Extreme events in discrete nonlinear lattices.

Abstract

We perform statistical analysis on discrete nonlinear waves generated through modulational instability in the context of the Salerno model that interpolates between the integrable Ablowitz-Ladik (AL) equation and the nonintegrable discrete nonlinear Schrödinger equation. We focus on extreme events in the form of discrete rogue or freak waves that may arise as a result of rapid coalescence of discrete breathers or other nonlinear interaction processes. We find power law dependence in the wave amplitude distribution accompanied by an enhanced probability for freak events close to the integrable limit of the equation. A characteristic peak in the extreme event probability appears that is attributed to the onset of interaction of the discrete solitons of the AL equation and the accompanied transition from the local to the global stochasticity monitored through the positive Lyapunov exponent of a nonlinear map.

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arXiv:0901.3480v2 [nlin.PS] 2 Feb 2009
Extreme events in discrete nonlinear lattices
A. Maluckov1, Lj. Hadˇ zievski2, N. Lazarides3,4, and G. P. Tsironis3
1Faculty of Sciences and Mathematics, Department of Physics, P. O. Box 224, 18001 Niˇ s, Serbia
2Vinˇ ca Institute of Nuclear Sciences, P. O. Box 522, 11001 Belgrade, Serbia
3Department of Physics, University of Crete, and Institute of Electronic Structure and Laser,
Foundation for Research and Technology – Hellas, P. O. Box 2208, 71003 Heraklion, Greece
4Department of Electrical Engineering, Technological Educational Institute of Crete,
P. O. Box 140, Stavromenos, 71500, Heraklion, Crete, Greece
(Dated: February 2, 2009)
We perform statistical analysis on discrete nonlinear waves generated though modulational insta-
bility in the context of the Salerno model that interpolates between the intergable Ablowitz-Ladik
(AL) equation and the nonintegrable discrete nonlinear Schr¨ odinger (DNLS) equation. We focus
on extreme events in the form of discrete rogue or freak waves that may arise as a result of rapid
coalescence of discrete breathers or other nonlinear interaction processes. We find power law depen-
dence in the wave amplitude distribution accompanied by an enhanced probability for freak events
close to the integrable limit of the equation. A characteristic peak in the extreme event probability
appears that is attributed to the onset of interaction of the discrete solitons of the AL equation and
the accompanied transition from the local to the global stochasticity monitored through the positive
Lyapunov exponent of a nonlinear map.
PACS numbers: 63.20.Ry; 47.20.Ky; 05.45+a
Introduction.- The motivation of the present work
stems from observations of the sudden appearance of ex-
tremely large amplitude sea waves referred to as rogue or
freak waves [1]. These waves appear very suddenly in rel-
atively calm seas, reach amplitudes of over 20m and may
destroy or sink small as well as large vessels [2]. The-
oretical analysis of ocean freak waves has been linked
to nonlinearities in the waver wave equations, studied
though the nonlinear Schr¨ odinger (NLS) equation and
shown that the probability of their appearance is not in-
significant [3]. A scenario for freak wave generation in
NLS is through a Benjamin-Feir (modulational) instabil-
ity, resulting in self-focusing effects and subsequent for-
mation of freak waves [4]. Modulational instability (MI)
induces local exponential growth in the wave train ampli-
tude [5, 6] that has been confirmed experimentally and
numerically [7].
Intriguingly, there are completely different physical
systems that possess the required nonlinear characteris-
tics which favour the appearance of rogue waves. Recent
observation of optical rogue waves in a microstructured
optical fiber was reported [8] in a regime near the thresh-
old of soliton-fission supercontinuum generation, i.e., in
a region where MI plays a key role in the dynamics. A
generalized NLS equation was used successfully to model
the generation of optical rogue waves while, additionally,
control and manipulation of rogue soliton formation was
also discussed [9]. The mechanism of the rogue waves
creation, or, more generally of extreme events, has be-
come an issue of principal interest in various other con-
texts as well, since rogue waves can signal catastrophic
phenomena such as an earthquake, a thunderstorm, or a
severe financial crisis. Knowledge of the probability of
occurrence of extreme events and the capability to pre-
dict the time at which such an event may take place is of
a great value. Such events are usually rare, and they ex-
hibit ”extreme-value” statistics, typically characterized
by heavy-tailed probability distributions.
tal observation of optical rogue-wave-like fluctuations in
fiber Raman amplifiers show that the probability distri-
bution of their peak power follows a power law [10].
