A ring resonator made of a silica-based optical fiber is a paradigmatic system for the generation of dissipative localized structures or dissipative solitons. We analyze the effect of the non-instantaneous nonlinear response of the fused silica or the Raman response on the formation of localized structures. After reducing the generalized Lugiato–Lefever to a simple and generic bistable model with a nonlocal Raman effect, we investigate analytically the formation of moving temporal localized structures. This reduction is valid close to the nascent bistability regime, where the system undergoes a second-order critical point marking the onset of a hysteresis loop. The interaction between fronts allows for the stabilization of temporal localized structures. Without the Raman effect, moving temporal localized structures do not exist, as shown in M. G. Clerc, S. Coulibaly, and M. Tlidi, Phys. Rev. Res. 2, 013024 (2020). The detailed derivation of the speed and the width associated with these structures is presented. We characterize numerically in detail the bifurcation structure and stability associated with the moving temporal localized states. The numerical results of the governing equations are in close agreement with analytical predictions.

1
L. A.
Lugiato
and
R.
Lefever
,
Phys. Rev. Lett.
58
,
2209
(
1987
).
2
A. J.
Scroggie
,
W. J.
Firth
,
G. S.
McDonald
,
M.
Tlidi
,
R.
Lefever
, and
L. A.
Lugiato
,
Chaos Solitons Fractals
4
,
1323
(
1994
).
3
F.
Leo
,
S.
Coen
,
P.
Kockaert
,
S. P.
Gorza
,
P.
Emplit
, and
M.
Haelterman
,
Nat. Photon.
4
,
471
(
2010
).
4
S.
Coen
,
M.
Tlidi
,
P.
Emplit
, and
M.
Haelterman
,
Phys. Rev. Lett.
83
,
2328
(
1999
).
5
V.
Odent
,
M.
Tlidi
,
M. G.
Clerc
,
P.
Glorieux
, and
E.
Louvergneaux
,
Phys. Rev. A
90
,
011806(R)
(
2014
).
6
Z.
Liu
,
M.
Ouali
,
S.
Coulibaly
,
M. G.
Clerc
,
M.
Taki
, and
M.
Tlidi
,
Opt. Lett.
42
,
1063
(
2017
).
7
K.
Panajotov
,
M. G.
Clerc
, and
M.
Tlidi
,
Eur. Phys. J. D
76
,
176
(
2017
).
8
N.
Akhmediev
et al.,
J. Opt.
18
,
063001
(
2016
).
9
S.
Coulibaly
,
M.
Taki
,
A.
Bendahmane
,
G.
Millot
,
B.
Kibler
, and
M. G.
Clerc
,
Phys. Rev. X
9
,
011054
(
2019
).
10
L. A.
Lugiato
,
F.
Prati
, and
M.
Brambilla
,
Nonlinear Optical Systems
(
Cambridge University Press
,
2015
).
11
Y. K.
Chembo
,
D.
Gomila
,
M.
Tlidi
, and
C. R.
Menyuk
, “
Theory and applications of the Lugiato-Lefever equation
,”
Eur. Phys. J. D
71
,
299
(
2017
).
12
M.
Tlidi
,
K.
Staliunas
,
K.
Panajotov
,
A. G.
Vladimirov
, and
M. G.
Clerc
,
Philos. Trans. R. Soc. A
372
,
20140101
(
2014
).
13
Nonlinear Dynamics: Materials, Theory and Experiments, edited by M. Tlidi and M. G. Clerc (Springer Proceedings in Physics, 2016), Vol. 173.
14
M. G.
Clerc
,
M. A.
Ferré
,
S.
Coulibaly
,
R. G.
Rojas
, and
M.
Tlidi
,
Opt. Lett.
42
,
2906
(
2017
).
15
T. W.
Hansch
,
Rev. Mod. Phys.
78
,
1297
(
2006
).
16
P.
Del’Haye
,
A.
Schliesser
,
O.
Arcizet
,
T.
Wilken
,
R.
Holzwarth
, and
T. J.
Kippenberg
,
Nature
450
,
1214
(
2007
).
17
T. J.
Kippenberg
,
R.
Holzwarth
, and
S. A.
Diddams
,
Science
332
,
555
(
2011
).
18
F.
Ferdous
,
H.
Miao
,
D. E.
Leaird
,
K.
Srinivasan
,
J.
Wang
,
L.
Chen
,
L. T.
Varghese
, and
A. M.
Weiner
,
Nat. Photon.
5
,
770
(
2011
).
19
Y. K.
Chembo
,
I. S.
Grudinin
, and
N.
Yu
,
Phys. Rev. A
92
,
043818
(
2015
).
20
M. G.
Clerc
,
S.
Coulibaly
, and
M.
Tlidi
,
Phys. Rev. Res.
2
,
013024
(
2020
).
21
M.
Tlidi
,
A. G.
Vladimirov
,
D.
Pieroux
, and
D.
Turaev
,
Phys. Rev. Lett.
103
,
103904
(
2009
).
22
K.
Panajotov
and
M.
Tlidi
,
Eur. Phys. J. D
59
,
67
(
2010
).
23
M.
Tlidi
,
K.
Panajotov
,
M.
