# Exciting extreme events in the damped and AC-driven NLS equation through
plane wave initial conditions

Sevastos Diamantidis  Department of Mathematics, University of the Aegean,
Karlovassi, GR 83200 Samos, Greece  Theodoros P. Horikis  Department of
Mathematics, University of Ioannina, Ioannina GR 45110, Greece  Nikos I.
Karachalios111Corresponding author. E-mail<EMAIL_ADDRESS>Department of
Mathematics, University of Thessaly, Lamia, GR 35100, Greece

###### Abstract

We investigate, by direct numerical simulations, the dynamics of the damped
and forced nonlinear Schrödinger (NLS) equation in the presence of a time
periodic forcing and for certain parametric regimes. It is thus revealed, that
the wave-number of a plane-wave initial condition dictates the number of
emerged Peregrine type rogue waves at the early stages of modulation
instability. The formation of these events gives rise to the same number of
transient “triangular” spatio-temporal patterns, each of which is reminiscent
of the one emerging in the dynamics of the integrable NLS in its semiclassical
limit, when supplemented with vanishing initial conditions. We find that the
$L^{2}$-norm of the spatial derivative and the $L^{4}$-norm detect the
appearance of rogue waves as local extrema in their evolution. The impact of
the various parameters and noisy perturbations of the initial condition in
affecting the above behavior is also discussed. The long time behaviour, in
the parametric regimes where the extreme wave events are observable, is
explained in terms of the global attractor possessed by the system and the
asymptotic orbital stability of spatially uniform continuous wave solutions.

The damped and AC-driven nonlinear Schrödinger equation, a prototypical model
for complex dynamical behavior, when supplemented with the simplest case of
plane wave initial conditions, exhibits behavior which supports the
universality of the Peregrine rogue wave and semiclassical type dynamics.
These dynamics may be used to explain extreme event formation in the presence
of non-integrable perturbations that describe damping and external forcing
effects.

##  I Introduction

Key element in the study of extreme events and rogue waves is the study of the
mechanisms for their generation and predicting the location and time of their
occurrence pre0 ; pre1 ; pre2 ; pre3 . This is particularly difficult as the
mechanisms that generate these waves are complex and highly nonlinear in
nature. Furthermore, as modulation instability (MI) ZO  is believed to be the
main mechanism for their formation EPeli ; Kharif1 ; Kharif2 , another
complication is added, as different initial states, under MI, will produce
different events in both amplitude, and location.

It is argued and justified by numerous experimental observations, that in deep
water and other physical systems like optics, plasmas hydro ; opt ; plasma
etc., rogue wave dynamics are governed by the nonlinear Schrödinger (NLS)
equation and its variants. These rogue wave events are modelled by the so-
called Peregrine soliton H_Peregrine , which is a rational solution of the NLS
equation, in its integrable form akh .

However, when additional phenomena are added, such as viscosity, wind effects,
opposing currents or higher order effects Ex1 ; Ex2 ; Ex3 ; Ex4 ; Ex5 ; Ex6 ;
Ex7 ; Ex8 , the integrable NLS system is insufficient to describe the
resulting dynamics. As such, additional, physically relevant terms, are
incorporated in the model in order to account for these different effects. In
this spirit, the present paper investigates once more, the wealth of the
dynamics arising from the variant of the NLS equation with linear damping and
a time-periodic forcing term:

$\displaystyle\mathrm{i}{{u}_{t}}+\frac{\nu}{2}{{u}_{xx}}+\sigma|u|^{2}u=-\mathrm{i}\gamma
u+\Gamma\exp(\mathrm{i}\Omega t).$ (1)

The parameter $\nu$ measures the second-order dispersion, $\sigma$ the
strength of the nonlinearity; both $\nu,\sigma>0$ are motivated by the
focusing integrable limit. The parameter $\gamma>0$ is the strength of the
linear damping (loss). The parameter $\Gamma>0$ measures the amplitude of
time-periodic forcing and $\Omega\in\mathbb{R}$ is its time frequency;
subscripts denote partial derivatives. The model is of significance in
numerous physical contexts, from nonlinear optics to plasma physics BoYu , and
is one of the prototypical partial differential equations exhibiting complex
spatiotemporal behavior nobe1 ; nobe2 ; NB86 . Generically, this is described
by the existence of a global attractor for the associated infinite dimensional
system. For the existence, finite dimensionality and regularity of the global
attractor we refer to Ghid88 ; Goubet1 ; Goubet2  (see also (RTem, , pg.
234-255)), XW95 ; PL95 ; NK2002  for the periodic initial-boundary value and
Cauchy problems of the damped and forced NLS in 1D and higher dimensional
spatial domains. The complicated structure of the global attractor has been
further elucidated by the analysis of the bifurcations the system undergoes Li
; Wig1 ; kai , particularly in the presence of a rich variety of soliton
solutions which have been approximated analytically Bar1 ; barashenkov ; Bar2
; Bar3 ; Boris4 ; Bar5 ; Bar6 ; Bar7 ; Bar8 . In many cases such soliton
solutions are the attractors of the system, in their stability regime.

In the present paper, it is further revealed that the richness of the dynamics
of (1) can be further enhanced by the emergence of extreme wave events
generated by the simplest continuous wave (cw) initial condition

$\displaystyle u(x,0)=u_{0}(x)=A\exp\left(-\frac{\mathrm{i}K\pi
x}{L}\right),\;\;K\in\mathbb{N},$ (2)

when the system is supplemented with periodic boundary conditions in the
fundamental interval $\mathcal{Q}=[-L,L]$,

$\displaystyle u(-L,t)=u(L,t).$ (3)

More precisely, we show that the initial condition (2) creates dynamical
effects, which are –to our knowledge– novel and can be relevant to mechanisms
for the generation of rogue waves. At this point, we have to stress that our
investigations also suggest that these effects may be highly unstable under
noisy perturbations of the specific initial condition (2), as it will be
explained below. These effects are the following:

In certain regimes for the parameters of the system, the initial condition (2)
dictates by its wave number $K$ and for suitable initial amplitudes $A$, the
emergence of $K$ Peregrine-type rogue waveforms at the initial stages of its
evolution. By a comparison against the profile of the analytical Peregrine
rogue wave (PRW) of the integrable limit ($\gamma=\Gamma=0$) it is judged that
the profiles of these PRW-type waveforms do not only have similar profiles,
but also share for a significant-time interval, growth and decay rates,
remarkably close to that of the PRW of the integrable limit. Their similarity
is further judged by examining the spectrum of the emerged structures which is
identical to the analytical PRWs. Another new dynamical effect we identified,
is the following: the PRW-type waveforms which emerge as first events in the
dynamics are on the top of $K$ triangular-type (“Christmas tree”)
spatiotemporal patterns BS . Despite the fact that the initial condition does
not decay at infinity, each one of these patterns possesses a structure
similar to the single spatiotemporal pattern formed in the case of the
integrable NLS, in its semi-classical limit BM1 ; BM2 ; Rev1_b . Hence, the
results suggest the universality of this dynamics even in the particular
example of the AC-damped and driven NLS (1), when supplemented with periodic
boundary conditions Rev1_b ; BioMan .

We discuss the potential structural robustness of the above dynamical
behavior, as the several parameters of the system are varied, and we find that
it persists up to certain threshold values for the parameters. We found that
is far from the dynamics of the integrable limit as it is not observed in this
limit when the same initial condition is posed. Furthermore, we examined the
stability of the observed patterns when the initial condition (2) is perturbed
by noise. We found that the noise perturbation destruct the previous
structures. The observed instability motivated us for a repetition of the
numerical experiment for the semiclassical limit NLS with the same parameters
and vanishing initial condition given in basis ; BM1 , perturbing now this
initial condition by noise. We found again an instability destructing the
characteristic triangular pattern of a “rogue waves lattice” which in the
presence of noise is replaced by the evolution of breather modes. As rogue
waves are highly unstable solutions PNJ , the findings suggest that the
presence of noise enhances the instability of the background sustaining the
rogue waveforms. Therefore, the numerical findings for the noisy perturbed
initial conditions suggest that the structured semiclassical type dynamics in
both cases of systems -the damped and AC-driven NLS and the conservative NLS
in its semiclassical limit-are very sensitive under noisy perturbations and
that spatiotemporal chaos dominates in their entire dynamical behaviour.

