PHYSICAL REVIEW E 86, 056601 (2012)
Observation of a hierarchy of up to fifth-order rogue waves in a water tank
A. Chabchoub,
1,*
N. Hoffmann,
1
M. Onorato,
2,3
A. Slunyaev,
4
A. Sergeeva,
4
E. Pelinovsky,
4
and N. Akhmediev
5
1
Mechanics and Ocean Engineering, Hamburg University of Technology, Eißendorfer Straße 42, 21073 Hamburg, Germany
2
Dipartimento di Fisica, Universit
`
a degli Studi di Torino, Torino 10125, Italy
3
Istituto Nazionale di Fisica Nucleare, INFN, Sezione di Torino, 10125 Torino, Italy
4
Department of Nonlinear Geophysical Processes, Institute of Applied Physics, Nizhny Novgorod, Russia
5
Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra, ACT 0200, Australia
(Received 19 July 2012; published 6 November 2012)
We present experimental observations of the hierarchy of rational breather solutions of the nonlinear
Schr
¨
odinger equation (NLS) generated in a water wave tank. First, five breathers of the infinite hierarchy
have been successfully generated, thus confirming the theoretical predictions of their existence. Breathers of
orders higher than five appeared to be unstable relative to the wave-breaking effect of water waves. Due to the
strong influence of the wave breaking and relatively small carrier steepness values of the experiment these results
for the higher-order solutions do not directly explain the formation of giant oceanic rogue waves. However, our
results are important in understanding the dynamics of rogue water waves and may initiate similar experiments in
other nonlinear dispersive media such as fiber optics and plasma physics, where the wave propagation is governed
by the NLS.
DOI: 10.1103/PhysRevE.86.056601 PACS number(s): 05.45.Yv, 47.20.Ky, 42.65.−k, 92.10.Hm
I. INTRODUCTION
The nonlinear Schr
¨
odinger equation (NLS) describes a
variety of nonlinear wave processes in physics, including
gravity waves on the surface of the deep ocean [1,2]. It
has been well established that the NLS has a hierarchy of
rational solutions that describe doubly localized structures
with infinitely increasing amplitude as its order increases
[3–7]. The lowest-order structure of this hierarchy is known as
the Peregrine breather [8]. Due to its localization both in time
and in space it is considered to be a prototype of a rogue wave in
the ocean [9] that has the remarkable property to “appear from
nowhere and disappear without a trace” [10]. All higher-order
structures of this family have the same property and in addition
have progressively increasing amplitudes [11]. As such, they
would describe rogue waves of significantly higher ratio of
peak to the background amplitude.
Theoretically, expressions for these solutions can be written
in general form for any order [4–7,12]. However, explicit
formulas become cumbersome with increasing order, and
the highest order to which a solution is presently known is
eight [13]. Difficulties in writing the solution are reoccurring
when they are observed experimentally. The Peregrine breather
has been observed recently in optics [14], in a water wave
tank [15], and in plasma [16]. The second-order structure,
or “superrogue wave,” has also been observed in the case of
deep-water waves [17]. Thus, fundamentally, the existence
of two lowest-order structures in experiment has been proven.
However, the higher-order structures of this hierarchy still have
to be shown to exist. This is not an easy task as these waves,
in addition to high amplitude, have higher steepness. In order
to keep the parameters of the structure within the allowed
limitations, one has to choose carefully the initial conditions
for their excitation.
*
amin.chabchoub@tuhh.de
In this work, we are able to demonstrate the existence of
higher-order rational solutions on a water surface up to fifth
order. This seems to be the maximum order which we can
achieve, at least with our existing experimental equipment.
This is a remarkable achievement taking into account the
approximate nature of modeling deep-water waves with the
NLS. Of course, we are far from claiming that these solutions
can describe real oceanic rogue waves [1,18–22]. However,
our results show clearly that nonlinearity may play a central
role in the dynamics of water waves. As such, these results can
be considered as significant progress in our understanding of
water wave dynamics.
II. THEORETICAL BACKGROUND
The NLS is one of the fundamental equations in theoretical
physics. Generally, it describes one-dimensional evolution
in time and space of weakly nonlinear wave packets in
optics, hydrodynamics, and plasmas and, more generally, in
nonlinear dispersive media [23,24]. In particular, it describes
the propagation of deep-water waves [25–27]. For this purpose,
the NLS can be written as
i
∂A
∂t
+ c
g
∂A
∂x
−
ω
0
8k
2
0
∂
2
A
∂x
2
−
ω
0
k
2
0
2
|
A
|
2
A = 0, (1)
where x and t are the spatial and time coordinates, respectively,
and ω
0
and k
0
denote the wave frequency and the wave number
of the carrier wave. The dispersion relation of linear deep-water
wave trains is given by ω
0
=
√
gk
0
, where g is the gravitational
acceleration. The wave group velocity is c
g
:=
dω
dk
|
k=k
0
=
ω
0
2k
0
.