Ferré
, and
M. G.
Clerc
,
Chaos
27
,
114312
(
2017
).
24
N.
Bloembergen
,
Nonlinear Optics
(
World Scientific
,
1996
).
25
G. P.
Agrawal
,
Nonlinear Fiber Optics
(
Springer
,
Berlin, Heidelberg
,
2000
).
26
L. D.
Landau
and
E. M.
Lifshitz
, Electrodynamics of Continuous Media, Course of Theoretical Physics Vol. 8 (Oxford, 1960).
29
F. M.
Atay
and
A.
Hutt
,
SIAM J. Appl. Dyn. Syst.
5
,
670
(
2006
).
30
J. D.
Murray
,
Mathematical Biology
(
Springer-Verlag
,
New York
,
2001
).
31
L. A.
Lugiato
,
F.
Prati
,
M. L.
Gorodetsky
, and
T. J.
Kippenberg
,
Philos. Trans. R. Soc. A
376
,
20180113
(
2018
).
32
M.
Tlidi
,
M.
Clerc
, and
K.
Panajotov
,
Philos. Trans. R. Soc. A (Lond.)
376
,
20180114
(
2018
).
33
K. J.
Blow
and
D.
Wood
,
IEEE J. Quantum Electron
25
,
2665
(
1989
).
34
Q.
Lin
and
G. P.
Agrawal
,
Opt. Lett.
31
,
3086
(
2006
).
35
M.
Tlidi
,
L.
Bahloul
,
L.
Cherbi
,
A.
Hariz
, and
S.
Coulibaly
,
Phys. Rev. A
88
,
035802
(
2013
).
36
P.
Parra-Rivas
,
D.
Gomila
, and
L.
Gelens
,
Phys. Rev. A
95
,
053863
(
2017
).
37
A. G.
Vladimirov
,
S. V.
Gurevich
, and
M.
Tlidi
,
Phys. Rev. A
97
,
013816
(
2018
).
38
A.
Hariz
,
L.
Bahloul
,
L.
Cherbi
,
K.
Panajotov
,
M.
Clerc
,
M. A.
Ferré
,
B.
Kostet
,
E.
Averlant
, and
M.
Tlidi
,
Phys. Rev. A
100
,
023816
(
2019
).
39
O.
Descalzi
,
J.
Cisternas
, and
H. R.
Brand
,
Phys. Rev. E
100
,
052218
(
2019
).
40
O.
Descalzi
and
H. R.
Brand
,
Chaos
30
,
043119
(
2020
).
41
J.
Knobloch
and
T.
Wagenknecht
,
Physica D
206
,
82
(
2005
).
42
A.
Yochelis
,
J.
Burke
, and
E.
Knobloch
,
Phys. Rev. Lett.
97
,
254501
(
2006
).
43
P.
Parra-Rivas
,
E.
Knobloch
,
D.
Gomila
, and
L.
Gelens
,
Phys. Rev. A
93
,
063839
(
2016
).
44
P.
Parra-Rivas
,
L.
Gelens
, and
F.
Leo
,
Phys. Rev. E
100
,
032219
(
2019
).
45
D.
Gomila
,
A. J.
Scroggie
, and
W. J.
Firth
,
Physica D
227
,
70
(
2007
).
46
M.
Tlidi
and
L.
Gelens
,
Opt. Lett.
35
,
270
(
2010
).
47
P.
Parra-Rivas
,
D.
Gomila
,
M. A.
Matías
,
S.
Coen
, and
L.
Gelens
,
Phys. Rev. A
89
,
043813
(
2014
).
48
P.
Parra-Rivas
,
D.
Gomila
,
L.
Gelens
, and
E.
Knobloch
,
Phys. Rev. E
97
,
042204
(
2018
).
49
M.
Tlidi
,
P.
Mandel
, and
R.
Lefever
,
Phys. Rev. Lett.
73
,
640
(
1994
).
50
P.
Parra-Rivas
,
D.
Gomila
,
E.
Knobloch
,
S.
Coen
, and
L.
Gelens
,
Opt. Lett.
41
,
2402
(
2016
).
51
E.
Doedel
,
H. B.
Keller
, and
J. P.
Kernevez
,
Int. J. Bifurc. Chaos
1
,
493
(
1991
).
52
E.
Doedel
,
H. B.
Keller
, and
J. P.
Kernevez
,
Int. J. Bifurc. Chaos
1
,
745
(
1991
).
53
P.
Coullet
,
C.
Elphick
, and
D.
Repaux
,
Phys. Rev. Lett.
58
,
431
(
1987
).
54
P.
Coullet
,
Int. J. Bifurc. Chaos
12
,
2445
(
2002
).
55
M. G.
Clerc
,
D.
Escaff
, and
V. M.
Kenkre
,
Phys. Rev. E
72
,
056217
(
2005
).
56
M. G.
Clerc
,
D.
Escaff
, and
V. M.
Kenkre
,
Phys. Rev. E
82
,
036210
(
2010
).
57
M. G.
Clerc
,
C.
Falcon
, and
E.
Tirapegui
,
Phys. Rev. Lett.
94
,
148302
(
2005
).
You do not currently have access to this content.