In this work we focus on the discrete counterparts of
rogue waves that may appear in nonlinear lattices as a
result of discrete soliton or breather induction and their
mutual interactions. Specifically we investigate the role
of integrability in the formation of discrete rogue waves
(DRW) and the resulting extreme event statistics. Their
appearance may affect dramatically the physical systems.
We use the Salerno model [11] that through a unique
parameter interpolates between a fully integrable discrete
lattice, viz. the Ablowitz-Ladik (AL) lattice [12] , and
the nonintegrable DNLS equation [13, 14]. One of the
basic questions to be addressed below is the probability
of occurrence of a DRW as a function of the degree of
integrability of the lattice and thus study the role of the
latter in the production of extreme lattice events [15].
The Salerno model.- The Salerno model (SM) is given
through the following set of equations
Experimen-
idψn
dt
= −(1 + µ|ψn|2)(ψn+1+ ψn−1) − γ|ψn|2ψn
(1)
where µ and γ are two nonlinearity parameters. When
µ = 0 the model becomes the DNLS equation while for
γ = 0 it reduces to AL. Several properties of the model
such as integrability [16] and stability of localized trav-
elling waves [17, 18] have been analyzed. Both the norm
N and the Hamiltonian H of the model are conserved
quantities. They are given by
N =
1
µ
?
n
ln|1 + µ|ψn|2|, (2)

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H =
?
n
?γ
µ2ln|1 + µ|ψn|2| −γ
µ|ψn|2− 2Re[ψnψ∗
n+1]
?
. (3)
It is also known that Eq. (1) exhibits MI, which may give
rise to stationary, spatially localized solutions in the form
of discrete breathers (DBs), i.e., periodic and spatially lo-
calized nonlinear excitations [19]. The MI induced DBs
appear in random lattice locations and may be mobile.
High-amplitude DBs tend to absorb low-amplitude ones,
resulting after some time in a small number of very high
amplitude excitations, which may get pinned at a specific
lattice site due to the Peierls-Nabarro potential barrier
in nonintegrable lattices [20]. In general, high-amplitude
DBs are virtual bottlenecks which slow down the relax-
ation processes in nonlinear lattices [21, 22], and it has
been proposed that they may serve as models for freak
waves [23]. The development of MI in Eq. (1) can be an-
alyzed with the linear stability analysis of its the plane
wave solutions perturbed by small phase and amplitude
perturbations [24]. The interplay between the on-site and
intersite nonlinear terms (i.e., according to the variation
of their relative strength through µ and γ), may change
MI properties and, consequently, the conditions for the
DBs to exist in the lattice [25]. The SM has recently
found applications in modelling Bose-Einstein conden-
sates of dipolar atoms in a strong periodic potential [26],
dilute Bose-Einstein condensates trapped in a periodic
potential [27], and even biological systems [28].
For later convenience in the numerical simulation, the
variables ψnin Eq. (1) are rescaled as ψn= ξn/√µ, so
that in terms of ξnthe dynamic equations read
idξn(t)
dt
= −(1 + |ξn(t)|2)(ξn+1+ ξn−1) − Γ|ξn(t)|2ξn,(4)
where Γ = γ/µ. Therefore, the whole two-dimensional
parameter space (γ,µ) can be scaled by µ = 1, leaving
γ as a free parameter. With that scaling we may go as
close to the DNLS limit as we want to, by simply let Γ to
attain very large values. However, the exact DNLS limit
µ = 0 has to be calculated separately.
Statistics of extreme events.- We integrate numerically
the system of Eqs. (4) with periodic boundary conditions
using a sixth order Runge-Kutta algorithm with fixed
time-stepping ∆t = 10−4. We started simulations with
different initial conditions (the plane wave, uniform back-
ground with white noise and Gaussian noise) which gave
similar results. Here we present calculations in which
the initial condition is uniform, ξn= 1 for any n, with
the addition of a small amount of white noise to accel-
erate the development of the MI. The uniform solution
is chosen in the interval where it is known from linear
stability analysis that it is unstable. By varying the non-
linearity parameters we identify broadly three regimes of
DRWs that are shown as spatiotemporal evolutions in
Fig. 1. For the purely integrable AL lattice (Γ = 0) DBs
are mobile and essentially noninteracting; as a result we
do not observe significant formation of high DRWs (Fig.
