In this paper, we undertake a systematic exploration of soliton turbulent phenomena and the emergence of extreme rogue waves within the framework of the one-dimensional fractional nonlinear Schrödinger (FNLS) equation, which appears in many fields, such as nonlinear optics, Bose–Einstein condensates, plasma physics, etc. By initiating simulations with a plane wave modulated by small noise, we scrutinized the universal regimes of non-stationary turbulence through various statistical indices. Our analysis elucidates a marked increase in the probability of rogue wave occurrences as the system evolves within a certain range of Lévy index α, which can be ascribed to the broadened modulation instability bandwidth. This heightened probability of extreme rogue waves is corroborated through multiple facets, including wave-action spectrum, fourth-order moments, and probability density functions. However, it is crucial to acknowledge that a decrease in α also results in a reduction in the propagation speed of solitons within the system. Consequently, only high-amplitude solitons with non-zero background are observed, and the occurrence of collisions that could generate higher-amplitude rogue waves is suppressed. This introduces an inverse competitive mechanism: while a lower α expands the bandwidth of modulation instability, it concurrently impairs the mobility of solitons. Our findings contribute to a deeper understanding of the mechanisms driving the formation of rogue waves in nonlinear fractional systems, offering valuable insights for future theoretical and experimental studies.

1
R. E.
Peierls
, “
Zur kinetischen Theorie der Wärmeleitungen in Kristallen
,”
Ann. Phys.
3
,
1055
1101
(
1929
).
2
S.
Nazarenko
,
Wave Turbulence
(
Springer
,
Berlin
,
2011
).
3
V. E.
Zakharov
, “
Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid
,”
Eur. J. Mech. B
18
,
327
344
(
1999
).
4
V. E.
Zakharov
, “
Turbulence in integrable systems
,”
Stud. Appl. Math.
122
,
219
(
2009
).
5
D. S.
Agafontsev
and
V. E.
Zakharov
, “
Integrable turbulence and formation of rogue waves
,”
Nonlinearity
28
,
2791
(
2015
).
6
D. S.
Agafontsev
and
V. E.
Zakharov
, “
Integrable turbulence generated from modulational instability of cnoidal waves
,”
Nonlinearity
29
,
3551
(
2016
).
7
M.
Onorato
,
D.
Proment
,
G.
El
,
S.
Randoux
, and
P.
Suret
, “
On the origin of heavy-tail statistics in equations of the nonlinear Schrödinger type
,”
Phys. Lett. A
380
,
3173
(
2016
).
8
J. M.
Soto-Crespo
,
N.
Devine
, and
N.
Akhmediev
, “
Integrable turbulence and rogue waves: Breathers or solitons
?”
Phys. Rev. Lett.
116
,
103901
(
2016
).
9
N.
Akhmediev
,
J. M.
Soto-Crespo
, and
N.
Devine
, “
Breather turbulence versus soliton turbulence: Rogue waves, probability density functions, and spectral features
,”
Phys. Rev. E
94
,
022212
(
2016
).
10
A. A.
Gelash
and
D. S.
Agafontsev
, “
Strongly interacting soliton gas and formation of rogue waves
,”
Phys. Rev. E
98
,
042210
(
2018
).
11
G.
Roberti
,
G.
El
,
S.
Randoux
, and
P.
Suret
, “
Early stage of integrable turbulence in the one-dimensional nonlinear Schrödinger equation: A semiclassical approach to statistics
,”
Phys. Rev. E
100
,
032212
(
2019
).
12
A.
Gelash
,
D.
Agafontsev
,
V.
Zakharov
,
G.
El
,
S.
Randoux
, and
P.
Suret
, “
Bound state soliton gas dynamics underlying the spontaneous modulational instability
,”
Phys. Rev. Lett.
123
,
234102
(
2019
).
13
D. S.
Agafontsev
,
S.
Randoux
, and
P.
Suret
, “
Extreme rogue wave generation from narrowband partially coherent waves
,”
Phys. Rev. E
103
,
032209
(
2021
).
14
Z.-Y.
Sun
,
X.
Yu
, and
Y. J.
Feng
, “
Coexistence of Gaussian and non-Gaussian statistics in vector integrable turbulence
,”
Phys. Rev. E
108
,
054211
(
2023
).
15
T.
Congy
,
G. A.
El
,
G.
Roberti
,
A.
Tovbis
,
S.
Randoux
, and
P.
Suret
, “
Statistics of extreme events in integrable turbulence
,”
Phys. Rev. Lett.
132
,
207201
(
2024
).
16
P.
Suret
,
S.
Randoux
,
A.