It is well known that the existence of localized modes as solitons and
breathers, can be proved by the construction of variational problems which may
identify the localized structures as extrema of suitable energy functionals.
This approach has been successfully applied in continuous, as well as in
discrete models JSh ; TTao ; Wein99 ; Pan1 . In this regard, we observed the
following dynamical feature: the square of the $L^{2}$-norm of the spatial
derivative

$\displaystyle D(t)=\int_{\mathcal{Q}}|u_{x}(x,t)|^{2}dx,$ (4)

and the $L^{4}$-functional

$\displaystyle||u(t)||_{L^{4}}^{4}=\int_{\mathcal{Q}}|u(x,t)|^{4}dx,$ (5)

detect the emergence of the rogue waves as local extrema in their evolution in
a time interval of enhanced instability, exactly at the time of their
appearance. We found that only these functionals exhibit this detection
property of extreme events as opposed to either of the standard conserved
quantities of the integrable limit, namely the momentum,

$\displaystyle
M(t)=\mathrm{Im}\int_{\mathcal{Q}}u(x,t)\overline{u_{x}}(x,t)dx,$ (6)

the power (also called mass or wave action),

$\displaystyle P(t)=||u(t)||^{2}_{L^{2}}=\int_{\mathcal{Q}}|u(x,t)|^{2}dx,$
(7)

the Hamiltonian energy,

$\displaystyle
H(t)=\nu\int_{\mathcal{Q}}|u_{x}(x,t)|^{2}dx-\sigma\int_{\mathcal{Q}}|u(x,t)|^{4}dx.$
(8)

The quantities (6)-(8) are three of the conserved quantities in an infinite
hierarchy in the integrable case. In fact it was the conservation of the
Hamiltonian energy of the integrable limit which motivated us to investigate
the behaviour of its components $D(t)$ and $||u(t)||_{L^{4}}^{4}$. For $D(t)$,
the manifestation of local extrema can be explained by the steep gradients of
the Peregrine-type waveforms. For $||u(t)||_{L^{4}}^{4}$, the balance laws
justify that it is suitably vertically translated above $D(t)$ at least for a
finite time interval, explaining the detection of rogue waves at the same
times. These observations can be potentially useful in guiding future studies
for the theoretical detection of extreme wave events by suitably formulated
variational problems involving the aforementioned functionals, rationalizing
further the numerical findings. Let us remark, that both functionals are
physically relevant as defining the kinetic and potential energy respectively,
and their behavior was used extensively in studying the asymptotic state of
noisy induced MI Rad1 . The functional $||u(t)||_{L^{4}}^{4}$ is also known as
the interaction Hamiltonian, being relevant in studying soliton interactions
BCai .

Finally, the long-time asymptotics of the solutions, as observed by the
involved functionals, are analytically justified in the parametric regimes we
are interested in, by the global asymptotic orbital stability of cw solutions.
The analytical arguments combine the linear stability analysis of spatially
uniform, cw-solutions (see Theorem III.2 which follows the lines of
barashenkov ), with the well known results on the existence of a global
attractor possessed by the associated infinite dimensional dynamical system
and the notion of orbital stability (see Theorem III.4). We yet stress that
while in general, the global attractor for the model has been proved to posses
a complicated structure, in certain regimes of our study it can be
characterized by an orbitally asymptotically stable cw-state. These arguments
prove that the observed instabilities in the parametric regimes of rogue waves
emergence are only transient in the presence of this state. The results of
relevant numerical experiments were found to be in excellent agreement with
the analytical predictions.

The presentation of the paper is the following. In Section II, we report the
numerical results on the observation of rogue-wave dynamics. In Section III we
discuss the relevant balance laws of (1) for functionals, recall some useful
estimates for our analysis and justify the functional type diagnostics for the
detection of rogue-waves. In Section IV, we explain the asymptotic behavior of
the solutions observed in terms of the orbital stability of spatially uniform
cw solutions. The last Section IV, briefly discusses our conclusions and
sketches a plan for future investigations.

##  II Numerical Results

###  II.1 Emergence of extreme events and spatiotemporal patterns.

We start the presentation of the numerical results by discussing the
spatiotemporal dynamics of the system, varying the wave number $K\geq 2$ of
the initial condition (2), keeping its amplitude fixed, $A=1$; the case $K=1$
will be discussed at the end of Section III.3.

Fig. 1 depicts the spatiotemporal evolution of the density $|u(x,t)|^{2}$ of
the solution for various cases of the initial wave number $K\geq 2$. The
parameters for Eq. (1) are $\nu=\sigma=1$, $\gamma=0.01$, $\Gamma=0.1$,
$\Omega=-2$ and $L=50$.

Figure 1: (Color Online) Dynamics of the cw-initial condition (2) for $A=1$,
varying its wavenumber $K$. The panels show the spatiotemporal evolution of
the density $|u|^{2}$ for each $K$. Other parameters for the problem (1)-(3):
$\nu=\sigma=1$, $\gamma=0.01$, $\Gamma=0.1$, $\Omega=-2$ and $L=50$.

Each one of the panels of Fig. 1 is associated with one of the snapshots
portrayed in Fig. 2, showing the profiles of the first localized structures
emerged in the contour plots of Fig. 2.

Figure 2: (Color Online) Snapshots of the density $|u|^{2}$ of the solution,
at the times of the emergence of the PRW-type waveforms (first events in the
dynamics). These correspond to the first events shown in the contour plots and
the related parameter values of Fig. 1.

Combining the results presented in both figures, the following dynamical
features are revealed:

  1. 1.

The first events in the spatiotemporal dynamics (first “spots” observed in the
contour plots of Fig. 1), correspond to localized waveforms possessing extreme
amplitude (at least eight-times higher than the initial one). The initial wave
number $K$ gives rise to $K$-localized waveforms, which are formed on the top
of a finite background, as shown in the snapshots of Fig. 2.

  2. 2.

Subsequently, the dynamics are manifested by a formation of transient
$K$-patterns with the following characteristics: each one is a distinct
spatiotemporal region separated by nonlinear caustics which bound the pattern
of the transient spatiotemporal oscillations. Particularly, a structured
“lattice” of extreme events occurs, occupying a triangular region formed
between the caustics. Progressively, their amplitude is decreasing, as we may
conjecture due to the effect of damping. This is evident in the cases
$K=2,3,4$. Each of these patterns is reminiscent of the pattern forming in the
case of the integrable NLS in its semiclassical limit

$\mathrm{i}\epsilon{{u}_{t}}+\frac{\epsilon^{2}}{2}{{u}_{xx}}+|u|^{2}u=0,$ (9)

for $\epsilon\rightarrow 0$ BM1 ; BM2 , when supplemented with vanishing
boundary conditions. We recall in Figure 3,

Figure 3: (Color Online) Spatiotemporal evolution of the density $|u|^{2}$ for
the integrable NLS in its semiclassical limit (9) when $\epsilon=0.02$, and
the initial condition is (10).

the characteristic spatiotemporal pattern of the evolution of the initial
condition

$u(x,0)=A(x)\exp\bigg{[}\frac{\mathrm{i}}{\epsilon}\Phi(x)\bigg{]},\;\;A(x)=\exp(-x^{2}),\;\;\Phi^{\prime}(x)=-\tanh
x,$ (10)

for $\epsilon=0.02$ in (9). This single lattice, as it was analysed in BM1 ;
BM2 ; Rev1_b , is occupied by PRW structures defined by spots which correspond
to the poles of the tritronquée solution of the Painlevé-I equation. A similar
single pattern was also observed in the case of the damped and forced NLS for
algebraically or exponentially vanishing initial data when the driver is
spatiotemporally localized $\lim_{|x|^{2}+|t|^{2}\rightarrow\infty}f(x,t)=0$,
D1 , and explained in terms of decaying estimates in D2 , but differs
dramatically form the pattern observed in ZNA2019  in the dynamics of (1)
initiated from the same vanishing initial conditions. Another interesting
feature observed in the damped and forced case, is that the points of
intersection of the caustics are points of wave interference, occupied by
localized structures of large amplitude.

  3. 3.

The “area” occupied by the above “Christmas tree” patterns BS , is decreasing
as the initial wave number is increasing; a compression of the triangular
regions is observed in the case $K=9$. However, this is only an effect of the
increased excited modes against the fixed spatial region compressing the
aforementioned pattern formation. This can be clarified further, when the
numerical experiments for increased length $L$ of the spatial interval and
some large $K$ are presented. The left panel of Fig. 4 shows the contour plot
of the spatiotemporal evolution of the density when $K=8$ and the same set of
parameters as in Fig. 2, but for $L=200$.

Figure 4: (Color Online) Dynamics of the cw-initial condition (2) for $A=1$,
$K=8$, $L=200$. The left panel shows the spatiotemporal evolution of the
density for $x\in[-L,L]$ and $t\in[0,60]$. The right panel offers a
magnification of the patterns observed in the left panel, for $x\in[-50,50]$
and $t\in[0,60]$. Other parameters for the problem (1)-(3) are:
$\nu=\sigma=1$, $\gamma=0.01$, $\Gamma=0.1$, $\Omega=-2$.

We clearly observe the formation of $K=8$-copies of the triangular pattern
with the characteristics discussed above, as it is further clarified by the
magnification offered in the right panel of Fig. 4 (showing two of them within
the region $x\in[-50,50]$, $t\in[0,60]$). This study also reveals that the
emergence of the above patterns is robust against variations with respect to
the half-length $L$.

  4. 4.

