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Periodic Wave Trains in Nonlinear Media: Talbot Revivals,
Akhmediev Breathers, and Asymmetry Breaking
Georgi Gary Rozenman,1,2 Wolfgang P. Schleich ,3,4 Lev Shemer ,5and Ady Arie 2
1Raymond and Beverly Sackler School of Physics and Astronomy,
Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
2School of Electrical Engineering, Iby and Aladar Fleischman Faculty of Engineering,
Tel Aviv University, Tel Aviv 69978, Israel
3Institut für Quantenphysik and Center for Integrated Quantum Science and Technology (IQST),
Universität Ulm, 89081 Ulm, Germany
4Hagler Institute for Advanced Study at Texas A&M University, Texas A&M AgriLife Research,
Institute for Quantum Science and Engineering (IQSE), and Department of Physics and Astronomy,
Texas A&M University, College Station, Texas 77843-4242, USA
5School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
(Received 1 March 2022; accepted 7 April 2022; published 27 May 2022)
We study theoretically and observe experimentally the evolution of periodic wave trains by utilizing
surface gravity water wave packets. Our experimental system enables us to observe both the amplitude
and the phase of these wave packets. For low steepness waves, the propagation dynamics is in the linear
regime, and these waves unfold a Talbot carpet. By increasing the steepness of the waves and the
corresponding nonlinear response, the waves follow the Akhmediev breather solution, where the higher
frequency periodic patterns at the fractional Talbot distance disappear. Further increase in the wave
steepness leads to deviations from the Akhmediev breather solution and to asymmetric breaking of the
wave function. Unlike the periodic revival that occurs in the linear regime, here the wave crests exhibit self
acceleration, followed by self deceleration at half the Talbot distance, thus completing a smooth transition
of the periodic pulse train by half a period. Such phenomena can be theoretically modeled by using the
Dysthe equation.
DOI: 10.1103/PhysRevLett.128.214101
Introduction.—The optical Talbot effect, discovered by
Henry Fox Talbot in the 19th century [1], is the revival of a
periodic light pattern at periodic distances (Talbot distances
denoted xT¼2T2=λ, where Tand λare the period of the
diffraction grating and optical wavelength) from the input
plane [2]. Moreover, higher frequency periodic patterns
appear at fractional Talbot distances. The fractional Talbot
distance is given by pxT=q, where pand qare prime
numbers and the oscillation frequency is qtimes faster [3].
Because of the quantum mechanical wave nature of
particles, such diffraction effects have also been observed
with matter waves [4–6], which are similar to those in the
case of light waves [7].
Ever since, this phenomenon in quantum physics, which
is closely related to the Talbot effect, has been extensively
studied in many quantum mechanical systems [8–18].For
instance, a quantum carpet can be constructed from an
evolution of the wave function of a single particle in an
infinite square well [19–21].
The evolution of periodic wave trains was also ex-
tended to the nonlinear regime. Specifically, Akhmediev
[22–24] derived an analytic solution, known as Akhmediev
breather, to the nonlinear Schrödinger equation (NLSE)
with third order nonlinear response, which was recently
studied experimentally in a cubic nonlinear optical medium
[25,26]. One of the interesting predictions [27], which was
not observed experimentally until now, is that when the
nonlinear effects become significant, the faster oscillations
at the fractional Talbot distances disappear.
In this Letter, we study the evolution of periodic wave
trains for different levels of nonlinearity, by using surface
gravity water wave pulses. The linear wave equation for
these waves is analogous to the Schrödinger equation for
quantum wave packets, and to the paraxial Helmholtz
equation for optical beams [28–34]. The nonlinear terms of
the Schrödinger equations can be controlled by setting the
steepness of the initial wave packet. At low steepness, the
propagation dynamics along the test section is approxi-
mately linear, thus we observe the familiar Talbot carpet,
including the higher frequency oscillations at fractional
Talbot distances. For higher steepness, we observe the
disappearance of these fractional orders, in correspondence
with Akhmediev’s solution. But when we further increase
the steepness, asymmetric breaking of the wave packet is
observed, i.e., the wave crests accelerate and then decele-
rate at half the Talbot distance leading to a shift of the pulse
PHYSICAL REVIEW LETTERS 128, 214101 (2022)
Editors' Suggestion
0031-9007=22=128(21)=214101(6) 214101-1 © 2022 American Physical Society
train by half a period. This behavior is significantly
different with respect to the familiar disappearance and
revival of the wave crests in the linear case. This pheno-
menon can no longer be described using the analytic
Akhmediev breather, but is in agreement with numerical
predictions based on the Dysthe equation [32,35].