Gelash
,
D. S.
Agafontsev
,
B.
Doyon
, and
G. A.
El
, “
Soliton gas: Theory, numerics, and experiments
,”
Phys. Rev. E
109
,
061001
(
2024
).
17
M.
Li
,
X. Z.
Zhu
, and
T.
Xu
, “
Integrable turbulence and statistical characteristics of chaotic wave field in the Kundu-Eckhaus equation
,”
Phys. Rev. E
109
,
014204
(
2024
).
18
A.
Costa
,
A. R.
Osborne
,
D. T.
Resio
,
S.
Alessio
,
E.
Chrivì
,
E.
Saggese
,
K.
Bellomo
, and
C. E.
Long
, “
Soliton turbulence in shallow water ocean surface waves
,”
Phys. Rev. Lett.
113
,
108501
(
2014
).
19
P.
Walczak
,
S.
Randoux
, and
P.
Suret
, “
Optical rogue waves in integrable turbulence
,”
Phys. Rev. Lett.
114
,
143903
(
2015
).
20
P.
Suret
,
R. E.
Koussaifi
,
A.
Tikan
,
C.
Evain
,
S.
Randoux
,
C.
Szwaj
, and
S.
Bielawski
, “
Single-shot observation of optical rogue waves in integrable turbulence using time microscopy
,”
Nat. Commun.
7
,
13136
(
2016
).
21
A.
Tikan
,
S.
Bielawski
,
C.
Szwaj
,
S.
Randoux
, and
P.
Suret
, “
Single-shot measurement of phase and amplitude by using a heterodyne time-lens system and ultrafast digital timeholography
,”
Nat. Photon.
12
,
228
(
2018
).
22
R. E.
Koussaifi
,
A.
Tikan
,
A.
Toffoli
,
S.
Randoux
,
P.
Suret
, and
M.
Onorato
, “
Spontaneous emergence of rogue waves in partially coherent waves: A quantitative experimental comparison between hydrodynamics and optics
,”
Phys. Rev. E
97
,
012208
(
2018
).
23
A. R.
Osborne
,
D. T.
Resio
,
A.
Costa
,
S.
Ponce de León
, and
E.
Chirivì
, “
Highly nonlinear wind waves in Currituck sound: Dense breather turbulence in random ocean waves
,”
Ocean Dyn.
69
,
187
(
2019
).
24
A. E.
Kraych
,
D.
Agafontsev
,
S.
Randoux
, and
P.
Suret
, “
Statistical properties of the nonlinear stage of modulation instability in fiber optics
,”
Phys. Rev. Lett.
123
,
093902
(
2019
).
25
I.
Redor
,
E.
Barthélemy
,
N.
Mordant
, and
H.
Michallet
, “
Analysis of soliton gas with large-scale video-based wave measurements
,”
Exp. Fluids
61
,
216
(
2020
).
26
P.
Suret
,
A.
Tikan
,
F.
Bonnefoy
,
F.
Copie
,
G.
Ducrozet
,
A.
Gelash
,
G.
Prabhudesai
,
G.
Michel
,
A.
Cazaubiel
,
E.
Falcon
,
G.
El
, and
S.
Randoux
, “
Nonlinear spectral synthesis of soliton gas in deep-water surface gravity waves
,”
Phys. Rev. Lett.
125
,
264101
(
2020
).
27
V. E.
Zakharov
and
A. B.
Shabat
, “
Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media
,”
Zh. Eksp. Teor. Fiz.
61
,
118
134
(
1971
). [Sov. Phys. JETP 34, 62069 (1972)].
28
A.
Hasegawa
and
Y.
Kodama
,
Solitons in Optical Communications
(
OxfordUniversity Press
,
1995
).
29
A. R.
Osborne
,
Nonlinear Ocean Waves
(
Academic Press
,
2009
).
30
L.
Pitaevskii
and
S.
Stringari
,
Bose-Einstein Condensation
(
Oxford University Press
,
2003
).
31
Y. S.
Kivshar
and
G. P.
Agrawal
,
Optical Solitons: From Fibers to Photonic Crystals
(
Academic Press
,
2013
).
32
B. A.
Malomed
,
D.
Mihalache
,
F.
Wise
, and
L.
Torner
, “
Spatiotemporal optical solitons
,”
J. Opt. B: Quantum Semiclass. Opt.
7
,
R53
(
2005
).
33
J.
Yang
,
Nonlinear Waves in Integrable and Nonintegrable Systems
(
SIAM
,
2010
).
34
B. A.
Malomed
,
Multidimensional Solitons
(
American Institute of Physics Publishers
,
2022
).