Do the first extreme events share characteristics of the PRW? To answer this
question we shall implement fitting arguments already used in BS ; All2 ;
ZNA2019 , against the analytical PRW of the integrable NLS:

$\displaystyle u_{\mbox{\tiny
PS}}(x,t;P_{0})=\sqrt{P_{0}}\left[1-\frac{4\left(1+\frac{2\mathrm{i}t}{\Lambda}\right)}{1+\frac{4x^{2}}{K_{0}^{2}}+\frac{4t^{2}}{\Lambda^{2}}}\right]\exp\left(\frac{\mathrm{it}}{\Lambda}\right).$
(11)

Recall that $P_{0}$ is the amplitude (power) of the continuous background
supporting the PRW (11), and the parameters $\Lambda=\frac{1}{\sigma\,P_{0}}$,
$K_{0}=\sqrt{\nu\,\Lambda}$. We denote the space-time translations of (11) by
$u_{\mbox{\tiny PS}}(x-x^{*},t-t^{*};P_{0})$. The location $x^{*}$, time
$t^{*}$ and power $P_{0}$, are numerically calculated: The power of its
background $P_{0}$ is locked, so that the amplitude of the analytical PRW (11)
coincides with the maximum amplitude of the extreme localized structures
portrayed in 5 attained at time $t^{*}$. The position $x^{*}$ is detected so
that the center of the analytical PRW coincides with the center of a selected
extreme event (from the group of the $K$-identical localized structures).

Figure 5: (Color Online) Top panels: Comparison of the right PRW-type waveform
observed in upper left panel of 1 [continuous (blue) curve], against the
space-time translation of the analytical PRW of the integrable limit
$u_{\mbox{\tiny PS}}(x-22.3,t-22.5;1.01)$ where $K_{0}=0.70$ and
$\Lambda=0.99$ [dashed (red) curve]. The left panel compares the profiles. The
right panel shows a comparison of the evolution of the density of their peaks
$|u(22.3,t)|^{2}$ and $u_{\mbox{\tiny PS}}(0,t-22.5;1.01)$ respectively, for
$t\in[18,24]$. Bottom panel: comparison of the spectrum of the PRW-type
waveform of the top panel, against the spectrum of the corresponding
analytical PRW.

The left panel of Fig. 5 justifies the similarity of the profile of the right
extreme event of maximum amplitude attained at $t^{*}=22.5$ [continuous (blue)
curve] to the profile of the PRW 11 $u_{\mbox{\tiny PS}}(x-22.3,t-22.5;1.01)$,
with $K_{0}=0.70$ and $\Lambda=0.99$. The inset shows a detail around the left
of the symmetric minima; the slopes of the two curves show that the extreme
event for the damped and forced NLS should posses, around the core, a spatial
decay rate close to that of the PRW. The right panel of Fig. 5 depicts the
comparison of the evolution of the density of the centers of the two waves;
their time growth is very close for $t\in[18,21]$, their coincidence becomes
almost exact for $t\in[21.5,23.5]$ and afterwards we observe their divergence.
Thus, the emerged waves for the damped and forced model posses for significant
time intervals the algebraic in time, growth and decay rate of a PRW.
Additional evidence is provided in the bottom panel of Fig. 5 comparing the
spectrum of the emerged event and the analytical PRW, suggesting an almost
exact agreement.

  5. 5.

Dynamics of the integrable limit $\gamma=\Gamma=0$. The above “Christmas tree”
pattern formation is not exhibited in the integrable limit but only in the
damped and forced case of (1), for certain parametric regimes (see below).
This is verified in Fig. 6 showing the spatiotemporal dynamics in this limit
and for the rest of parameters fixed as in Fig. 1.

Figure 6: (Color Online) Contour plots of the spatiotemporal evolution of the
density in the case of the integrable limit $\Gamma=\gamma=0$, for two cases
of initial wave-numbers. Values of the rest of the parameters: $A=1$,
$\nu=\sigma=1$, $L=60$.

In both cases, the dynamics are almost identical, characterized by the
emergence of the vertical “first line” of extreme events, separating the
spatiotemporal domain in two distinct regions: before this line, and for
$t\lesssim 30$ the manifestation of MI is weak, while from the “line” and
above $t\gtrsim 30$, the manifestation of MI is characterized again by the
occurrence of extreme events; the line itself resembles a spatially periodic
state reminiscent of an Akhmediev breather (see (Akhb2016, , Fig 3, pg.
103901-3) for similar patterns).

While in the integrable limit, we comment on the robustness of the numerical
scheme and its outcomes. As complicated dynamics are discussed, it is
important to make sure that numerically induced complex behavior HA ; AH , due
to the numerical scheme used is not present, perturbing artificially its
homoclinic structure. This is done through checking the behavior of the
conserved quantities in the integrable limit $\gamma=\Gamma=0$. Indeed, in
Fig. 7 the first three integrals of motion $M(t)$ [Eq. (6)], $P(t)$ [Eq. (7)]
and $H(t)$ [Eq. (8)] of the unperturbed, integrable NLS equation are shown to
be constant throughout the evolution. As such, our findings are numerically
supported with no artificially induced phenomena entering the dynamics.

Figure 7: (Color Online) The evolution of the first three conserved quantities
$P(t)$ [square of the $L^{2}$-norm or wave action (black solid curve), $M(t)$
[momentum (dashed red curve)] and $H(t)$ [Hamiltonian (dotted blue curve)] for
the integrable equation Eq. (1) with $\gamma=\Gamma=0$, for different values
of $K$. The rest of the parameters are as in Fig. 6. No numerical
discrepancies are present as all remain constant for the duration of the
evolution.

  6. 6.

Robustness of the dynamics with respect to parameters. An important question
we examine is the potential robustness of the dynamics as the various
parameters are varied. Generically, it persists up to certain thresholds
identified for the parameters, as suggested by the results presented in Figure
8. Concerning variations of the damping strength and amplitude of the forcing,
we found that the phenomenon emerges when $\Gamma>\Gamma_{\mathrm{thresh}}$
(for suitable fixed $\gamma$) and persists for
$\gamma<\gamma_{\mathrm{thresh}}$ (for suitable fixed $\Gamma$). Particularly,
we found that the dynamics are sensitive for small increments of the damping
strength $\gamma\sim\mathcal{O}(10^{-3})$, as shown in the top left panel for
$\gamma=0.02$ where the effects are weaken. Yet in this regime, the emergence
of the triangular patterns persists up to a certain threshold
$\gamma_{\mathrm{thresh}}$. Similarly, the threshold
$\Gamma_{\mathrm{thresh}}\sim\mathcal{O}(10^{-3})$ was found relatively small
for the given fixed set of parameters; for $\Gamma<\Gamma_{\mathrm{thresh}}$
the dynamics were close to that of the integrable limit. For increased values
of $\Gamma$, the amplitude of the first extreme events, as well as, the
amplitude and the spatiotemporal density of the extreme waves within the
triangular regions (and on the caustics) is increased, as shown in the top
middle panel corresponding to the case of $\Gamma=0.8$. Relevant thresholds
$\sim\mathcal{O}(1)$ were identified for the rest of parameters. For
$\Omega<\Omega_{\mathrm{thresh}}$ the dynamics are like the one observed for
large $\Gamma$, while for $\Omega>\Omega_{\mathrm{thresh}}$ approaches that of
the integrable limit, as shown in the top right panel for $\Omega=20$; this is
explained as in the limit of large $\Omega$, the time-period of the
oscillations which is dictated by the frequency of the driver should tend to
zero. A behavior similar to the one of large $\Omega$ is observed for
$\sigma>\sigma_{\mathrm{thresh}}$ and larger amplitudes of the initial
condition $A>A_{\mathrm{thresh}}$ where the effects are enhanced; see the
bottom left and middle panels for $\sigma=1.5$ and $A=1.5$, respectively. The
bottom right panel for $\nu=3$ verifies that for increased
$\nu>\nu_{\mathrm{thresh}}$, the triangular patterns are disordered due to the
increased dispersion.

  7. 7.

Robustness of the spatiotemporal patterns under noisy perturbation of the
initial condition. We also examine the stability of the observed
spatiotemporal patterns when the plane wave initial condition (2) is perturbed
by noise. The left panel of Figure 9 depicts the dynamics for the damped and
forced NLS (1) for the same set of parameters as in the top left panel of
Figure 1, and for the initial condition (2) with $K=2$ perturbed by random
additive noise of $1\%$. The results suggest that the “Christmas tree”
patterns of Figure 1 are unstable under noisy perturbations. In the presence
of noise the resulting dynamics are now reminiscent to that of the noisy
induced MI (Rad1, , Fig. 2, pg. 234102-3).

This instability motivated us to perform a numerical study for the
semiclassical limit NLS (9) with the same parameters as in the study of Figure
3, but with the initial condition (10) which created the triangular pattern
BM1 ; BM2 , now perturbed by the same noise as above. We observe that even in
the semiclassical limit NLS, the rogue-wave lattice observed previously is
unstable; The new pattern suggests the replacement of the above lattice by the
formation of breather type dynamics, while the whole pattern is more
reminiscent of the evolution observed in BioMan .

Both results suggest that the presence of noise enhances the instability
effects on the background supporting the rogue waves, an effect which was
analysed in PNJ .

Figure 8: (Color Online) Contour plots of the spatiotemporal evolution of the
density, when a parameter is varied while the rest of them are fixed as in
Fig. 1.

Figure 9: (Color Online) Left panel: Dynamics of the cw-initial condition (2)
for $A=1$, and wavenumber $K=2$ when perturbed by random additive noise of
$1\%$ for the damped and forced NLS (1), with parameters parameters
$\nu=\sigma=1$, $\gamma=0.01$, $\Gamma=0.1$, $\Omega=-2$ and $L=50$ (same as
in Figure 1). Right panel: Dynamics of the integrable NLS in the semiclassical
limit (9) for $\epsilon=0.02$, when the initial condition (10) is perturbed by
random additive noise of $1\%$.