1a). In the vicinity of the AL limit, i.e. for small Γ
FIG. 1: Evolution of the scaled amplitudes |ξn| for a lattice
of size N = 101, with Γ (µ and γ in the DNLS case), is shown
on the figure. The initial conditions for all cases are ξn = 1
for any n (uniform) plus a small amount of white noise.
(Γ ∼ 0.1), there is an onset of weak interaction of the
localized modes of the AL lattice leading to a significant
increase in DRW formation that are mobile (Fig. 1b,c).
In this regime the DBs are highly mobile indicating that
DB merging could be responsible for creation of high-
amplitude localized waves. For Γ >> 0.1 on the other
hand, DNLS-type behavior dominates the SM and local-
ized structures that are initially created through the MI
become easily trapped in the lattice (Fig. 1d).
The three regimes mentioned previously are probed
by calculating the time-averaged height distributions Ph.
We first define the forward (backward) height at the
n−th site as the difference between two successive min-
imum (maximum) and maximum (minimum) values of
|ξn(t)|. We use then both the forward and the backward
heights for the calculation of the local height distribution;
after spatial averaging the latter results in the height
probability densities (HPDs) shown in Fig. 2. We note
that the tails of the HPDs are related to extreme events
and the appearance of DRWs. For Γ finite the HPDs are
sharply peaked but have extended tails indicating that
extreme events are more than several times as large as
the mean distribution height. In the DNLS limit (Γ ≫ 1)
the obtained HPD is very close to the Rayleigh distribu-
tion whose tails decay very fast [29], indicating negligible
probability for the occurrence of extreme events (dotted
curve in Fig. 2). In all the other cases the decay of the
tails of the HPDs is much slower.
In order to probe further the onset of extreme dis-
crete events we employ the practice used in water waves
and define a DRW as one that has a height greater than
hth = 2.2hs, with hs being the significant wave height.
The latter is defined as the average height of the one-
third higher waves in the height distribution. As a result,
the probability of occurrence of extreme DRW events

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?
?
?
?
?
?
?
?
?
?
?
FIG. 2: The normalized height probability density Ph(h) for
several values of Γ and for the DNLS limit (with γ = 6).
The line with slope −1 is added to assist comparisons and
corresponds to Ph ∼ 1/h. Approximately vertical drop cor-
responds to the DNLS limit with an exponential tail. The
increase of Γ(µ = 1) leads to the decrease of the slope and
appearance of plateau on the Ph curve; the latter increases
the extreme event probability leading a maximum at Γ = 0.07
(Fig. 3.)
Pee = Ph(h > hth) is obtained by integration of the
(normalized) HPD from h = hthup to infinity. By eval-
uating several HPDs as those in Fig. 2 we may estimate
the probability of occurrence of DRWs Peeas a function
of the parameter Γ (the results are shown in Fig. 3). We
note that the probability for the occurrence of a DRW
has a certain value in the AL case, subsequently peaks for
small values of Γ and decays precipitously when Γ >> 1.
This behavior of the probability Pee is compatible with
the DB picture outlined earlier, viz. in the very weakly
nonintegrable regime the AL modes may interact leading
to DB fusion and DRW generation. On the other hand,
as nonintegrability becomes stronger, the scattering of
the AL modes is more chaotic leading to a suppression
of DRW formation.
Map approach.- In order to probe deeper on the for-
mation of DRWs we substitute ψn= φnexp(−iωt) into
Eq. (1), with φna real-valued function of the lattice site
n, and obtain the stationary equation
ωφn+ (1 + µ|φn|2)(φn+1+ φn−1) + γ|φn|2φn= 0, (5)
which can be transformed in the two-dimensional map
xn+1= −ω + γx2
n
1 + µx2
n
xn− yn,yn+1= xn, (6)
where we have defined xn = φn and yn = φn−1. Eqs.