35
F.
Copie
,
S.
Randoux
, and
P.
Suret
, “
The physics of the one-dimensional nonlinear Schrödinger equation in fiber optics: Rogue waves, modulation instability and self-focusing phenomena
,”
Rev. Phys.
5
,
100037
(
2020
).
36
N. N.
Akhmediev
,
V. M.
Eleonskii
, and
N. E.
Kulagin
, “
Exact first-order solutions of the nonlinear Schrödinger equation
,”
Theor. Math. Phys.
72
,
809
818
(
1987
).
37
N.
Akhmediev
,
A.
Ankiewicz
, and
J. M.
Soto-Crespo
, “
Rogue waves and rational solutions of the Hirota equation
,”
Phys. Rev. E
80
,
026601
(
2009
).
38
E. A.
Kuznetsov
, “
Solitons in a parametrically unstable plasma
,”
Sov. Phys. Dokl.
22
,
507-508
(
1977
).
39
T.
Kawata
and
H.
Inoue
, “
Inverse scattering method for the nonlinear evolution equations under nonvanishing conditions
,”
J. Phys. Soc. Jpn.
44
,
1722
(
1978
).
40
Y.-C.
Ma
, “
The perturbed plane-wave solutions of the cubic Schrödinger equation
,”
Stud. Appl. Math.
60
,
43
(
1979
).
41
D. H.
Peregrine
, “
Water waves, nonlinear Schrödinger equations and their solutions
,”
J. Aust. Math. Soc. B
25
,
16
(
1983
).
42
A. A.
Gelash
and
V. E.
Zakharov
, “
Superregular solitonic solutions: A novel scenario for the nonlinear stage of modulation instability
,”
Nonlinearity
27
,
R1
(
2014
).
43
B.
Kibler
,
J.
Fatome
,
C.
Finot
,
G.
Millot
,
F.
Dias
,
G.
Genty
,
N.
Akhmediev
, and
J. M.
Dudley
, “
The Peregrine soliton in nonlinear fibre optics
,”
Nat. Phys.
6
,
790
(
2010
).
44
A.
Chabchoub
,
N. P.
Hoffmann
, and
N.
Akhmediev
, “
Rogue wave observation in a water wave tank
,”
Phys. Rev. Lett.
106
,
204502
(
2011
).
45
H.
Bailung
,
S. K.
Sharma
, and
Y.
Nakamura
, “
Observation of Peregrine solitons in a multicomponent plasma with negative ions
,”
Phys. Rev. Lett.
107
,
255005
(
2011
).
46
A.
Chabchoub
,
N.
Hoffmann
,
M.
Onorato
, and
N.
Akhmediev
, “
Super rogue waves: Observation of a higher-order breather in water waves
,”
Phys. Rev. X
2
,
011015
(
2012
).
47
A.
Chabchoub
,
N.
Hoffmann
,
M.
Onorato
,
A.
Slunyaev
,
A.
Sergeeva
,
E.
Pelinovsky
, and
N.
Akhmediev
, “
Observation of a hierarchy of up to fifth-order rogue waves in a water tank
,”
Phys. Rev. E
86
,
056601
(
2012
).
48
T. B.
Benjamin
and
J. E.
Feir
, “
The disintegration of wave trains on deep water part 1. Theory
,”
J. Fluid Mech.
27
,
417
(
1967
).
49
V. E.
Zakharov
and
A. A.
Gelash
, “
Nonlinear stage of modulation instability
,”
Phys. Rev. Lett.
111
,
054101
(
2013
).
50
M.
Närhi
,
B.
Wetzel
,
C.
Billet
,
S.
Toenger
,
T.
Sylvestre
,
J.-M.
Merolla
,
R.
Morandotti
,
F.
Dias
,
G.
Genty
, and
J. M.
Dudley
, “
Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability
,”
Nat. Commun.
7
,
13675
(
2016
).
51
D. R.
Solli
,
C.
Ropers
,
P.
Koonath
, and
B.
Jalali
, “
Optical rogue waves
,”
Nature
450
,
1054
1057
(
2007
).
52
N.
Akhmediev
,
A.
Ankiewicz
, and
M.
Taki
, “
Waves that appear from nowhere and disappear without a trace
,”
Phys. Lett. A
373
,
675
(
2009
).
53
N.
Akhmediev
,
J. M.
Soto-Crespo
, and
A.
Ankiewicz
, “
Extreme waves that appear from nowhere: On the nature of rogue waves
,”
Phys. Lett. A
373
,
2137
(
2009
).
54
F.