##  III Behavior of functionals in the regime of rogue waves and asymptotic
behavior

###  III.1 Conserved quantities, balance laws and attractors

We start the analytical considerations by discussing the conserved quantities
and energy balance laws for (1). These laws will enable us to justify the
behavior of the functionals which detect the emergence of rogue waves as
extrema of functionals. We also recall that from the energy balance laws,
suitable parametric estimates can be derived for the solutions, leading to the
well known result of the existence of a global attractor for the corresponding
infinite dimensional dynamical system Ghid88 ; Goubet1 ; Goubet2 . This it
will be particularly useful in analysing the long-time asymptotics for certain
parametric regimes of interest.

The problem is formulated in the Sobolev spaces of $2L$\- periodic functions
on the fundamental interval $\mathcal{Q}=[-L,L]$. For the shake of
completeness, we recall their definition:

$\displaystyle H^{k}_{\mathrm{per}}(\mathcal{Q})$ $\displaystyle=$
$\displaystyle\\{u:\mathcal{Q}\rightarrow\mathbb{C},\;\;u\;\mbox{and}\;\frac{\partial^{j}u}{\partial
x^{j}}\in L^{2}(\mathcal{Q}),\;\;j=1,2,...,k;$ (12) $\displaystyle
u(x),\;\;\mbox{and}\;\;\frac{\partial^{j}u}{\partial x^{j}}(x)\;\mbox{for
$j=1,2,...,k-1$, are $2L$-periodic}\\}.$

The change of variables $u\rightarrow u\exp(-\mathrm{i}\Omega t)$, transforms
the non-autonomous (1), to its autonomous form:

$\displaystyle\mathrm{i}{{u}_{t}}+\frac{\nu}{2}{{u}_{xx}}+\sigma|u|^{2}u=\Omega
u-\mathrm{i}\gamma u+\Gamma.$ (13)

We proceed to the study of the behavior of various functionals for the
solutions. First, it is important to recall that the functional

$\displaystyle I(t)=\frac{\exp(2\gamma
t)}{2L}\operatorname{Im}\int_{\mathcal{Q}}{u\left(x,t\right)\overline{{{u}_{x}}}\left(x,t\right)dx},$
(14)

is a conserved quantity for (1), implying that the momentum functional $M(t)$
defined in Eq. (6) decays exponentially:

$\displaystyle M(t)=M(0)\exp{(-2\gamma
t)},\;\;\;\lim_{t\rightarrow\infty}M(t)=0,$ (15)

as it was proved in Bar8 .

Figure 10: (Color Online) Top row: Evolution of the momentum functional $M(t)$
for two cases of the wave number $K$ of the initial condition (2). Left panel
for $K=2$ and right panel for $K=8$. The continuous (blue curve) traces the
analytical exponential decaying law (15), and the dashed (red) curve the
numerical evolution of $M(t)$.

The consderation of the momentum $M(t)$ is important for the justification of
the numerical results on the complex dynamics discussed above: Recalling again
HA ; AH , numerical complex (even chaotic) behavior may be produced by
numerical schemes which do not preserve the conservation laws of the system.
This is not the case in our numerical results for the damped and forced case.
Fig. 10 depicts the comparison of the evolution of the momentum $M(t)$ as
given analytically by (15) [continuous (blue) curve], against its numerical
values [dashed (red) curve] for two cases of wave numbers $K=2$ and $K=8$; the
rest of parameters are fixed as in the study of Fig. 1. The system is
integrated for $t\in[0,200]$, and in both cases of wave numbers we observe the
excellent agreement of the numerical evolution of the momentum with the
analytical prediction. In the same point of view we may examine the behavior
of some other conserved quantities of the integrable limit in the damped and
forced NLS (1). The power $P(t)$ satisfies when $\gamma,\Gamma>0$ the estimate

$\displaystyle P(t)\leq P(0)\exp(-\gamma
t)+\frac{2\Gamma^{2}L}{\gamma^{2}}[1-\exp(-\gamma t)],\;\;\mbox{for
all}\;\;t>0.$ (16)

From the estimate (16), one may extract uniform boundedness of the solutions
of (1) in the $L^{2}$-norm. For the other fundamental conserved quantity of
the integrable limit, the Hamiltonian energy, it can be shown that in the case
of the damped and forced NLS (1), it satisfies the balance law

$\displaystyle\frac{1}{4}\frac{d}{dt}H(t)=\mathrm{Re}\int_{\mathcal{Q}}\mathrm{i}\gamma
u\bar{u}_{t}\,dx+\Omega\mathrm{Re}\int_{\mathcal{Q}}u\bar{u}_{t}dx-\Gamma\mathrm{Re}\int_{\mathcal{Q}}\bar{u}_{t}dx.$
(17)

Next, we may consider the functional

$\displaystyle
F(t)=\frac{\nu}{4}\int_{\mathcal{Q}}|u_{x}|^{2}dx-\frac{\sigma}{4}\int_{\mathcal{Q}}|u|^{4}dx-\frac{\Omega}{2}\int_{\mathcal{Q}}|u|^{2}dx+\Gamma\mathrm{Re}\int_{\mathcal{Q}}udx,$
(18)

for which, the following differential inequality holds:

$\displaystyle\frac{d}{dt}F(t)+\gamma
F(t)\leq\frac{\gamma\Omega}{2}||u||_{L^{2}}^{2}+\frac{3\gamma\sigma}{8L}||u||_{L^{2}}^{4}+\frac{9\gamma\sigma^{2}}{4\nu}||u||_{L^{2}}^{6}.$
(19)

The differential inequality (19) if combined with the uniform in time
estimates for $P$, it also provides uniform in time estimates in the
$H^{1}_{\mathrm{per}}(\mathcal{Q})$-norm. Then, one may use the energy balance
equation (17) and the energy method of JB04 , XW95  (together with the
compactness of of the embedding $H^{1}_{\mathrm{per}}(\mathcal{Q})\subset
L^{2}(\mathcal{Q})$), to prove the following result.

###### Theorem III.1

Let $\nu,\sigma,\gamma>0$ and $\Gamma,\Omega\in\mathbb{R}$. Then, for the
dynamical system $\varphi_{t}:H^{1}_{\mathrm{per}}(\mathcal{Q})\rightarrow
H^{1}_{\mathrm{per}}(\mathcal{Q})$ defined from equation (1) when supplemented
with the periodic boundary conditions (3), there exists a global attractor
$\mathcal{A}$ in $H^{1}_{\mathrm{per}}(\mathcal{Q})$.

The global attractor is invariant under the flow, i.e.,
$\varphi_{t}(\mathcal{A})=\mathcal{A}$, for all $t\geq 0$, and

$\displaystyle\mathrm{dist}(\varphi_{t}(u_{0}),\mathcal{A})\rightarrow
0,\;\;\mbox{as $t\rightarrow\infty$, for all $u_{0}\in
H^{1}_{\mathrm{per}}(\mathcal{Q})$}.$ (20)

The distance in (20) is defined as a distance between a point $u$ and a set
$\mathcal{S}$ of $H^{1}_{\mathrm{per}}(\mathcal{Q})$:

$\displaystyle\mathrm{dist}(u,\mathcal{S})=\mathrm{inf}_{w\in\mathcal{S}}||u-w||_{H^{1}_{\mathrm{per}}(\mathcal{Q})}.$
(21)

For the parametric regimes we are interested in our simulations, the structure
of the global attractor will be revealed in the next section.

###  III.2 Rogue waves detected by extrema of functionals.

Although the momentum functional $M(t)$ involves the gradient of the solution
$u_{x}$, it does not capture the manifestation of the PRW-extreme events,
which are characterized by the emergence of steep gradients. Hence, we examine
numerically the evolution of the functionals $P(t)$ and $H(t)$, as well as,
the components of the Hamiltonian $D(t)$ and $||u(t)||_{L^{4}}^{4}$ given in
Eq. (4) and Eq. (5), respectively.

The top panels of Fig. 11 and Figure 12 depict the dynamics of the functionals
$D(t)$ [continuous (black) curve] and of $||u(t)||^{4}_{L^{4}}$ [dashed (red)
curve], for two cases of the wave number of the initial condition, $K=2$ and
$K=4$ respectively. The rest of parameters are still fixed as in the study of
Fig. 1. The bottom panels of Figure 11 and Figure 12 depict the dynamics of
the power $P(t)$ [continuous (black) curve] and of the Hamiltonian $H(t)$
[dashed (red) curve].

Figure 11: (Color Online) Top left panel: evolution of $D(t)$ (see (4))
[continuous (black) curve] and of $||u(t)||^{4}_{L^{4}}$ [dashed (red) curve],
for wave number $K=2$ of the initial condition (2), when $t\in[0,30]$. Top
right panel: Same as above but for $t\in[0,200]$. Bottom left panel: Evolution
of the power $P(t)$ [continuous (black) curve] and of the Hamiltonian $H(t)$
[dashed (red) curve], for $K=2$, when $t\in[0,30]$. Bottom right panel: Same
as above for $t\in[0,1000]$. The insets portray a zoom-in of the evolution
when $t\in[900,1000]$.

