math. 66, 678 (2009).
P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives, Springer-Verlag, Berlin, Heidelberg 2009.
For the generic case of