Fedele
, “
Rogue waves in oceanic turbulence
,”
Phys. D
237
,
2127
(
2008
).
55
K.
Dysthe
,
H. E.
Krogstad
, and
P.
Müller
, “
Oceanic rogue waves
,”
Annu. Rev. Fluid Mech.
40
,
287
(
2008
).
56
C.
Kharif
and
E.
Pelinovsky
, “
Physical mechanisms of the rogue wave phenomenon
,”
Eur. J. Mech. B. Fluids
22
,
603
(
2003
).
57
N.
Akhmediev
and
E.
Pelinovsky
, “
Editorial-introductory remarks on discussion & debate: Rogue waves-towards a unifying concept
?”
Eur. Phys. J. Spec. Top.
185
,
1
(
2010
).
58
M.
Onorato
,
S.
Residori
,
U.
Bortolozzo
,
A.
Montina
, and
F. T.
Arecchi
, “
Rogue waves and their generating mechanisms in different physical contexts
,”
Phys. Rep.
528
,
47
(
2013
).
59
B.
Kibler
,
J.
Fatome
,
C.
Finot
,
G.
Millot
,
F.
Dias
,
G.
Genty
,
N.
Akhmediev
, and
J. M.
Dudley
, “
The peregrine soliton in nonlinear fibre optics
,”
Nat. Phys.
6
,
790
(
2010
).
60
A. N.
Ganshin
,
V. B.
Efimov
,
G. V.
Kolmakov
,
L. P.
Mezhov-Deglin
, and
P. V. E.
McClintock
, “
Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium
,”
Phys. Rev. Lett.
101
,
065303
(
2008
).
61
W. M.
Moslem
, “
Langmuir rogue waves in electron-positron plasmas
,”
Phys. Plasmas
18
,
032301
(
2011
).
62
Y.
Bludov
,
V.
Konotop
, and
N.
Akhmediev
, “
Matter rogue waves
,”
Phys. Rev. A
80
,
33610
(
2009
).
63
Z.
Yan
,
V.
Konotop
, and
N.
Akhmediev
, “
Three-dimensional rogue waves in nonstationary parabolic potentials
,”
Phys. Rev. E
82
,
036610
(
2010
).
64
D.
Laveder
,
T. T.
Passot
,
P.
Sulem
, and
G.
Sánchez-Arriaga
, “
Rogue waves in Alfvénic turbulence
,”
Phys. Lett. A
375
,
3997
4002
(
2011
).
65
Z.
Yan
, “
Financial rogue waves
,”
Commun. Theor. Phys.
54
,
947
(
2010
).
66
J. M.
Dudley
,
F.
Dias
,
M.
Erkintalo
, and
G.
Genty
, “
Instabilities, breathers and rogue waves in optics
,”
Nat. Photonics
8
,
755
(
2014
).
67
A.
Armaroli
,
C.
Conti
, and
F.
Biancalana
, “
Rogue solitons in optical fibers: A dynamical process in a complex energy landscape
?”
Optica
2
,
497
(
2015
).
68
T. I.
Lakoba
, “
Effect of noise on extreme events probability in a one-dimensional nonlinear Schrödinger equation
,”
Phys. Lett. A
379
,
1821
(
2015
).
69
L.
Wang
and
Z.
Yan
, “
Rogue wave formation and interactions in the defocusing nonlinear Schrödinger equation with external potentials
,”
Appl. Math. Lett.
111
,
106670
(
2021
).
70
L.
Wang
and
Z.
Yan
, “
Data-driven rogue waves and parameter discovery in the defocusing nonlinear Schrödinger equation with a potential using the PINN deep learning
,”
Phys. Lett. A
404
,
127408
(
2021
).
71
Z.-Y.
Sun
and
X.
Yu
, “
Nearly integrable turbulence and rogue waves in disordered nonlinear Schrödinger systems
,”
Phys. Rev. E
103
,
062203
(
2021
).
72
M. V.
Flamarion
,
E.
Pelinovsky
, and
E.
Didenkulova
, “
Non-integrable soliton gas: The Schamel equation framework
,”
Chaos, Solitons, Fractals
180
,
114495
(
2024
).
73
N.
Laskin
, “
Fractional quantum mechanics
,”
Phys. Rev. E
62
,
3135
(
2000
).
74
S.
Longhi
, “
Fractional Schrödinger equation in optics
,”
Opt. Lett.
40
,
1117
1120
(
2015
).
75
B.
Guo
,
Y.
Han
, and
J.
Xin
, “
Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation
,”
Appl. Math. Comput.
204
,
468
477
(
2008
).
76
C.
Klein
,
C.