Scheme.—The observation of the linear and nonlinear
Talbot effect is performed in a 5 m long, 0.4 m wide, and
0.2 m deep water-wave tank with a computer controlled
wave maker, see Fig. 1. A wave energy absorbing beach is
placed at the other end of the water tank. To eliminate the
effect of the beach, precise measurements of the water
surface elevation are carried out by wave gauges at
distances not exceeded 4.5 m from the wave maker.
For surface gravity water waves with high steepness, we
apply the spatial version of the Dysthe equations [32,35,36]
∂A
∂ξþi∂2A
∂τ2þijAj2Aþ8εjAj2∂A
∂τ
þ2εA2∂A
∂τþ4iεA∂Φ
∂τ
Z¼0¼0;ð1Þ
4∂2Φ
∂τ2þ∂2Φ
∂Z2¼0ðZ<0Þ;
∂Φ
∂Z
Z¼0¼∂jAj2
∂τ;∂Φ
∂Z
Z→−∞ ¼0ð2Þ
for the normalized amplitude envelope Aðτ;ξÞand the
envelope of the self-induced velocity potential Φðτ;ZÞin
the moving frame.
Here the scaled dimensionless variables ξ,τ, and Zare
related to the propagation coordinate x, the time t, and the
vertical coordinate zby ξ≡ε2k0x,τ≡εω0ðx=cg−tÞ, and
Z≡εk0z. The carrier wave number k0and the angular
carrier frequency ω0satisfy the deep-water dispersion
relation ω2
0¼k0g, with gbeing the gravitational acce-
leration, and define the group velocity cg≡ω0=2k0.
Moreover, Φ≡ϕ=ðω0a2
0Þdenotes the dimensionless velo-
city potential.
We recall that for moderate nonlinearites, Eq. (1) reduces
to a nonlinear Schrödinger equation [30]
i∂A
∂ξ¼∂2A
∂τ2þjAj2Að3Þ
for which a general Akhmediev solution of the cubic
NLSE, expressed in terms of the Jacobi elliptic functions
cn, sn, and dn is given [37] by
Aðτ;ξÞ¼καðτÞdnðκξ;1=κÞþi
ksnðκξ;1=κÞ
1−αðτÞcnðκξ;1=κÞexpðiξÞ;ð4Þ
where αðτÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½1=ð1þκÞ
pcnðffiffiffiffiffi
2κ
pτ;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½ðκ−1Þ=2κ
pÞand
κ>1is a modification parameter [37].
The complex amplitude envelope A≡jAjexpðiφÞdeter-
mines the variation in time and space of the surface
elevation
ηðt; xÞ≡a0Aðt; xÞcos ½k0x−ω0t;ð5Þ
including the carrier wave, where a0is the maximum
amplitude of the envelope.
For observing both the linear and nonlinear Talbot effect,
the wave maker at x¼0is prescribed to generate the
temporal multilobe periodic surface elevation in the form of
the Akhmediev wave
Aðτ;0Þ¼καðτÞ
1−αðτÞð6Þ
with the period Δτ¼ð4=ffiffiffi
κ
pÞKðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½ðκ−1Þ=2k
pÞ, where
KðmÞis the complete elliptic integral of the first kind [38].
Linear Talbot effect.—First, we study the propagation
dynamics of water waves induced by this initial profile
Aðτ;0Þ, Eq. (6), in a linear regime, when the wave steepness
ε¼k0a0is low, ε<0.1. In this case the envelope Aðτ;ξÞ
obeys the equation [30,39]
i∂A
∂ξ¼∂2A
∂τ2ð7Þ
that is similar to the one-dimensional time-dependent
Schrödinger equation of a free particle. However, the roles
of time and space are interchanged.
We measure the elevation ηðt; xÞat 40 spatial locations
and 7200 temporal points (with an averaging of 10). In
Fig. 2we present the amplitude jAðt; xÞj observed (a) in our
setup together with the simulations (d) and (g) based on
Eq. (7) for ε¼0.026. We observe that the revival of the
periodic pattern occurs at xT¼ð4.08 0.10Þm, which
FIG. 1. Experimental setup for observing the linear and non-
linear Talbot effect with surface gravity water waves.
PHYSICAL REVIEW LETTERS 128, 214101 (2022)
214101-2
agrees with the corresponding simulations xTs¼4.12 m
and analytical calculation xTa¼4.05 m[40]. In addition,
the fractional Talbot effect having the period 1
2xTis also
observed around x¼ð1.90 0.10Þm.