Figure 12: (Color Online) Top left panel: evolution of $D(t)$ (see (4))
[continuous (black) curve] and of $||u(t)||^{4}_{L^{4}}$ [dashed (red) curve],
for wave number $K=4$ of the initial condition (2), when $t\in[0,30]$.
Parameters: $\gamma=0.01$, $\Gamma=0.1$, $\Omega=-2$. Top right panel: Same as
above but for $t\in[0,200]$. Bottom left panel: Evolution of the power $P(t)$
[continuous (black) curve] and of the Hamiltonian $H(t)$ [dashed (red) curve],
for $K=4$, when $t\in[0,30]$. Bottom right panel: Same as above for
$t\in[0,200]$.

We observe important differences but also similarities between the behavior of
$D(t)$ and $||u(t)||^{4}_{L^{4}}$, against $P(t)$ and $H(t)$. The differences
are the following:

  * $\blacktriangleright$

The functionals $D(t)$ and $||u(t)||^{4}_{L^{4}}$ detect the emergence of
extreme events at the same time $t^{*}$ of their occurrence, as it can be
easily inferred by a comparison of the top panels of Figures 11-12 and Figures
1-2 (top and bottom left panels). This detection is manifested by the _local
maxima possessed by $D(t)$ and $||u(t)||^{4}_{L^{4}}$, at the time of
occurrence of the rogue waves._ Furthermore, both functionals detect the time
interval where a transient enhanced instability occurs, accompanied by the
emergence of extreme events, while afterwards, both functionals demonstrate
stabilization manifested by a converging behavior.

  * $\blacktriangleright$

The power $P(t)$ and the Hamiltonian $H(t)$ do not exhibit extrema at the time
$t^{*}$ of occurrence of the extreme events. Both functionals exhibit (modulo
oscillations) a uniform with respect to time converging behavior, without
detecting the interval of the above enhanced transient instability.

The simulations suggest some other features:

  * $\blacktriangleright$

Prior to the detection of the emergence of the first rogue wave accompanied by
the manifestation of the first local maximum, the functional $D(t)$ is almost
constant for a finite time interval ($[0,T_{s}]$ ($T_{s}\approx 5$ in the case
$K=2$ and $T_{s}\approx 10$ in the case $K=4$). This is the time interval
where the instability is not yet strongly manifested; the initial condition
has the form of a plane wave and within this time interval, the solution
resembles closely a plane wave. For instance, we may assume that for short
time intervals prior the manifestation of the instability, like $[0,T_{s}]$,
the solution can be approximated as

$\displaystyle u(x,t)\approx Ae^{i\theta(x,t)},\;\;\theta(x,t)\approx
kx-\omega t,\;\;k\approx\frac{K\pi}{L},\;\;\omega\approx\Omega.$ (22)

The approximation (22) implies that $D(t)\approx\frac{2A^{2}K^{2}\pi^{2}}{L}$,
for $t\in[0,T_{s}]$. For $t\gtrsim T_{s}$, the instability growth becomes
exponential and reaches a local maximum due to the presence of the first rogue
wave. The subsequent appearance of rogue waves is detected within the time
interval where $D(t)$ exhibits its large amplitude oscillations. The
appearance of the local maxima of $D(t)$ is explained by the steep gradients
of the emerged rogue waves.

  * $\blacktriangleright$

For $t\in[0,T_{s}]$ and within the time interval of the enhanced instability
(where the local maxima appear of both functionals), the graph
$||u(t)||^{4}_{L^{4}}$ is above the graph of $D(t)$. We also note the
decreasing oscillatory behavior of $||u(t)||^{4}_{L^{4}}$ prior to the
appearance of its first local maximum. Both effects can be explained by the
balance law for the Hamiltonian $H(t)$ given in Eq. (17) and the approximation
(22): Substitution of (22) into the right-hand side of (17) yields that

$\displaystyle\frac{1}{4}\frac{d}{dt}H(t)\approx-2\gamma A\omega
L+\frac{A\Gamma\omega}{K}\sin(kL)\sin(\omega t)\approx-2\gamma A\Omega L.$
(23)

For the parameter values $\gamma=0.01$, $\Gamma=0.1$, $\Omega=-2$ considered
in the numerical studies of Figures 11 and 12, the right-hand side of (23) is
positive for $t\in(0,T_{s}]$; Thus for $t\in(0,T_{s}]$, the Hamiltonian $H(t)$
is increasing as found in the numerical simulations. Returning to (23), by
integration with respect to time for arbitrary $t\in(0,T_{s}]$, we get

$\displaystyle H(t)$ $\displaystyle\approx$ $\displaystyle H(0)-8\gamma
A\omega Lt+\frac{16A\Gamma\sin(kL)}{k^{2}}\cos{\omega
t}-\frac{8A\Gamma\sin(kL)}{k^{2}}$ (24) $\displaystyle\approx$ $\displaystyle
H(0)-8\gamma A\Omega Lt,\;\;t\in(0,T_{s}).$

For the given set of parameters for Figs. 11 and 12, the right-hand side of
(24) captures qualitatively the superposition of linear and oscillatory growth
of $H(t)$ observed in the numerics for the short time intervals $(0,T_{s}]$.
Furthermore, from (24) and the definition of the Hamiltonian (8), we see that
$||u(t)||^{4}_{L^{4}}$ is given by

$\displaystyle||u(t)||_{L^{4}}^{4}\approx D(t)-H(0)+8\gamma A\Omega Lt.$ (25)

Comparing again the graphs of $D(t)$ and $||u(t)||_{L^{4}}^{4}$ in Figs. 11
and 12, we see that for the given set of parameters the approximative formula
(25) captures the translation of $||u(t)||_{L^{4}}^{4}$ over $D(t)$ and its
decrease prior to the time interval of enhanced instability, since $H(0)<0$
and $\Omega<0$. Importantly, this translation explains that the extrema of
$||u(t)||_{L^{4}}^{4}$ occur at the same times as of $D(t)$.

  * $\blacktriangleright$

We also examined the behavior of another important quantity, the variance

$\displaystyle V(t)=\int_{\mathcal{Q}}|x|^{2}|u(x,t)|^{2}dx,$ (26)

which may exhibit a similar behavior as $P(t)$ due to the bound

$\displaystyle V(t)\leq L^{2}P(t),$ (27)

and the estimate (16). Indeed, we observe in Figure 13 a converging behavior
of $V(t)$ for the same set of parameters and wavenumbers of the initial
condition (2) as in the study of Figures figure 11 and 12.

Figure 13: (Color Online) Parameters: $\gamma=0.0$, $\Gamma=0.1$, $\Omega=-2$.
Time evolution of the variance $V(t)$ for the initial condition (2), when
$K=2$ [continuous (black) curve] and $K=4$ [red (dahsed) curve]. Left panel:
time evolution for $t\in[0,200]$. Right panel: time evolution for
$t\in[0,1000]$]. The insets portray a zoom of the evolution for $t\in[0,30]$
(upper inset) and $t\in[900,100]$ (bottom inset).

Summarizing all the above, we found the following: Neither of the main
conserved quantities of the integrable limit $\gamma=\Gamma=0$, may detect the
emergence of rogue waves as extreme events in the transient dynamics of the
initial condition (2) for the damped and forced NLS (1) $\gamma,\Gamma>0$.
Instead, _each part of the Hamiltonian energy $D(t)$ and
$||u(t)||_{L^{4}}^{4}$ detects the emergence of rogue waves as local maxima at
the times of their appearance._

We conclude with a general comment. The diagnostic tools $D(t)$ and
$||u(t)||_{L^{4}}^{4}$ may be used as general tools for the detection of
extreme events in NLS-type equations. While we will explore further this
effect elsewhere, we depict the evolution of these functionals for the
integrable NLS equation in Fig. 14 with the rest of parameters fixed as in the
study of Fig. 11, corresponding to the spatiotemporal dynamics depicted in the
left panel Fig. 6.

Figure 14: (Color Online) The relative study of Fig. 11 in the case of the
integrable limit $\gamma=\Gamma=0$ (see the dynamics depicted in the left
panel of Fig. 6).

What is important to note here, is that the two curves are self-similar and
only differ in height consistently. This is due to the conservation of energy
which can be rewritten as

$\displaystyle||u(t)||_{L^{4}}^{4}=D(t)-H(0).$

Thus, while the Hamiltonian is constant throughout the evolution and negates
any extreme events prediction, this is not the case for its components as we
also analysed for the damped and forced case.

###  III.3 Long-time asymptotics in the regime of rogue waves

In order to understand the large-time behavior of the solutions as suggested
by above functionals, it is important to discuss the stability of stationary
states of (1). We will carry out a stability analysis following essentially
the lines of barashenkov . We have set for simplicity $\mu=\nu=1$ in (1), and
summarize the scenarios for modulational stability/instability in the
following proposition.

###### Proposition III.2

Spatially uniform continuous wave solutions of the form

$u_{s}(t)=\phi_{0}e^{\mathrm{i}\Omega
t},\quad\phi_{0}\in\mathbb{C},\;\;\Omega\in\mathbb{R},$ (28)

when they exist for (1), posses the following modulational, linear stability
properties:

  1. 1.