The surface elevation of water waves η(x,t) can be calculated
from the NLS variable A(x,t). To second order in steepness it
is given by
η(x,t) = Re{A
(
x,t
)
exp
[
i
(
k
0
x − ω
0
t
)
]
}
+Re
1
2
k
0
A
2
(
x,t
)
exp
[
2i
(
k
0
x − ω
0
t
)
]
. (2)
056601-1
1539-3755/2012/86(5)/056601(6) ©2012 American Physical Society
A. CHABCHOUB et al. PHYSICAL REVIEW E 86, 056601 (2012)
When dealing with exact solutions, it is convenient to use
the dimensionless form of the NLS,
iψ
T
+ ψ
XX
+ 2|ψ|
2
ψ = 0, (3)
which is obtained from Eq. (1) using the rescaled variables:
T =−
ω
0
8
t, X = (x − c
g
t)k
0
= xk
0
−
ω
0
2
t, ψ =
√
2k
0
A.
(4)
Here, X is the coordinate in a frame moving with the group
velocity, and T is the rescaled time.
Generally, the NLS has a hierarchy of breather solutions
localized in space and time [3,8,10]. They pulsate only once,
thus representing a class of solutions that can be considered a
model for oceanic rogue waves. The j-order rational solution
of the NLS in general form, also known as the jth Akhmediev-
Peregrine solution, can be written in the form
ψ
j
(
X,T
)
= ψ
0
(
−1
)
j
+
G
j
+ iH
j
D
j
exp(2i|ψ
0
|
2
T ), (5)
where the background amplitude is ψ
0
and the polynomials
G
j
(X,T ), H
j
(X,T ), and D
j
(X,T ) can be found in Ref. [11]
for a few lowest-order solutions. In particular, for the Peregrine
breather, G
1
= 4, H
1
= 16|ψ
0
|
2
T , and D
1
= 1 + 4|ψ
0
|
2
X
2
+
16|ψ
0
|
4
T
2
. The corresponding expressions for higher-order
solutions are progressively more complicated. To give an
example, the expressions for the eighth-order solution written
similarly would require 60 printed pages [13]. We assume that
the polynomials G
j
, H
j
, and D
j
for j = 3,4,5 are known
from Refs. [3,5,6] and will not copy them here. The amplitude
profiles for rational solutions up to fourth order are illustrated
in Fig. 1. The central amplitude is equal to (2j + 1)ψ
0
.It
increases progressively with j.
Experimental explorations of this class of breather solutions
started recently with the Peregrine breather, observed first in
fiber optics [14], in a water tank [15], and in plasma [16]. The
second-order solution was observed only in the case of water
waves [17]. In the present work, we took further steps in order
FIG. 1. (Color online) Higher-order Akhmediev-Peregrine ratio-
nal solutions of the NLS with the background amplitude ψ
0
= 1. At
X = 0andT = 0 the maximal amplitude amplification of the j th
solution is 2j + 1.
to observe third-, fourth-, and fifth-order solutions in a water
wave tank.
III. OBSERVATIONS OF THE HIGHER-ORDER
ROGUE WAVES
The present experiments were conducted in a 15 m ×
1.6m× 1.5 m water wave tank. An illustration and technical
details of the tank can be found in Ref. [15]. The single-flap
wave maker is computer controlled to generate the desired
wave shapes and heights. To avoid wave reflections, an ab-
sorbing beach is installed at the opposite end. All experiments
are conducted in deep-water conditions, with the ratio of the
water depth h of 1 m to the wavelength being much larger
than unity. The water surface elevation at any given point is
measured by a capacitance wave gauge with a sensitivity of
1.06 V/cm and a sampling frequency of 0.5 kHz, placed at a
distance 9 m from the flap.
In order to generate rational breathers representing giant
rogue waves in the tank, one has to fix the initial amplitude a
0
and the steepness ε = a
0
k
0
of the carrier wave. Once the ampli-
tude and the steepness are determined, it is straightforward to
find the wave number k
0
and to derive the wave frequency from
the linear dispersion relation, i.e., ω
0
=
√
gk
0
=
√
gε
a
0
. Then,
using the relation ψ
0
= a
0
√
2k
0
and inverting the scaling in
Eq. (4), the analytical solution can be converted to dimensional
form.