Sparber
, and
P.
Markowich
, “
Numerical study of fractional nonlinear Schrödinger equations
,”
Proc. R. Soc. A
470
,
20140364
(
2014
).
77
B. A.
Malomed
, “
Optical solitons and vortices in fractional media: A mini-review of recent results
,”
Photonics
8
,
353
(
2021
).
78
B. A.
Malomed
, “
Basic fractional nonlinear-wave models and solitons
,”
Chaos
34
,
022102
(
2024
).
79
Y.
Zhang
,
X.
Liu
,
M. R.
Belić
,
W.
Zhong
,
Y.
Zhang
, and
M.
Xiao
, “
Propagation dynamics of a light beam in a fractional Schrödinger equation
,”
Phys. Rev. Lett.
115
,
180403
(
2015
).
80
S.
Liu
,
Y.
Zhang
,
B. A.
Malomed
, and
E.
Karimi
, “
Experimental realizations of the fractional Schrödinger equation in the temporal domain
,”
Nature Comm.
14
,
222
(
2023
).
81
C.
Huang
and
L.
Dong
, “
Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice
,”
Opt. Lett.
41
,
5636
5639
(
2016
).
82
L.
Zhang
,
C.
Li
,
H.
Zhong
,
C.
Xu
,
D.
Lei
,
Y.
Li
, and
D.
Fan
, “
Propagation dynamics of super-Gaussian beams in fractional Schrödinger equation: From linear to nonlinear regimes
,”
Opt. Exp.
24
,
14406
14418
(
2016
).
83
X.
Yao
and
X.
Liu
, “
Off-site and on-site vortex solitons in space-fractional photonic lattices
,”
Opt. Lett.
43
,
5749
5752
(
2018
).
84
M.
Chen
,
S.
Zeng
,
D.
Lu
,
W.
Hu
, and
Q.
Guo
, “
Optical solitons, self-focusing, and wave collapse in a space-fractional Schrödinger equation with a Kerr-type nonlinearity
,”
Phys. Rev. E
98
,
022211
(
2018
).
85
J.
Xie
,
X.
Zhu
, and
Y.
He
, “
Vector solitons in nonlinear fractional Schrödinger equations with parity-time-symmetric optical lattices
,”
Nonlinear Dyn.
97
,
1287
(
2019
).
86
L.
Zeng
and
J.
Zeng
, “
Preventing critical collapse of higher-order solitons by tailoring unconventional optical diffraction and nonlinearities
,”
Commun. Phys.
3
,
26
(
2020
).
87
S.
Kumar
,
P.
Li
, and
B. A.
Malomed
, “
Domain walls in fractional media
,”
Phys. Rev. E
106
,
054207
(
2022
).
88
M.
Zhong
,
L.
Wang
,
P.
Li
, and
Z.
Yan
, “
Spontaneous symmetry breaking and ghost states supported by the fractional P T-symmetric saturable nonlinear Schrödinger equation
,”
Chaos
33
,
013106
(
2023
).
89
M.
Zhong
,
Y.
Chen
,
Z.
Yan
, and
B. A.
Malomed
, “
Suppression of soliton collapses, modulational instability and rogue-wave excitation in two-Lévy-index fractional Kerr media
,”
Proc. R. Soc. A
480
,
20230765
(
2024
).
90
O.
Ciaurri
,
L.
Roncal
,
P. R.
Stinga
,
J. L.
Torrea
, and
J. L.
Varona
, “
Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications
,”
Adv. Math.
330
,
688
738
(
2018
).
91
M. I.
Molina
, “
The fractional discrete nonlinear Schrödinger equation
,”
Phys. Lett. A
384
,
126180
(
2020
).
92
M.
Zhong
,
B. A.
Malomed
, and
Z.
Yan
, “
Dynamics of discrete solitons in the fractional discrete nonlinear Schrödinger equation with the quasi-Riesz derivative
,”
Phys. Rev. E
110
,
014215
(
2024
).
93
M.
Zhong
,
B. A.
Malomed
,
J.
Song
, and
Z.
Yan
, “
Two-dimensional fractional discrete NLS equations: Dispersion relations, rogue waves, fundamental and vortex solitons
,”
Stud. Appl. Math.
154
,
e70001
(
2024
).
94
L.
Zhang
,
Z.
He
,
C.
Conti
,
Z.
Wang
,
Y.
Hu
,
D.
Lei
,
Y.
Li
, and
D.
Fan
, “
Modulational instability in fractional nonlinear Schrödinger equation
,”
Commun. Nonlinear Sci. Numer. Simul.
48
,
531
540
(
2017
).
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