    1. (a)

When $\Omega>0$, they are stable with respect to small perturbations of the
form $y(x,t)=z(x)e^{\lambda t}$, $z(x)=e^{-\mathrm{i}kx}$, i.e.,
modulationally stable, for all $k\in\mathbb{R}$, if
$|\phi_{0}|^{2}\leq\Omega/3$.

    2. (b)

When $\Omega<0$ and $|\phi_{0}|^{2}<\gamma$, there exists a stability band
$I_{k^{2}}=[0,2(-\Omega+|\phi_{0}|^{2})]\cup[2(-\Omega+3|\phi_{0}|^{2}),\infty)$,
such that $u_{s}$ is modulationally stable if $k^{2}\in I_{k^{2}}$. Otherwise
is unstable.

  2. 2.

When $\Omega>0$ and $|\phi_{0}|^{2}>\Omega/3$, we distinct between the
following cases:

    1. (a)

They are unstable if $|\phi_{0}|^{2}>\max\left\\{\gamma,\Omega/2\right\\}$.

    2. (b)

When $\Omega/3<|\phi_{0}|^{2}<\Omega/2$, they are unstable if
$\gamma<\Omega/2$.

_Proof:_ _A. Existence_. Substituting (28) into (1), we see that $\phi_{0}$
satisfies the equation,

$-\phi_{0}\Omega+|\phi_{0}|^{2}\phi_{0}=\Gamma-\mathrm{i}\gamma\phi_{0}.$

Now, writing $\phi_{0}=\alpha
e^{\mathrm{i}\theta},\alpha\in\mathbb{R},\theta\in(-\pi,\pi]$, the above
equation leads to

$\alpha^{3}-\alpha\Omega=\Gamma e^{-\mathrm{i}\theta}-\mathrm{i}\gamma\alpha.$

Separating the real and imaginary parts, gives us the system

$\displaystyle\alpha^{3}-\alpha\Omega$ $\displaystyle=\Gamma\cos\theta,$
$\displaystyle\gamma\alpha$ $\displaystyle=-\Gamma\sin\theta.$

By eliminating $\theta$, we get

$\alpha^{6}-2\alpha^{4}\Omega+(\gamma^{2}+\Omega^{2})\alpha^{2}=\Gamma^{2},$

and setting $\rho=\alpha^{2}$, we end up with a cubic algebraic equation

$F(\rho;\gamma,\Gamma,\Omega)\equiv\rho^{3}-2\Omega\rho^{2}+(\gamma^{2}+\Omega^{2})\rho-\Gamma^{2}=0.$
(29)

Substituting $\rho=-q,q>0$, will give us the sum of four negative terms.
Hence, equation (29) can have either three positive real solutions or two
complex-conjugate solutions and one real, positive. Any positive root of
equation (29) defines a solution

$\displaystyle\phi_{0}=\sqrt{\rho}e^{\mathrm{i}\theta},\;\;\tan\theta=\frac{\gamma}{\Omega-\rho}.$
(30)

Through the substitution $\rho=y+2\Omega/3$ (see (Zwin, , Sec. 2.3.2)),
equation (29) can be transformed to the “reduced” form

$y^{3}+py+q=0,$ (31)

where $p=\gamma^{2}-\Omega^{2}/3$ and
$q=2/3\Omega(\gamma^{2}+\Omega^{2}/9)-\Gamma^{2}$.

Analysis of a such cubic equation suggests the distinction between two cases,
$p\geq 0$ and $p<0$.

  * $\blacktriangleright$

If $p\geq 0$, then $\gamma^{2}\geq\Omega^{2}/3$ and the derivative of the
equation (31) will be positive. Hence, (31) is monotone increasing and
therefore, has one real solution.

  * $\blacktriangleright$

If $p<0$, then $\gamma^{2}<\Omega^{2}/3$ and the number of the real roots can
be determined by the sign of the following quantity

$\displaystyle
Q=\left(\frac{\gamma^{2}-\Omega^{2}/3}{3}\right)^{3}+\left(\frac{\Omega}{3}\big{(}\gamma^{2}+\frac{\Omega^{2}}{9}\big{)}-\frac{\Gamma^{2}}{2}\right)^{2}.$
(32)

If $Q<0,$ there are three real roots and if $Q>0,$ there is one. Further
analysis shows that there exist three roots if
$\Gamma_{-}(\gamma)\leq\Gamma\leq\Gamma_{+}(\gamma)$, where

$\Gamma_{\pm}(\gamma,\Omega)=\left\\{\frac{2\Omega}{3}\left(\gamma^{2}+\frac{\Omega^{2}}{9}\right)\pm\frac{2}{3}\sqrt{\frac{1}{3}\left(\frac{\Omega^{2}}{3}-\gamma^{2}\right)^{3}}\right\\}^{1/2},$

and one root otherwise.

Summarizing all the above, we conclude for the existence of stationary
solutions, with the following cases:

  * $\blacktriangleright$

For $\gamma^{2}\geq\frac{\Omega^{2}}{3}$, one real root.

  * $\blacktriangleright$

For $\gamma^{2}<\frac{\Omega^{2}}{3}$, $\begin{cases}\text{three real roots
if}\quad\Gamma_{-}(\gamma)\leq\Gamma\leq\Gamma_{+}(\gamma),\\\ \text{one real
root, otherwise}.\end{cases}$

In the case where 3 positive roots exist, there are three branches of
solutions defined by the critical points of $F$ (29), namely,

$\rho_{\pm}(\gamma,\Omega)=\frac{2\Omega}{3}\pm\frac{1}{3}\sqrt{\Omega^{2}-3\gamma^{2}}.$
(33)

The branches are:
$0<|\phi_{0}|^{2}\leq\rho_{-}(\gamma),\rho_{-}(\gamma)<|\phi_{0}|^{2}<\rho_{+}(\gamma)$
and $|\phi_{0}|^{2}\geq\rho_{+}(\gamma)$.

_B. Stability_. Next, we examine the stability of these solutions.
Substituting, $\tilde{u}(x,t)=e^{i\Omega t}u(x,t)$ into equation (1) we obtain

$\mathrm{i}u_{t}-u\Omega+\frac{1}{2}u_{xx}+|u|^{2}u=\Gamma-\mathrm{i}\gamma
u.$ (34)

Let $\xi(x,t)$ be a small perturbation. We consider

$u(x,t)=u_{s}(x)+\xi(x,t),$ (35)

where $u_{s}(x)$ is a stationary solution of equation (34). Substituting (35)
into (34), linearizing and separating the real and imaginary parts, yields the
following system

$J(y_{t}+\gamma y)=Hy,$ (36)

where

$J=\begin{pmatrix}0&-1\\\ 1&0\\\ \end{pmatrix},\quad\\\
y(x,t)=\begin{pmatrix}\text{Re}\leavevmode\nobreak\ \xi\\\
\text{Im}\leavevmode\nobreak\ \xi\end{pmatrix},\quad\\\
H=\begin{pmatrix}-\frac{1}{2}\partial_{x}^{2}+\Omega-3u_{R}^{2}-u_{I}^{2}&-2u_{R}u_{I}\\\
-2u_{R}u_{I}&-\frac{1}{2}\partial_{x}^{2}+\Omega-
u_{R}^{2}-3u_{I}^{2}\end{pmatrix}.$

Note that, $u_{s}(x)=u_{R}+\mathrm{i}u_{I}$. Substituting
$y(x,t)=z(x)e^{\lambda t}$ into (36), yields the eigenvalue problem

$Hz(x)=\mu Jz(x),$

where $\mu=\lambda+\gamma$. The solution $u_{s}(x)$ will be stable if the
above system does not have eigenvalues $\mu$ with real part greater that
$\gamma$. In the case of the homogeneous solution $u_{s}(x)=\phi_{0}$, the
eigenvalue $\mu$ and the eigenvector $z(x)$ can be found explicitly. Now, by
writing $z(x)=z_{0}e^{-\mathrm{i}kx}$ and $\phi_{0}=u_{R}+\mathrm{i}u_{I}$, we
get

$(H_{k}-\mu J)z_{0}=0,$

where

$H_{k}=\begin{pmatrix}\frac{k^{2}}{2}+\Omega-3u_{R}^{2}-u_{I}^{2}&-2u_{R}u_{I}\\\
-2u_{R}u_{I}&\frac{k^{2}}{2}+\Omega-u_{R}^{2}-3u_{I}^{2}\end{pmatrix}.$

For non-trivial solutions, we require the determinant of $(H_{k}-\mu J)$ to be
zero, implying the equation

$\displaystyle-\mu^{2}=\left(\frac{k^{2}}{2}+\Omega-|\phi_{0}|^{2}\right)\left(\frac{k^{2}}{2}+\Omega-3|\phi_{0}|^{2}\right)\equiv
G(k^{2}).$ (37)

  * $\blacktriangleright$

When $\Omega>0$, for $|\phi_{0}|^{2}\leq\Omega/3,G(k^{2})\geq 0$. Hence,
$\text{Re}\mu=0$ and the solution is modulationally stable.

