We investigate the properties of time-dependent dissipative solitons for a cubic complex Ginzburg–Landau equation stabilized by nonlinear gradient terms. The separation of initially nearby trajectories in the asymptotic limit is predominantly used to distinguish qualitatively between time-periodic behavior and chaotic localized states. These results are further corroborated by Fourier transforms and time series. Quasiperiodic behavior is obtained as well, but typically over a fairly narrow range of parameter values. For illustration, two examples of nonlinear gradient terms are examined: the Raman term and combinations of the Raman term with dispersion of the nonlinear gain. For small quintic perturbations, it turns out that the chaotic localized states are showing a transition to periodic states, stationary states, or collapse already for a small magnitude of the quintic perturbations. This result indicates that the basin of attraction for chaotic localized states is rather shallow.

1.
N. J.
Zabusky
and
M. D.
Kruskal
,
Phys. Rev. Lett.
18
,
240
(
1965
).
2.
A. C.
Newell
,
Solitons in Mathematics and Physics
(
Society for Industrial and Applied Mathematics
,
Philadelphia, PA
,
1985
).
3.
M. C.
Cross
and
P. C.
Hohenberg
,
Rev. Mod. Phys.
65
,
851
(
1993
).
4.
Dissipative Solitons: From Optics to Biology and Medicine, edited by N. Akhmediev and A. Ankiewicz (Springer, Heidelberg, 2008).
5.
N.
Akhmediev
,
J. M.
Soto-Crespo
, and
H. R.
Brand
,
Phys. Lett. A
377
,
968
(
2013
).
6.
R.
Heinrichs
,
G.
Ahlers
, and
D. S.
Cannell
,
Phys. Rev. A
35
,
2761
(
1987
).
7.
E.
Moses
,
J.
Fineberg
, and
V.
Steinberg
,
Phys. Rev. A
35
,
2757
(
1987
).
8.
P.
Kolodner
,
D.
Bensimon
, and
C. M.
Surko
,
Phys. Rev. Lett.
60
,
1723
(
1988
).
9.
J. J.
Niemela
,
G.
Ahlers
, and
D. S.
Cannell
,
Phys. Rev. Lett.
64
,
1365
(
1990
).
10.
P.
Kolodner
,
Phys. Rev. A
44
,
6448
(
1991
).
11.
P.
Kolodner
,
Phys. Rev. A
44
,
6466
(
1991
).
12.
H. H.
Rotermund
,
S.
Jakubith
,
A.
von Oertzen
, and
G.
Ertl
,
Phys. Rev. Lett.
66
,
3083
(
1991
).
13.
A.
von Oertzen
,
A. S.
Mikhailov
,
H. H.
Rotermund
, and
G.
Ertl
,
J. Phys. Chem. B
102
,
4966
(
1998
).
14.
H.
Kuwayama
and
S.
Ishida
,
Sci. Rep.
3
,
2272
(
2013
).
15.
P. R.
Bauer
,
A.
Bonnefont
, and
K.
Krischer
,
Sci. Rep.
5
,
16312
(
2015
).
16.
V. B.
Taranenko
,
K.
Staliunas
, and
C. O.
Weiss
,
Phys. Rev. A
56
,
1582
(
1997
).
17.
G.
Slekys
,
K.
Staliunas
, and
C. O.
Weiss
,
Opt. Commun.
149
,
113
(
1998
).
18.
E. A.
Ultanir
,
G. I.
Stegeman
,
D.
Michaelis
,
C. H.
Lange
, and
F.
Lederer
,
Phys. Rev. Lett.
90
,
253903
(
2003
).
19.
A. F. J.
Runge
,
N. G. R.
Broderick
, and
M.
Erkintalo
,
Optica
2
,
36
(
2015
).
20.
J. S.
Peng
,
M.
Sorokina
,
S.
Sugavanam
,
N.
Tarasov
,
D. V.
Churkin
,
S. K.
Turitsyn
, and
H. P.
Zeng
,
Commun. Phys.
1
,
20
(
2018
).
21.
J.
Peng
and
H.
Zeng
,
Commun. Phys.
2
,
34
(
2019
).
22.
O.
Thual
and
S.