Next, using the Hilbert transform [41,42], we extract the
phase k0x−ω0tþφðt; xÞof the surface elevation
η¼ηðt; xÞ. After removing the carrier phase k0x−ω0t,
the complete space-time profile of φðt; xÞis presented in
Fig. 3(a) together with the corresponding simulations,
shown in Figs. 3(d) and 3(g). Thus, we have successfully
observed the linear Talbot effect for both the amplitude and
the phase.
Nonlinear Talbot effect.—We further study the Talbot
effect in the nonlinear regime. For this purpose, the wave
maker at x¼0is again prescribed to generate the temporal
multilobe periodic surface elevation given by Eq. (6) but
now with a relatively large steepness ε¼0.18 where
nonlinear effects come into play. The intensity distribution
of the solution with κ¼2is displayed in Fig. 2(b)
alongside the theoretical results, predicted by Eq. (1),
shown in Fig. 2(e). We observe that the solution exhibits
the self-imaging Talbot carpet which appears at Talbot
distances, and it is π-phase shifted. The wave dynamics for
ε¼0.18 is governed by the nonlinear response. The
solution corresponds to the eigenmode of NLSE, given
by Eq. (4) and it propagates in a nonlinear medium. We
experimentally observe that for such nonlinear waves, the
fractional revivals, which are clearly seen in Figs. 2(a),2(d),
and 2(g) are absent, as was recently predicted in a
theoretical study [27]. However, one can see that even at
intermediate nonlinearities, the analytical solution based on
Eq. (3), which is shown in Fig. 2(h) does not describe
accurately the experimental result shown in 2(b), whereas
the simulation based on Eq. (1) in Fig. 2(e) yields a more
x[m]
-10 -5 0 5 10
1
4
2
0.0
0.5
1.5
1.0
(a)
0
2
8
6
(b)
0
3
6
9
A[mm]
-10 -5 0 5 10
1
4
2
-10 -5 0 5 10
1
4
2
(c)
x[m]
-10 -5 0 5 10
1
4
2
0.0
0.5
1.5
1.0
(d)
0
2
8
6
(e)
0
3
6
9
A[mm]
-10 -5 0 5 10
1
4
2
-10 -5 0 5 10
1
4
2
(f)
x[m]
-10 -5 0 5 10
1
4
2
0.0
0.5
1.5
1.0
(g)
0
2
8
6
(h)
0
3
6
9
A[mm]
-10 -5 0 5 10
1
4
2
-10 -5 0 5 10
1
4
2
(i)
t-x/cg[s]
t-x/cg[s]t-x/cg[s]
FIG. 2. Space-time profile of the envelope amplitude Aðt; xÞinduced by the Akhmediev wave, Eq. (6), with κ¼2for linear, nonlinear,
and highly nonlinear waves. (a) a0¼1.5mm and ω0¼13 ½rad=s(ε¼0.026). (b) a0¼8mm and ω0¼15 ½rad=s(ε¼0.18).
(c) a0¼9mm and ω0¼17 ½rad=s(ε¼0.265). The theoretical plots in (d)–(f) are simulations based on Eq. (1) for the corresponding
parameters in (a)–(c), i.e., steepness of ε¼0.026, 0.18, 0.265. (g) Simulation based on the linear Schrödinger equation, Eq. (7) for
ε¼0.026, (h) analytical plot for ε¼0.18 based on Eq. (4), (i) numerical plot for ε¼0.265 based on Eq. (3).
PHYSICAL REVIEW LETTERS 128, 214101 (2022)
214101-3
accurate prediction. For instance, it indicates the uneven
gap in the formed dark canals at x¼2m. The mentioned
differences are further strengthened by observing the
longitudinal differences in the phase pattern with the
experimental result shown in Fig. 3(a). The simulation
based on Eq. (1) and shown in Fig. 3(b) predicts these
characteristics while the analytical result shown in Fig. 3(h)
has shorter longitudinal lobes.
Finally, we increase the wave steepness to ε¼0.265 and
study the evolution of periodic wave trains in this highly
nonlinear regime. As seen in Fig. 2(c), we observe extra-
ordinary behaviour of wave packets which has not been
reported before either theoretically or experimentally [27].