  * $\blacktriangleright$

When $\Omega<0$, we consider

$\displaystyle G^{\prime}(k^{2})=\frac{k^{2}}{2}+\Omega-2|\phi_{0}|^{2},$ (38)

where the prime denotes differentiation with respect to the variable $k^{2}$.
The critical point is

$\displaystyle k^{2}_{crit}=2(2|\phi_{0}|^{2}-\Omega)>0,$ (39)

since $\Omega<0$, and is a global minimum since

$\displaystyle G^{\prime\prime}(k^{2})=\frac{1}{2}>0.$ (40)

Next, we find that $G_{min}=G(k^{2}_{crit})=-|\phi_{0}|^{4}$. Hence,
$\text{Re}\mu=|\phi_{0}|^{2}$ and the solution will be unstable for
$|\phi_{0}|^{2}>\gamma$.

In the case where $|\phi_{0}|^{2}<\gamma$, we have that $G(k^{2})\geq 0$ for
$k^{2}\in I_{k^{2}}$ and the solution is modulationaly stable.

  * $\blacktriangleright$

When $\Omega>0$ and $|\phi_{0}|^{2}>\Omega/3$, we distinguish between the
following cases:

    1. 1.

If $|\phi_{0}|^{2}>\Omega/2$, we work again with (38)-(39)-(40) as in the case
$\Omega<0$, to show that for $|\phi_{0}|^{2}>\gamma$ the solution is unstable.

    2. 2.

For $\Omega/3<|\phi_{0}|^{2}<\Omega/2,$ the minimum occurs at $k^{2}=0$ and is
equal to $G_{min}=(|\phi_{0}|^{2}-\Omega)(3|\phi_{0}|^{2}-\Omega)$. Therefore,
the solution is unstable for

$-(|\phi_{0}|^{2}-\Omega)(3|\phi_{0}|^{2}-\Omega)>\gamma^{2}.$

Further analysis will show that the inequality above is valid when

$|\phi_{0}|^{2}_{-}<|\phi_{0}|^{2}<|\phi_{0}|^{2}_{+},$

where

$|\phi_{0}|^{2}_{\pm}=\frac{2\Omega}{3}\pm\frac{1}{3}\sqrt{\Omega^{2}-3\gamma^{2}}.$

Note that the amplitudes $|\phi_{0}|^{2}_{\pm}$ are equivalent to the critical
points of $F$, (33). Moreover, since $|\phi_{0}|^{2}<\Omega/2$ then
$|\phi_{0}|^{2}_{-}<\Omega/2$ and after some calculations, we may conclude
that this instability may occur in the region $\gamma<\Omega/2$. $\Box$

#### Rationalization of the asymptotic behavior for
$\gamma=0.01$,$\Gamma=0.1$, $\Omega=-2$.

Proposition III.2 is particularly useful in explaining the long-time
asymptotics observed in Fig. 11. For this purpose, we recall the notion of
orbital stability and orbital asymptotic stability. The positive orbit of the
flow $\varphi_{t}$ starting at $u_{0}$ at $t_{0}$ is the set
$\mathcal{O}^{+}(u_{0})=\cup_{t\geq t_{0}}\phi_{t}(u_{0})$.

###### Definition III.3

The solution $u_{*}(t)\in H^{1}_{\mathrm{per}}(\mathcal{Q})$ is said to be
orbitally stable, if given $\epsilon>0$, there exists some
$\delta=\delta(\epsilon)>0$, such that for any other solution $u(t)$
satisfying
$||u^{*}(t_{0})-u(t_{0})||_{H^{1}_{\mathrm{per}}(\mathcal{Q})}<\delta$, then
$\mathrm{dist}(u(t),\mathcal{O}^{+}(u^{*}(t_{0}))<\epsilon$ for $t>t_{0}$. It
is said to be asymptotically orbitally stable if it is orbitally stable and in
addition, there exists a constant $\tilde{\delta}>0$ such that if
$||u^{*}(t_{0})-u(t_{0})||_{H^{1}_{\mathrm{per}}(\mathcal{Q})}<\tilde{\delta}$,
then
$\lim_{t\rightarrow\infty}\mathrm{dist}(u(t),\mathcal{O}^{+}(u^{*}(t_{0}))=0$.

Combining Theorem III.1 and Proposition III.2 with Definition III.3, we have

###### Theorem III.4

Let $\nu=\sigma=1$ and assume that the parameters $\gamma>0$,
$\Gamma,\Omega\in\mathbb{R}$ are chosen so that there exists a unique
modulationally stable cw solution $u^{s}$ of the form (28). Then it should be
globally, orbitally asymptotically stable.

_Proof:_ Let $\hat{u}(t)$ denote the solution starting at $t_{0}=0$ from the
initial condition $\hat{u}_{0}=u^{s}(0)+\xi(x,0)$. When the unique cw solution
$u_{s}$ is modulationally stable, we have $\lambda<0$, and as its small
perturbations decay, the solution $\hat{u}(t)=\varphi_{t}(\hat{u}_{0})$ comes
arbitrarily close to $u_{s}$ in the sense

$\displaystyle\mathrm{dist}\left(\hat{u}(t),\mathcal{O}^{+}(u^{s}(0))\right)=\mathrm{dist}\left(\varphi_{t}(\hat{u}_{0}),\mathcal{O}^{+}(u^{s}(0))\right)<\epsilon.$
(41)

However, according to Theorem III.1, there exists a unique global attractor
$\mathcal{A}$ attracting all initial conditions $u_{0}$, independently of the
size of their $H^{1}_{\mathrm{per}}(\mathcal{Q})$\- norm and thus the limiting
relation (20) holds. Hence, for all initial conditions $u_{0}\in
H^{1}_{\mathrm{per}}(\mathcal{Q})$, there exists $T_{0}(u_{0})>0$ such that
the solutions $u(t)=\varphi_{t}(u_{0})$,with $u(0)=u_{0}$ satisfy

$\displaystyle\mathrm{dist}(\varphi_{t}(u_{0}),\mathcal{A})<\epsilon,\;\;\mbox{for
all $t>T_{0}$}.$ (42)

Since (42) holds also for the initial condition $\hat{u}_{0}$ and equation (1)
is actually autonomous due to (13), the uniqueness of the attractor and (41)
imply that $\mathcal{A}=\mathcal{O}^{+}(u^{s}(0))$. Yet, (20) shows that

$\displaystyle\lim_{t\rightarrow\infty}\mathrm{dist}\left(\varphi_{t}(u_{0}),\mathcal{O}^{+}(u^{s}(0))\right)=0,$

and that $u^{s}$ is globally orbitally asymptotically stable. This asymptotic
orbital stability should hold even in the case where a specific stability band
$I_{k^{2}}$ exists as in Proposition (III.2) $1(b)$, where $G(k^{2})\geq 0$
($\mathrm{Re}\mu=0$) when $k^{2}\in I_{k^{2}}$ and $G(k^{2})<0$ when
$k^{2}\in\mathbb{R}^{+}\setminus I_{k^{2}}$ ($\mathrm{Re}\mu\neq 0$): even in
the latter case, where $\mathrm{Re}\mu\neq 0$, yet
$\mathcal{A}=\mathcal{O}^{+}(u^{s}(0))$ should attract all initial conditions
and the divergence from $u_{s}$ should be only transient; after finite time of
divergence from $u_{s}$, the initial condition should approach
$\mathcal{A}=\mathcal{O}^{+}(u^{s}(0))$. $\Box$

For the parameters $\gamma=0.01$, $\Gamma=0.1$ and $\Omega=-2$ used in the
simulations, the algebraic equation (29) has only one real root

$\displaystyle\rho_{s}\approx 0.0025=|\phi_{0}^{s}|^{2}.$ (43)

This is in accordance with the analysis of Proposition III.2, since for the
above choice of parameters the quantity $Q$ defined in (32) is found to equal
$Q\approx 0.003>0$. Thus, there exists a unique stationary solution defined by
(28), (30) and (43),

$\displaystyle u_{s}(t)=\phi_{0}^{s}e^{\mathrm{i}\Omega
t},\;\;\phi^{s}_{0}=\sqrt{\rho_{s}}e^{\mathrm{i}\theta_{s}},\;\;\tan\theta_{s}=\frac{\gamma}{\Omega-\rho_{s}}.$

with $\sqrt{\rho_{s}}\approx 0.05,\;\;\tan\theta_{s}\approx-0.005$. Now, this
small amplitude cw $u_{s}$ satisfies the stability criterion of Theorem III.2
1(b), since $\Omega<0$ and $|\phi_{0}^{s}|^{2}<\gamma$. Then, applying Theorem
III.4, we extract its global asymptotic stability. Thus, all initial
conditions should converge to the small amplitude cw $u_{s}$, and this also
holds for the initial condition (2) with $A=1$ and $K=2$, as shown by the
convergence of the energy and norm quantities discussed in Figures 11 and 12.

We present a numerical illustration of the results stated in Proposition III.2
and Theorem III.4. Figure 15 shows the spatiotemporal behavior of the
solutions when $\nu=\sigma=1$, $\gamma=0.1$, $\Gamma=1$, $\Omega=1$. The
initial condition is (2) with $A=1$, $K=2$ and $L=50$. For this set of
parameters there exist again one spatially uniform cw solution defined by the
real root of the equation (29)

$\rho_{u}=1.69=|\phi_{0}^{u}|^{2}.$

Figure 15: (Color Online) Parameters (1)-(3): $\nu=\sigma=1$, $\gamma=0.1$,
$\Gamma=1$, $\Omega=1$. Dynamics of the cw-initial condition (2) for $A=1$,
$K=2$, and $L=50$. The left panel shows the spatiotemporal evolution of the
density for $x\in[-40,40]$ and $t\in[0,15]$. The right panel offers a
magnification of the patterns observed in the left panel, for $x\in[-10,10]$
and $t\in[200,250]$.

