In this paper, we introduce a class of novel --hyperbolic-function potentials composed of the Dirac and hyperbolic functions, supporting fully real energy spectra in the non-Hermitian Hamiltonian. The threshold curves of symmetry breaking are numerically presented. Moreover, in the self-focusing and defocusing Kerr-nonlinear media, the -symmetric potentials can also support the stable peakons, keeping the total power and quasi-power conserved. The unstable -symmetric peakons can be transformed into other stable peakons by the excitations of potential parameters. Continuous families of additional stable numerical peakons can be produced in internal modes around the exact peakons (even unstable). Further, we find that the stable peakons can always propagate in a robust form, remaining trapped in the slowly moving potential wells, which opens the way for manipulations of optical peakons. Other significant characteristics related to exact peakons, such as the interaction and power flow, are elucidated in detail. These results will be useful in explaining the related physical phenomena and designing the related physical experiments.
REFERENCES
1.
A.
Hasegawa
and Y.
Kodama
, Solitons in Optical Communications
(Oxford University Press
, Oxford
, 1995
).2.
N.
Akhmediev
and A.
Ankiewicz
, Solitons: Nonlinear Pulses and Beams
(Chapman and Hall
, London
, 1997
).3.
Y. S.
Kivshar
and G.
Agrawal
, Optical Solitons: From Fibers to Photonic Crystals
(Academic Press
, San Diego, CA
, 2003
).4.
5.
L.
Pitaevskii
and S.
Stringari
, Bose-Einstein Condensation
(Oxford University Press
, Oxford
, 2003
).6.
C. M.
Bender
and S.
Boettcher
, Phys. Rev. Lett.
80
, 5243
(1998
). 7.
Z.
Musslimani
, K. G.
Makris
, R.
El-Ganainy
, and D. N.
Christodoulides
, Phys. Rev. Lett.
100
, 030402
(2008
). 8.
B.
Bagchi
and R.
Roychoudhury
, J. Phys. A
33
, L1
(2000
). 9.
B.
Bagchi
, C.
Quesne
, and M.
Znojil
, Mod. Phys. Lett. A
16
, 2047
(2001
). 10.
Z.
Ahmed
, Phys. Lett. A
282
, 343
(2001
). 11.
B.
Bagchi
and C.
Quesne
, Phys. Lett. A
301
, 173
(2002
). 12.
Z.
Shi
, X.
Jiang
, X.
Zhu
, and H.
Li
, Phys. Rev. A
84
, 053855
(2011
). 13.
S.
Nixon
, L.
Ge
, and J.
Yang
, Phys. Rev. A
85
, 023822
(2012
). 14.
V.
Achilleos
, P.
Kevrekidis
, D.
Frantzeskakis
, and R.
Carretero-González
, Phys. Rev. A
86
, 013808
(2012
). 15.
Z.
Yan
, Philos. Trans. R. Soc. A
371
, 20120059
(2013
). 16.
Y.
Lumer
, Y.
Plotnik
, M. C.
Rechtsman
, and M.
Segev
, Phys. Rev. Lett.
111
, 263901
(2013
). 17.
Z.
Yan
, Z.
Wen
, and C.
Hang
, Phys. Rev. E
92
, 022913
(2015
). 18.
Z.-C.
Wen
and Z.
Yan
, Phys. Lett. A
379
, 2025
(2015
). 19.
Z.
Yan
, Z.
Wen
, and V. V.
Konotop
, Phys. Rev. A
92
, 023821
(2015
). 20.
A.
Guo
, G.
Salamo
, D.
Duchesne
, R.
Morandotti
, M.
Volatier-Ravat
, V.
Aimez
, G.
Siviloglou
, and D.
Christodoulides
, Phys. Rev. Lett.
103
, 093902
(2009
). 21.
C. E.
Rüter
, K. G.
Makris
, R.
El-Ganainy
, D. N.
Christodoulides
, M.
Segev
, and D.
Kip
, Nat. Phys.
6
, 192
(2010
). 22.
A.
Regensburger
, C.
Bersch
, M.-A.
Miri
, G.
Onishchukov
, D. N.
Christodoulides
, and U.
Peschel
, Nature
488
, 167
(2012
). 23.