To determine the boundary conditions for the wave maker,
the dimensionalized analytical solution of the NLS (5) is
translated along the tank in order to observe the position of
maximal amplitude of the rational solutions (x = 0) closer
to the beach. Then the expression for the surface elevation,
Eq. (2), is used to calculate the initial condition at the position
of the paddle, which is x =−9 m. Such an arrangement
produces the maximal amplitude of the breather at a distance
of 9 m from the flap.
In order to calibrate our wave-generating equipment, first,
we measured the response function of the flap. Namely, we
generated a pure sinusoidal wave with a given amplitude at
the output of the computer signal controlling the flap and
measured the water wave amplitude in the tank. The plot of
water wave amplitude versus the amplitude of the computer
signal is shown in Fig. 2. It can be seen that within the range
of amplitudes we are dealing with, this plot is linear. The
corresponding coefficient of this linear response function has
been used in the computer program in further experiments.
The first observation of a Peregrine soliton evolution in
water waves was reported in Ref. [15]. It was done at a
fixed frequency ω
0
= 10.68 s
−1
, an amplitude of a
0
= 0.01
m, and a water depth value of h = 1 m in the tank. In
order to confirm that the Peregrine breather does exist for
a wider range of parameters of the experimental setup, we
repeated these observations. Figure 3 shows another example
of a measured Peregrine breather. Here, the steepness of the
carrier is ε = 0.09, and the background wave has an elevation
amplitude of 0.5 cm. Therefore, the frequency is ω
0
=
13.29 s
−1
. Then, the maximal amplitude amplification is 3,
providing an amplitude of the rogue wave of 1.5 cm.
The first experimental demonstration of the second-order
Akhmediev-Peregrine soliton was reported earlier in Ref. [17].
056601-2
OBSERVATION OF A HIERARCHY OF UP TO FIFTH- ... PHYSICAL REVIEW E 86, 056601 (2012)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
0.5
1
1.5
2
2.5
Amplitude of the computer signal
Amplitude of the generated waves (cm)
FIG. 2. (Color online) Response function of the flap for a fixed
steepness value ε = 0.1andafixeddepthh = 1m.
Figure 4 shows another observation for the parameter values
of a
0
= 3 mm and k
0
= 13.33 m
−1
; hence, ω
0
= 11.43 s
−1
.
As predicted in theory, the carrier wave reaches its maximal
amplitude amplification of 5 at the center of the breather.
Generally, in order to generate higher-order breather solutions
in a water wave tank, the carrier parameters have to be
carefully chosen to avoid significant distortions related to wave
breaking. As the local steepness of the wave in this case is
higher than for the Peregrine breather, we have to reduce
carrier steepness to 0.04 and its amplitude to 3 mm. Thus,
the maximum amplitude of the second-order breather reaches
15 mm.
0 2 4 6 8 10 12
0
0
1.5
1.5
Time (s)
Surface elevation (cm)
10 mm
10 mm
FIG. 3. (Color online) Experimental observation of the Peregrine
soliton at x = 0 for carrier parameters a
0
= 5mmandε = 0.09 (blue
top curve). For comparison, the theoretical prediction at the same
position is shown by the red bottom curve.
0 5 10 15 20 25 30 35 40
0
0
1.5
1.5
Time (s)
Surface elevation (cm)
6mm
6mm
FIG. 4. (Color online) Experimental demonstration of the second-
order rational breather at x = 0 for carrier wave parameters a
0
= 3
mm and ε = 0.04 (blue top curve). For comparison, the theoretical
curve at the same spatial position is shown by the red bottom curve.
Figure 5 presents experimental demonstrations of the third-
order breather solution. These are observed for several steep-
ness values of the background wave, namely, 0.04 [Fig. 5(a)],
0.03 [Fig. 5(b)], and 0.02 [Fig. 5(c)], and carrier amplitudes of
0.05 cm [Fig. 5(a)], 0.1 cm [Fig. 5(b)], and 0.2 cm [Fig. 5(c)],
respectively. For these amplitudes, experimental curves are
reasonably well described by the theory with an amplification
factor of 7 being reached at the peak of the breather for all three
cases. Relatively small background amplitudes are crucial
for these observations. The curves are becoming asymmetric
when the background steepness increases, as can be seen from
Fig. 5(c). The presence of such an asymmetry when increasing
the carrier steepness was also observed for the Peregrine
solution in Ref. [28] and for the second-order solution in
0 5 10 15 20 25 30 35 40
0
0.35
0
0.35
Time (s)
Surface elevation (cm)
0 6 12 18 24
0
0
0.7
0.7
Time (s)
0 6 12 18 24
0
0
1.4
1.4
Time (s)
2mm
2mm
4mm
4mm
1mm
1mm
(a)
(b) (c)
FIG. 5. (Color online) Three experimental observations (blue top
curves) of the third-order rational breather at x = 0 for carrier wave
parameters (a) a
0
= 2mmandε = 0.04, (b) a
0
= 1mmandε = 0.03,
and (c) a
0
= 0.5mmandε = 0.02. For comparison, the theoretical
predictions at the same positions are shown by the red bottom curves.