Namely, at half the Talbot distance, around x¼2–3m the
envelope wave amplitude Aðt; xÞbreaks asymmetrically
and new canals are formed. In that particular region, each
lobe slightly self-accelerates and each new emerging lobe
slightly decelerates back to the same group velocity as the
lobes found at the origin. The fractional disappearance and
revivals, observed in Fig. 2(a) for low steepness, are absent,
as depicted in Fig. 2(b). Instead, we observe a continuous
shift of the wave crests by half of the period. These results
are successfully verified by solving Eqs. (1) and (2)
numerically, as shown in Fig. 2(f), and compared to the
numerical solution of the third-order nonlinear Schrödinger
equation, that is Eq. (3) [27], shown in Fig. 2(i). Thus,
Eqs. (1) and (2), which include high-order terms pro-
portional to ε, describe accurately the dynamics of the
[rad]
π/2
-π/2
0
π/4
-π/4
[rad]
π/2
-π/2
0
π/4
-π/4
[rad]
π/2
-π/2
0
π/4
-π/4
1
4
2
-10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10
[rad]
π/2
-π/2
0
π/4
-π/4
[rad]
π/2
-π/2
0
π/4
-π/4
[rad]
π/2
-π/2
0
π/4
-π/4
1
4
2
-10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10
t-x/c [s]
g
[rad]
π/2
-π/2
0
π/4
-π/4
t-x/c [s]
g
[rad]
π/2
-π/2
0
π/4
-π/4
t-x/c [s]
g
[rad]
π/2
-π/2
0
π/4
-π/4
1
4
2
-10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10
x[m]
x[m]
x[m]
(a) (b) (c)
)f()e()d(
)i()h()g(
FIG. 3. Space-time profile of the envelope phase φðt; xÞinduced by the Akhmediev wave, Eq. (6), with κ¼2for linear, nonlinear, and
highly nonlinear waves. (a) a0¼1.5mm and ω0¼13 ½rad=s(ε¼0.026). (b) a0¼8mm and ω0¼15 ½rad=s(ε¼0.18).
(c) a0¼9mm and ω0¼17 ½rad=s(ε¼0.265). The theoretical plots in (d)–(f) are simulations based on Eq. (1) for the corresponding
parameters in (a)–(c), i.e., steepness of ε¼0.026, 0.18, 0.265. (g) Simulation based on the linear Schrödinger equation, Eq. (7) for
ε¼0.026, (h) analytical plot for ε¼0.18 based on Eq. (4), (i) numerical plot for ε¼0.265 based on Eq. (3).
PHYSICAL REVIEW LETTERS 128, 214101 (2022)
214101-4
envelope amplitude Aðt; xÞin the nonlinear regime, in
particular the Talbot effect. These asymmetric properties
are also observed in the phase in Fig. 3(c) and are modeled
numerically in 3(f) by solving Eqs. (1) and (2).
In conclusion, we have studied and observed the linear
and nonlinear Talbot effect for both the amplitude and the
phase of surface gravity water waves originating from the
Akhmediev wave. In addition, beyond the linear regime we
have observed the disappearance of the fractional revivals.
Finally, for even higher steepness of water waves, the
fractional Talbot effect (at x¼1.9m) is absent and a self-
accelerating pattern of the wave crest is observed, in
contrast to the familiar disappearance and revival pattern
in the linear case. We also emphasize that our experimental
setup is not limited to freely propagating waves and it is
possible to study this phenomena in the presence of a linear
potential [30,31].
We anticipate that the evolution of periodic patterns that
we observed here should occur for other types of waves,
e.g., optical waves, where at low intensity they will exhibit
revivals at integer and fractional Talbot distances, followed
by the disappearance of the higher periodic structures at
fractional distances for a Kerr nonlinear medium, and
finally deviations from the Akhemediev breather solution
when higher order nonlinear terms come into play.
We thank Maxim A. Efremov, Matthias Zimmermann,
and Anatoliy Khait for fruitful discussions help and
support, and Tamir Ilan for technical support and advice.
We also thank the CLEO-2021 conference where we first
presented the results that led to this Letter [43]. This work is
supported by DIP, the German-Israeli Project Cooperation
(AR 924/1-1, DU 1086/2-1) supported by the DFG, the
Israel Science Foundation (Grants No. 1415/17, No. 508/
19). W. P. S. is grateful to Texas A&M University for a
Faculty Fellowship at the Hagler Institute for Advanced
Study at the Texas A&M University as well as to the Texas
A&M AgriLife Research. The research of the IQST is
financially supported by the Ministry of Science, Research
and Arts Baden-Württemberg.
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1−λ2=a2
pÞ. In surface gravity waves this distance
is given in temporal units by tT¼T=ð1−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−T2=Δt2
pÞ,
where Tis the period of the carrier wave and Δtis the period
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PHYSICAL REVIEW LETTERS 128, 214101 (2022)
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