G.
Castaldi
, S.
Savoia
, V.
Galdi
, A.
Alù
, and N.
Engheta
, Phys. Rev. Lett.
110
, 173901
(2013
). 24.
A.
Regensburger
, M.-A.
Miri
, C.
Bersch
, J.
Näger
, G.
Onishchukov
, D. N.
Christodoulides
, and U.
Peschel
, Phys. Rev. Lett.
110
, 223902
(2013
). 25.
B.
Peng
, Ş. K.
Özdemir
, F.
Lei
, F.
Monifi
, M.
Gianfreda
, G. L.
Long
, S.
Fan
, F.
Nori
, C. M.
Bender
, and L.
Yang
, Nat. Phys.
10
, 394
(2014
). 26.
A. A.
Zyablovsky
, A. P.
Vinogradov
, A. A.
Pukhov
, A. V.
Dorofeenko
, and A. A.
Lisyansky
, Phys. Usp.
57
, 1063
(2014
). 27.
P.-Y.
Chen
and J.
Jung
, Phys. Rev. Appl.
5
, 064018
(2016
). 28.
K.
Takata
and M.
Notomi
, Phys. Rev. Appl.
7
, 054023
(2017
). 29.
H.
Hodaei
, A. U.
Hassan
, S.
Wittek
, H.
Garcia-Gracia
, R.
El-Ganainy
, D. N.
Christodoulides
, and M.
Khajavikhan
, Nature
548
, 187
(2017
). 30.
B.
Liu
, H.-F.
Zhang
, R.-X.
Zhong
, X.-L.
Zhang
, X.-Z.
Qin
, C.
Huang
, Y.-Y.
Li
, and B. A.
Malomed
, Phys. Rev. A
99
, 053602
(2019
). 31.
E. A.
Ultanir
, G. I.
Stegeman
, and D. N.
Christodoulides
, Opt. Lett.
29
, 845
(2004
). 32.
K. G.
Makris
, R.
El-Ganainy
, D. N.
Christodoulides
, and Z. H.
Musslimani
, Phys. Rev. Lett.
100
, 103904
(2008
). 33.
C.-Q.
Dai
, X.-G.
Wang
, and G.-Q.
Zhou
, Phys. Rev. A
89
, 013834
(2014
). 34.
S.
Hu
, X.
Ma
, D.
Lu
, Z.
Yang
, Y.
Zheng
, and W.
Hu
, Phys. Rev. A
84
, 043818
(2011
). 35.
J.
Yang
, Opt. Lett.
39
, 5547
(2014
). 36.
D. A.
Zezyulin
and V. V.
Konotop
, Phys. Rev. A
85
, 043840
(2012
). 37.
B.
Midya
and R.
Roychoudhury
, Phys. Rev. A
87
, 045803
(2013
). 38.
H.
Cartarius
and G.
Wunner
, Phys. Rev. A
86
, 013612
(2012
). 39.
F.
Single
, H.
Cartarius
, G.
Wunner
, and J.
Main
, Phys. Rev. A
90
, 042123
(2014
). 40.
C. P.
Jisha
, L.
Devassy
, A.
Alberucci
, and V.
Kuriakose
, Phys. Rev. A
90
, 043855
(2014
). 41.
F. K.
Abdullaev
, Y. V.
Kartashov
, V. V.
Konotop
, and D. A.
Zezyulin
, Phys. Rev. A
83
, 041805
(2011
). 42.
N.
Moiseyev
, Phys. Rev. A
83
, 052125
(2011
). 43.
C. P.
Jisha
, A.
Alberucci
, V. A.
Brazhnyi
, and G.
Assanto
, Phys. Rev. A
89
, 013812
(2014
). 44.
H.
Wang
and D.
Christodoulides
, Commun. Nonlinear Sci. Numer. Simul.
38
, 130
(2016
). 45.
S. V.
Suchkov
, A. A.
Sukhorukov
, J.
Huang
, S. V.
Dmitriev
, C.
Lee
, and Y. S.
Kivshar
, Laser Photonics Rev.
10
, 177
(2016
). 46.
G.
Burlak
and B. A.
Malomed
, Phys. Rev. E
88
, 062904
(2013
). 47.