Fauve
,
J. Phys. France
49
,
1829
(
1988
).
23.
H. R.
Brand
and
R. J.
Deissler
,
Phys. Rev. Lett.
63
,
2801
(
1989
).
24.
R. J.
Deissler
and
H. R.
Brand
,
Phys. Rev. A
44
,
R3411
(
1991
).
25.
H. R.
Brand
,
P. S.
Lomdahl
, and
A. C.
Newell
,
Phys. Lett. A
118
,
67
(
1986
).
26.
H. R.
Brand
,
P. S.
Lomdahl
, and
A. C.
Newell
,
Physica D
23
,
345
(
1986
).
27.
R. J.
Deissler
and
H. R.
Brand
,
Phys. Lett. A
146
,
252
(
1990
).
28.
R. J.
Deissler
and
H. R.
Brand
,
Phys. Rev. Lett.
81
,
3856
(
1998
).
29.
H. P.
Tian
,
Z. H.
Li
,
J. P.
Tian
,
G. S.
Zhou
, and
J.
Zi
,
Appl. Phys. B
78
,
199
(
2004
).
30.
R. J.
Deissler
and
H. R.
Brand
,
Phys. Rev. Lett.
72
,
478
(
1994
).
31.
J. M.
Soto-Crespo
,
N.
Akhmediev
, and
A.
Ankiewicz
,
Phys. Rev. Lett.
85
,
2937
(
2000
).
32.
N.
Akhmediev
,
J. M.
Soto-Crespo
, and
G.
Town
,
Phys. Rev. E
63
,
056602
(
2001
).
33.
S. T.
Cundiff
,
J. M.
Soto-Crespo
, and
N.
Akhmediev
,
Phys. Rev. Lett.
88
,
073903
(
2002
).
34.
M.
Facão
and
M. I.
Carvalho
,
Phys. Rev. E
92
,
022922
(
2015
).
35.
M.
Facão
and
M. I.
Carvalho
,
Phys. Rev. E
96
,
042220
(
2017
).
36.
M. I.
Carvalho
and
M.
Facão
,
Phys. Rev. E
100
,
032222
(
2019
).
37.
O.
Descalzi
,
J.
Cisternas
, and
H. R.
Brand
,
Phys. Rev. E
100
,
052218
(
2019
).
38.
O.
Descalzi
and
H. R.
Brand
,
Chaos
30
,
043119
(
2020
).
39.
O.
Descalzi
,
C.
Cartes
, and
H. R.
Brand
,
Phys. Rev. E
103
,
L050201
(
2021
).
40.
O.
Descalzi
,
C.
Cartes
, and
H. R.
Brand
,
Phys. Rev. E
103
,
042215
(
2021
).
41.
O.
Descalzi
,
C.
Cartes
, and
H. R.
Brand
,
Phys. Rev. E
105
,
L062201
(
2022
).
42.
O.
Descalzi
,
M. I.
Carvalho
,
M.
Facão
, and
H. R.
Brand
,
Chaos
32
,
123107
(
2022
).
43.
O.
Descalzi
and
C.
Cartes
,
Chaos, Solitons Fractals
164
,
112703
(
2022
).
44.
R. J.
Deissler
,
Physica D
25
,
233
(
1987
).
45.
C.
Cartes
,
O.
Descalzi
, and
H. R.
Brand
,
Phys. Rev. E
85
,
015205
(
2012
).
46.
H. A.
Haus
,
E. P.
Ippen
, and
K.
Tamura
,
IEEE J. Quantum Electron.
30
,
200
(
1994
).
47.
C.-J.
Chen
,
P. K. A.
Wai
, and
C. R.
Menyuk
,
Opt. Lett.
20
,
350
(
1995
).
48.
B.
Fornberg
and
G. B.
Whitham
,
Philos. Trans. R. Soc. London, Ser. A
289
,
373
(
1978
).
49.
T.
Brabec
and
F.
Krausz
,
Phys. Rev. Lett.
78
,
3282
(
1997
).
50.
G. P.
Agrawal
,
Nonlinear Fiber Optics
(
Academic Press
,
Oxford
,
2013
).
You do not currently have access to this content.