Since $|\phi_{0}^{u}|^{2}>\max\left\\{\gamma,\Omega/2\right\\}=0.5$, the
solution is unstable according to the instability criterion of Proposition
III.2 $2(b)$. In the contour plot shown in the left panel for $t\in[0,16]$, we
still observe the emergence of extreme events on the top of two distinct
“tree”-patterns. However, they do not posses the structure observed in Figure
1 and 4. The system exhibits complex spatiotemporal behavior as clearly shown
in the right panel, for $t\in[200,250]$.

Figure 16: (Color Online) Parameters (1)-(3): $\nu=\sigma=1$, $\gamma=0.01$,
$\Gamma=0.1$, $\Omega=1$. Dynamics of the initial condition (45) for $a=1$,
$\kappa=2$, and $L=50$. The left panel shows the spatiotemporal evolution of
the density for $x\in[-L,L]$ and $t\in[0,15]$. The right panel depicts the
spatiotemporal evolution, for $x\in[-L,L]$ and $t\in[60,250]$.

A second numerical example considers the case of parameters $\nu=\sigma=1$,
$\gamma=0.01$, $\Gamma=0.1$, $\Omega=1$. For these parameters, the quantity
$Q$ defined in (32) is found $Q\approx-0.0003<0$ and the equation (29) has
three real roots

$\displaystyle\rho^{s}_{1}=0.01=|\phi_{0}^{s,1}|^{2},\;\;\rho^{u}_{2}=0.98=|\phi_{0}^{u,2}|^{2},\;\;\rho^{u}_{3}=1=|\phi_{0}^{u,3}|^{2}.$
(44)

The above roots define three cw solutions of the form (28), namely,
$u^{s,1}_{s}$, $u^{u,2}_{s}$ and $u^{u,3}_{s}$ respectively. The solution
$u^{s,1}_{s}$ is asymptotically stable due to Theorem III.4 as it satisfies
$|\phi_{0}^{s,1}|^{2}<\Omega/3=1/3$, i.e., the stability criterion of
Proposition III.2 $1(a)$. The other two solutions $u^{u,2}_{s}$ and
$u^{u,3}_{s}$ are unstable since they satisfy the instability criterion
$|\phi_{0}^{u,2}|^{2},|\phi_{0}^{u,3}|^{2}>\Omega/2=0.5$ of Proposition III.2
$2(a)$. Figure 16 depicts the spatiotemporal evolution of the initial
condition

$\displaystyle u_{0}(x)=u_{s}^{1,s}(0)+a\exp\left(\frac{\mathrm{i}\kappa\pi
x}{L}\right),$ (45)

for $a=1$, $\kappa=2$, $L=50$. The initial condition (45) with $a=1$ is a
large amplitude perturbation of the cw solution $u_{s}(t)$. The instability
illustrated in the left panel, manifested again by the emergence of extreme
events, is transient, as predicted by Theorem III.4. In the right panel it is
shown that solution approaches the asymptotically stable cw $u_{s}^{s,1}$. The
convergence becomes even more evident as verified in Fig. 17, where the
evolution of the density of the solution $|u(x^{*},t)|^{2}$ at a specific
point $x=x^{*}$ is plotted. This $x^{*}$ is chosen to be the point where the
solution exhibits an extreme spot, here $x^{*}=10$, as seen in the left panel
of Fig. 16. The plot depicts the convergence of the density to the amplitude
of the asymptotically orbitally stable cw, as predicted.

Figure 17: (Color Online) Dynamics of the density $|u(x^{*},t)|^{2}$
[continuous (black) curve], for fixed $x^{*}=10$ and the initial condition and
parameters of Fig. 16. Asymptotic orbital stability of the cw $u_{s}^{s,1}$
with amplitude $\rho^{s}_{1}=0.01=|\phi_{0}^{s,1}|^{2}$ marked by the red
(dashed) line (see the discussion for the amplitudes given in (44)).

#### The case $K=1$ of the wave number of the initial condition (2).

We conclude with a comment for the case $K=1$ of the wave number of the plane
wave initial condition which may also explain the formation of the dendritic
patterns for higher wavenumbers. To assist us on this, we plot in Fig. 18 the
spatiotemporal evolution of the density $|u(x,t)|^{2}$ of the initial
condition (2) for $K=1$, while all other parameters are $\Omega=-2$,
$\gamma=0.01$, $\Gamma=0.1$, as in the case of the study of Fig. 1 and
$L=200$.

Figure 18: (Color Online) Parameters (1)-(3): $\nu=\sigma=1$, $\gamma=0.01$,
$\Gamma=0.1$, $\Omega=-2$. Dynamics of the initial condition (2) for $A=1$,
$K=1$, and $L=200$. The left panel shows the spatiotemporal evolution of the
density for $x\in[-40,40]$ and $t\in[0,160]$. The right panel depicts the
spatiotemporal evolution, for $x\in[150,180]$ and $t\in[0,160]$.

The two panels present two different regions. As one can readily see, the
patterns observed in Fig. 1 are missing and the dynamics are found to be
essentially different compared to the case $K\geq 2$. However, these figures
may be used to provide a useful understanding of the potential mechanism that
may lead to the formation of the dendritic patterns observed in the case
$K\geq 2$.

Indeed, noting that in the early stages of the evolution linear effects are
more dominant, a plane wave initial condition will behave as a solution to the
linear problem, and as such, will be subject to linear superposition. The
single maximum/minimum initial state evolves as above. If more maxima/minima
are added, the initial states around the minima/maxima will also undergo such
evolution, but now, constructive and destructive interference may allow for
more intricate patterns to form. The constructive interference will lead to
the peaks of the dendritic structure, while destructive interference will
leave some small amplitude waves on the constant background (seen in all
numerical simulations), which have no effect on the overall phenomenon.

We also note that the dynamics we observed when $K=1$ is no exception to the
findings of the article suggested by Proposition III.2 and Theorem III.4,
which seem to be rather sharp: Since the amplitude of the initial condition
satisfies $A^{2}=1>\gamma$, we expect, according to the instability case 1(b)
of Proposition III.2, transient modulational instabilities and then
convergence to the orbitally stable cw state of amplitude (43), as in the case
of $K=2$. On the other hand, when switching the initial amplitude in order to
satisfy the stability criterion 1(b) of Proposition III.2 $A^{2}<\gamma$ with
$k^{2}\in I_{k^{2}}$, we don’t expect transient instabilities prior the
convergence to the asymptotically stable cw state. This is indeed the case
when $A=0.09$ ($A^{2}=0.0081<\gamma=0.01$) and $K=1$ which corresponds to
$k^{2}=0.0002\in I_{k^{2}}$, as shown in the evolution of the initial
condition in Fig. 19.

Figure 19: (Color Online) Parameters (1)-(3): $\nu=\sigma=1$, $\gamma=0.01$,
$\Gamma=0.1$, $\Omega=-2$. Dynamics of the initial condition (2) for $A=0.09$,
$K=1$, and $L=200$. The panel shows the spatiotemporal evolution of the
density for $x\in[-20,20]$ and $t\in[0,160]$.

##  IV Conclusions

In this paper, we examined the dynamics of the linearly damped and time-
periodically driven Schrödinger equation, initiated by the simplest initial
condition, namely, a single mode plane wave. This gives rise to highly non-
trivial transient dynamics manifested by the birth of rogue waves reminiscent
of the Peregine soliton in terms of its temporal decay and spatial profile. We
also identified the formation of spatiotemporal patterns resembling the one
observed in the semiclassical Hamiltonian limit. Our explorations justified
that this behavior is far from the one of the integrable limit and discussed
its robustness as the parameters of the model are varied. We also found that
suitable functionals may detect the emerged rogue waves as their extrema. This
observation can be potentially useful in designing analytical pursuits
establishing the existence of extreme wave events by variational methods.

The asymptotic behavior of solutions for the parametric regimes examined in
the simulations was justified by the orbital stability of stationary states of
the model, aided by the existence of the global attractor of the system.

We also examined the stability of the patterns when the plane wave initial
condition is perturbed by noise. We found that the noisy perturbation of the
initial condition destructs the dynamics observed previously. Such an
instability was also found when repeating the experiment basis  for the
semiclassical limit NLS perturbing the relevant vanishing initial condition,
also by noise. These findings suggest that the semiclassical type dynamics are
unstable in the presence of noisy perturbations, and may inspire further
investigations concerning the physical observability of such structured
spatiotemporal patterns Rev1_b , when random effects are present.

We believe that our findings for the important example of the AC- damped and
driven NLS support further the universality of extreme wave dynamics in
systems where the NLS equation is the underlying model. Further studies may
include more general initial conditions in the form of a superposition of such
plane waves and more general norm-based diagnostic tools. Relevant
investigations are in progress, and results will be reported in upcoming
works.

## Acknowledgment

We would like to thank the referees for their constructive comments and
suggestions (including the one for the study of noisy perturbations), which
improved considerably the original version of the paper.

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