Y. V.
Bludov
, V. V.
Konotop
, and B. A.
Malomed
, Phys. Rev. A
87
, 013816
(2013
). 48.
R.
Fortanier
, D.
Dast
, D.
Haag
, H.
Cartarius
, J.
Main
, G.
Wunner
, and R.
Gutöhrlein
, Phys. Rev. A
89
, 063608
(2014
). 49.
D.
Dizdarevic
, D.
Dast
, D.
Haag
, J.
Main
, H.
Cartarius
, and G.
Wunner
, Phys. Rev. A
91
, 033636
(2015
). 50.
C.-Q.
Dai
, X.-F.
Zhang
, Y.
Fan
, and L.
Chen
, Commun. Nonlinear Sci. Numer. Simul.
43
, 239
(2017
). 51.
52.
53.
54.
55.
Z.
Yan
, Y.
Chen
, and Z.
Wen
, Chaos
26
, 083109
(2016
). 56.
Y.
Chen
, Z.
Yan
, and X.
Li
, Commun. Nonlinear Sci. Numer. Simul.
55
, 287
(2018
). 57.
Y.
Chen
and Z.
Yan
, Sci. Rep.
6
, 23478
(2016
). 58.
Y.
Chen
, Z.
Yan
, D.
Mihalache
, and B. A.
Malomed
, Sci. Rep.
7
, 1257
(2017
). 59.
Y.
Chen
and Z.
Yan
, Phys. Rev. E
95
, 012205
(2017
). 60.
Z.
Wen
and Z.
Yan
, Chaos
27
, 053105
(2017
). 61.
Z.
Yan
and Y.
Chen
, Chaos
27
, 073114
(2017
). 62.
Y.
Shen
, Z.
Wen
, Z.
Yan
, and C.
Hang
, Chaos
28
, 043104
(2018
). 63.
Y.
Chen
, Z.
Yan
, and W.
Liu
, Opt. Express
26
, 33022
(2018
). 64.
Y.
Chen
, Z.
Yan
, and D.
Mihalache
, Chaos
29
, 083108
(2019
). 65.
L.
Wang
, B. A.
Malomed
, and Z.
Yan
, Phys. Rev. E
99
, 052206
(2019
). 66.
B. A.
Malomed
, Physica D
399
, 108
(2019
). 67.
V. V.
Konotop
, J.
Yang
, and D. A.
Zezyulin
, Rev. Mod. Phys.
88
, 035002
(2016
). 68.
Y.
He
, B. A.
Malomed
, and D.
Mihalache
, Philos. Trans. R. Soc. A
372
, 20140017
(2014
). 69.
70.
R.
El-Ganainy
, K. G.
Makris
, M.
Khajavikhan
, Z. H.
Musslimani
, S.
Rotter
, and D. N.
Christodoulides
, Nat. Phys.
14
, 11
(2018
). 71.
D.
Christodoulides
and J.
Yang
, Parity-Time Symmetry and Its Applications
(Springer
, 2018
), Vol. 280.72.
73.
74.
M.
Zhong
, Y.
Chen
, Z.
Yan
, and S.-F.
Tian
, Phys. Rev. E
105
, 014204
(2022
). 75.
T.
Cazenave
, Semilinear Schrodinger Equations
(American Mathematical Society
, 2003
), Vol. 10.76.
J.
Yang
and T. I.
Lakoba
, Stud. Appl. Math.
118
, 153
(2007
). 77.
J.
Yang
, Nonlinear Waves in Integrable and Nonintegrable Systems
(SIAM
, 2010
).78.
H.
Cartarius
, D.
Haag
, D.
Dast
, and G.
Wunner
, J. Phys. A
45
, 444008
(2012
). 79.
A.
Rodrigues
, K.
Li
, V.
Achilleos
, P.
Kevrekidis
, D.
Frantzeskakis
, and C. M.
Bender
, Rom. Rep. Phys.
65
, 5
(2013
).80.
H.
Susanto
, R.
Kusdiantara
, N.
Li
, O.
Kirikchi
, D.
Adzkiya
, E.
Putri
, and T.
Asfihani
, Phys. Rev. E
97
, 062204
(2018
). © 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
You do not currently have access to this content.