(6) represent a real analytic area-preserving map [18, 30]
with the lattice index n playing the role of discrete ’time’.
The phase portraits of the map Eq. (6) for several
Γ-values are shown in Fig.
phase space consists of perfectly disconnected separatri-
ces while for non-zero Γ, the stable and unstable man-
ifolds intersect transversely, resulting in the generation
4. In the AL limit, the
?
?
?
?
Γ
FIG. 3: The normalized probability Pee = Ph(h ≥ hth) for
the occurrence of extreme events as a function of the integra-
bility parameter Γ. All data present averaged results of five
numerical measurements differing in the initial conditions.
of a homoclinic tangle. With increasing Γ the motion
near separatrices becomes exceedingly complicated and
the trajectories wander irregularly before approaching an
attracting set (Figs. 4b and 4c). Moreover, for any Γ ?= 0,
the position of separatrices in phase space changes in
time, resulting in overlapping of neighboring separatrices
and diffusion in those regions which have been traversed
by a separatrix. The sharp peak of the probability of
occurrence of extreme events Pee(h > hth) in the SM
(Fig. 3) can be associated with the opening of a stochas-
ticity web, when orbits fast explore all extended narrow
stochasticity regions leading to an anomalous relaxation
phase [16, 29]. This event signs the transition from the lo-
cal to global stochasticity [31] in SM. On the other hand,
the decrease of Pee(h > hth) for larger Γ’s is related to
the increasingly longer trapping time in more developed
stochasticity region.
The Melnikov analysis in the SM [18] shows that the
magnitude of the separatrix splitting and the consequent
development of stochasticity depends on the Γ/|ω| ra-
tio. The conjecture that Peeis associated with the com-
plexity of the phase portraits of the corresponding maps
implies that Pee should also depend on the Γ/|ω| ra-
tio. In our case |ω| is related to the modulation fre-
quency of the initially uniform solution U with the rela-
tion |ω| = (γ+2µ)U2+2, which, through the MI process
it transformed into a train of localized DB-like config-
urations.We have checked numerically that for fixed
ratio Γ/|ω| and different values of U and Γ we obtain the
same HPD. As a consequence, the probability of extreme
events Peeas a function of the Γ/|ω| is qualitatively the
same with that of Peeas a function of Γ shown in Fig. 3.
The degree of nonintegrability in the SM model can
be quantified by calculating the Lyapunov exponents of
the corresponding maps [32]. We have thus calculated
the maximum Lyapunov exponent L [31] for the map
Eq. (6), for the parameters used in the calculation of

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FIG. 4: Orbits started at different initial positions in the
neighborhood of map origin and corresponding the one-
dimensional Liapunov exponents.
the phase portrait shown in the left panels of Fig. 4. It
is observed that homoclinic orbits which correspond to
perfect separatrices are characterized by vanishing Lya-
punov exponent (Fig. 4a). With increasing stochasticity,
L tends to a finite positive value which generally depends
on the values of the parameters and the initial conditions
(Figs. 4a and 4b).
Conclusions.- The probability of occurrence of extreme
events Pee in the SM results from the competition be-
tween the self-focusing and the energy transport mecha-
nisms which are implicitly correlated with the degree of
integrability of the model [16]. Through modulational
instability and starting from a slightly perturbed uni-
form background we can generate high-amplitude local-
ized moving structures of the DB type that lead to the
formation of extreme events of DRW type. Depending
on their number, amplitude and life-time, they may pre-
vent of facilitate the energy flow in the lattice, affect-
ing thus the probability of extreme event formation Pee.
We find that the latter probability depends strongly on
Γ that affects the degree of integrability of the lattice:
DRW are much more probable very close to the inte-
grable SM limit rather than in the nonintegrable one. We
find a resonance-like maximum in Pee(Γ) that, through
a nonlinear map approach, is linked to separatrix break-
ing and the onset of global stochasticity. This regime
corresponds physically to weak interaction between the
quasi-integrable modes of the system.
A. M. and Lj.H. acknowledge support from the Min-
istry of Science of Serbia (Project 141034). One of us
(GPT) acknowledges discussions with Oriol Bohigas.
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