056601-3
A. CHABCHOUB et al. PHYSICAL REVIEW E 86, 056601 (2012)
0 5 10 15 20 25 30 35 40 45
1
2
3
4
5
6
7
8
9
Time (s)
Distance from the paddle (m)
FIG. 6. (Color online) Evolution of the third-order rogue wave
along the water wave tank for carrier parameters a
0
= 2mmand
ε = 0.04. The curves are measured at distances separated by 1 m
from each other.
Ref. [17]. A further increase of the background steepness
results in a breaking of the wave near the center of the breather.
Figure 6 shows the spatial evolution of the third-order
rational solution along the tank. The lowest curve is measured
at a distance of 1 m from the starting end of the tank; i.e.,
it is close to the paddle. The upper curve is measured at a
distance of 9 m from the paddle, i.e., at the location of the
maximum amplitude of the breather. The intermediate curves,
taken at distances separated by 1 m from each other, show
the growth of the central amplitude of the third-order breather
and the concentration of wave energy towards the central area
of the breather at the developed stage of evolution. They also
show that the whole localized formation moves with the group
velocity. The limited length of the tank does not allow us to
observe the complete evolution of the breather starting from
slightly perturbed sinusoidal wave. The distance required for
half of the evolution length until the maximum is reached can
be estimated as around 100 m. Thus, we had to start with
the initial condition at x =−9 m when the perturbation grew
noticeably. This would be the limitation for most of the water
tanks used in laboratories.
Figure 7 presents experimental observations of the fourth-
order breather solution. These experiments are done for
background amplitudes of 1 and 3 mm for steepness values
of 0.02 and 0.03, respectively. The expected wave amplitude
amplification factor of 9 is reached in these experiments re-
markably well. Unavoidable asymmetry of the curves appears
when increasing the steepness of the carrier.
The highest-order breather that we were able to observe so
far is the fifth-order solution. These results are shown in Fig. 8.
The theoretical amplification factor of the carrier amplitude
here is 11, which is a remarkable fact by itself, despite the
background wave amplitude being only 1 mm. Any attempts
to generate sixth- and higher-order breathers failed as the wave
breaking ruined the central part of the breather, thus reducing
0 10 20 30 40 50
0
0
0.9
0.9
Time (s)
0 10 20 30 40 50
0
0
2.7
2.7
Time (s)
Surface elevation (cm)
(a)
(b)
2mm
2mm
6mm
6mm
FIG. 7. (Color online) Two observations of the fourth-order
rational breather at x = 0 for carrier parameters (a) a
0
= 1mmand
ε = 0.02 in and (b) a
0
= 3mmandε = 0.03 (blue top curves)
compared to the theoretical prediction at the same position (red
bottom curves).
the central peak significantly. As a result, the whole wave
evolution has been greatly distorted.
IV. EXPERIMENTAL LIMITATIONS AND
BREAKING LIMITS
There are a few restrictions that have to be men-
tioned related to the experimental observations described
above. Higher-order solutions have a multipeak structure and
increased derivative relative to the first-order solution. Taking
this into account is essential when setting up higher-order
solutions. First, the spatial extension of the higher-order
breathers increases strongly with the order. The higher the
order of the breather is, the longer the space needed for the
FIG. 8. (Color online) (a) Fifth-order Akhmediev-Peregrine
rational solutions of the NLS with the background ψ
0
= 1.
(b) Observations of the fifth-order rational breather at x = 0 for carrier
wave parameters a
0
= 1mmandε = 0.01 (blue top curve) compared
to the theoretical prediction at the same position (red bottom curve).
056601-4
OBSERVATION OF A HIERARCHY OF UP TO FIFTH- ... PHYSICAL REVIEW E 86, 056601 (2012)
0 10 20 30 40 50 60
0
0
1.1
1.1
Time (s)
Surface elevation (cm)
2mm
2mm
FIG. 9. (Color online) Measurement of the fifth-order rational
breather which breaks before reaching the maximal amplitude
amplification at x = 0 for carrier parameters a
0
= 1mmandε = 0.02
(blue top curve) compared to the theoretical prediction at the same
position (red bottom curve).
breather to develop is; therefore, due to the limited size of
the tank, the higher order must be the initial wave amplitude
generated by the flap. For example, for the fifth-order solution,
in order to observe the solution at a distance 9 m from the flap,
the initial amplitude of the wave after the flap has to be as
high as about 10 times the amplitude of the underlying carrier
Stokes wave. Thus, the tank length provides a relatively small
part of the breather evolution. However, 9 m in the experiment
still correspond to 14 wavelengths, and we can clearly observe
the nonlinear evolution of the most essential central part of the
breather.
Second, significant local amplification of the wave am-
plitude of the breather means that the wave breaking may
occur before the highest amplitude is reached. Indeed, Fig. 9
shows that due to the breaking of the wave, the amplitude
amplification of 11 expected for the fifth-order solution is not
reached. This happens when the carrier steepness is increased
to 0.02, while the amplitude is 1 mm, i.e., the same as in
Fig. 8(b). Since the objective of the present study is to generate
higher-order solutions with the best possible accuracy, we tried
to keep the amplitudes at sufficiently low level. Thus, very
low steepness values of the carrier wave have been used. We
avoided exceeding criteria for local breaking during the whole
wave evolution. Controlling the steepness for each breather, we
found that spilling type breaking occurs for the experimentally
estimated threshold carrier steepness values ε
b
shown in
Table I.
Keeping the steepness values low did not allow us to reach
high amplitudes for breathers in absolute terms. Nevertheless,
the ratio of the peak amplitude to the background amplitude
was confirmed to agree with the theory. The latter increases
with the order of the solution. We should note that the wave
breaking is a process independent from breather generation.
The distance at which the breaking of an initially sinusoidal
wave occurs depends on the amplitude of this wave and
generally decreases with the amplitude.
TABLE I. Lowest threshold carrier wave steepness value for each
rational breather when wave breaking starts.
Rational solutions Threshold steepness value ε
b
First order 0.12
Second order 0.06
Third order 0.05
Fourth order 0.04
Fifth order 0.02
V. SUMMARY
We have confirmed, experimentally, that the higher-order
rational breather solutions of the NLS can be generated in
conditions of deep-water gravity waves. Up to fifth order
a breather can be observed in a 15 m tank without being
significantly distorted by the wave-breaking effect and the
limitations of the short tank.
The steepness of the carrier wave is the crucial parameter
in these observations. The smaller the carrier steepness is, the
better the agreement is with the theoretical NLS prediction.
For each observed rational breather we determined the carrier
steepness values when the corresponding amplified wave
starts to break. Due to this limitation, the absolute values
of peak amplitudes for higher-order solutions cannot be
significantly increased. However, the ratio of peak amplitude to
the background wave increases in accordance with the theory.
Clearly, the results of experimental observation of higher-
order breathers in the water tank cannot be directly applied
to explain giant ocean rogue waves. First, the probability of
their excitation in a chaotic wave field would be extremely
low. Second, wave-breaking phenomena would destroy them
well before they reached high amplitudes. Third, the steep-
ness values of our experiment are significantly smaller than
steepness values related to the ocean. Furthermore, there are
myriad other factors that differ between the ocean and an
ideal laboratory setup. Nevertheless, the mere fact that such
solutions can be observed proves the validity of the nonlinear
approach to the dynamics of water waves. It also shows that
the range of phenomena described by the NLS is significantly
richer than the simple world of solitons and small-amplitude
radiation waves. This world is not complete if we do not take
into account the whole hierarchy of breathers.
ACKNOWLEDGMENTS
A.C. would like to thank Pierre Gaillard for helpful
discussions. N.H., E.P., A.Se. and N.A. acknowledge the
support of the Volkswagen Stiftung. N.A. acknowledges
partial support of the Australian Research Council (Discovery
Project No. DP110102068). E.P. acknowledges support from
RFBR grant 11-05-00216, A.Sl. acknowledges support from
RFBR grant 11-02-00483, and A.Se. is supported by the
President’s grant MK-4378.2011.5. M.O. was supported by
ONR grant N000141010991 and by the European Union under
the project EXTREME SEAS (SCP8-GA-2009-234175). N.A.
acknowledges support from the Alexander von Humboldt
Foundation.
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