In this paper, we analyze the large-space and large-time asymptotic properties of the vector rogon-soliton and soliton-like solutions of the n-component nonlinear Schrödinger equation with mixed nonzero and zero boundary conditions. In particular, we find that these solutions have different decay velocities along different directions of the axis, that is, the solutions exponentially and algebraically decay along the positive and negative directions of the axis, respectively. Moreover, we study the change of the acceleration of soliton moving with the increase in time or distance along the characteristic line (i.e., soliton moving trajectory). As a result, we find that the product of the acceleration and distance square tends to some constant value as time increases. These results will be useful to better understand the related multi-wave phenomena and to design physical experiments.
Topics
Soliton solutions
REFERENCES
1.
A.
Osborne
, Nonlinear Ocean Waves and the Inverse Scattering Transform
(Elsevier
, New York
, 2010
).2.
C.
Kharif
, E.
Pelinovsky
, and A.
Slunyaev
, Rogue Waves in the Ocean
(Springer
, New York
, 2009
).3.
L.
Draper
, “Freak ocean waves
,” Weather
21
, 2
–4
(1966
).4.
K.
Dysthe
, H. E.
Krogstad
, and P.
Müller
, “Oceanic rogue waves
,” Annu. Rev. Fluid Mech.
40
, 287
–310
(2008
). 5.
D. R.
Solli
, C.
Ropers
, P.
Koonath
, and B.
Jalali
, “Optical rogue waves
,” Nature (London)
450
, 1054
(2007
). 6.
Y. V.
Bludov
, V. V.
Konotop
, and N.
Akhmediev
, “Matter rogue waves
,” Phys. Rev. A
80
, 033610
(2009
). 7.
Z.
Yan
, V. V.
Konotop
, and N.
Akhmediev
, “Three-dimensional rogue waves in nonstationary parabolic potentials
,” Phys. Rev. E
82
, 036610
(2010
).8.
Z.
Yan
, “Financial rogue waves
,” Commun. Theor. Phys.
54
, 947
(2010
). 9.
Z.
Yan
, “Vector financial rogue waves
,” Phys. Lett. A
375
, 4274
(2011
). 10.
M.
Shats
, H.
Punzmann
, and H.
Xia
, “Capillary rogue waves
,” Phys. Rev. Lett.
104
, 104503
(2010
). 11.
L.
Stenflo
and M.
Marklund
, “Rogue waves in the atmosphere
,” J. Plasma Phys.
76
, 293
(2010
). 12.
A. N.
Ganshin
, V. B.
Efimov
, G. V.
Kolmakov
, L. P.
Mezhov-Deglin
, and P. V.
McClintock
, “Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium
,” Phys. Rev. Lett.
101
, 065303
(2008
). 13.
S.
Toenger
, T.
Godin
, C.
Billet
, F.
Dias
, M.
Erkintalo
, G.
Genty
, and J. M.
Dudley
, “Emergent rogue wave structures and statistics in spontaneous modulation instability
,” Sci. Rep.
5
, 1
(2015
). 14.
J.
Kasparian
, P.
Béjot
, J. P.
Wolf
, and J. M.
Dudley
, “Optical rogue wave statistics in laser filamentation
,” Opt. Express
17
, 12070
(2009
). 15.
W. M.
Moslem
, P. K.
Shukla
, and B.
Eliasson
, “Surface plasma rogue waves
,” Europhys. Lett.
96
, 25002
(2011
). 16.
D.
Benney
and A. C.
Newell
, “The propagation of nonlinear wave envelopes
,” J. Math. Phys.
46
, 133
(1967
). 17.
L. A.
Ostrovskii
, “Propagation of wave packets and space-time self-focusing in a nonlinear medium
,” Sov. Phys. JETP
24
, 797
–800
(1967
).18.
19.
H. C.
Yuen
and B. M.
Lake
, “Nonlinear dynamics of deep-water gravity waves
,” Adv. Appl. Mech.
22
, 67
–229
(1982
).20.
N.
Akhmediev
, A.
Ankiewicz
, and M.
Taki
, “Waves that appear from nowhere and disappear without a trace
,” Phys. Lett. A
373
, 675
(2009
). 21.
N.
Akhmediev
, A.
Ankiewicz
, and J. M.
Soto-Crespo
, “Rogue waves and rational solutions of the nonlinear Schrödinger equation
,” Phys. Rev. E
80
, 026601
(2009
). 22.
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
). 23.
B.
Guo
, L.
Ling
, and Q.
Liu
, “Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions
,” Phys. Rev. E
85
, 026607
(2012
). 24.
Z.
Yan
, “Nonautonomous ‘rogons’ in the inhomogeneous nonlinear Schrödinger equation with variable coefficients
,” Phys. Lett. A
374
, 672
–679
(2010
). 25.
J.
He
, H.
Zhang
, L.
Wang
, K.
Porsezian
, and A.
Fokas
, “Generating mechanism for higher-order rogue waves
,” Phys. Rev. E
87
, 052914
(2013
). 26.
X. Y.
Wen
, Y.
Yang
, and Z.
Yan
, “Generalized perturbation (n, M)-fold Darboux transformations and multi-rogue-wave structures for the modified self-steepening nonlinear Schrödinger equation
,” Phys. Rev. E
92
, 012917
(2015
). 27.
B.
Yang
and J.-K.
Yang
, “Universal patterns of rogue waves
,” Physica D
419
, 132850
(2021
). 28.
D.
Peregrine
, “Water waves, nonlinear Schrödinger equations and their solutions
,” J. Aust. Math. Soc. B, Appl. Math.
25
, 16
(1983
). 29.
A.
Ankiewicz
, D. J.
Kedziora
, and N.
Akhmediev
, “Rogue wave triplets
,” Phys. Lett. A
375
, 2782
(2011
). 30.
D. J.
Kedziora
, A.
Ankiewicz
, and N.
Akhmediev
, “Circular rogue wave clusters
,” Phys. ReV. E
84
, 056611
(2011
). 31.
D. J.
Kedziora
, A.
Ankiewicz
, and N.
Akhmediev
, “Triangular rogue wave cascades
,” Phys. Rev. E
86
, 056602
(2012
). 32.
D. J.
Kedziora
, A.
Ankiewicz
, and N.
Akhmediev
, “Classifying the hierarchy of nonlinear-Schrödinger-equation rogue-wave solutions
,” Phys. Rev. E
88
, 013207
(2013
). 33.
A.
Ankiewicz
and N.
Akhmediev
, “Multi-rogue waves and triangular numbers
,” Rom. Rep. Phys.
69
, 104
(2017
).34.
L.
Ling
, L. C.
Zhao
, Z. Y.
Yang
et al., “Generation mechanisms of fundamental rogue wave spatial-temporal structure
,” Phys. Rev. E
96
, 022211
(2017
). 35.
L. C.
Zhao
and J.
Liu
, “Rogue-wave solutions of a three-component coupled nonlinear Schrödinger equation
,” Phys. Rev. E
87
, 013201
(2013
). 36.
L.
Ling
, B.
Guo
, and L. C.
Zhao
, “High-order rogue waves in vector nonlinear Schrödinger equations
,” Phys. Rev. E
89
, 041201
(2014
). 37.
L.
Ling
, L. C.
Zhao
, and B.
Guo
, “Darboux transformation and classification of solution for mixed coupled nonlinear Schrödinger equations
,” Commun. Nonlinear Sci. Numer. Simul.
32
, 285
(2016
). 38.
L. C.
Zhao
, B.
Guo
, and L.
Ling
, “High-order rogue wave solutions for the coupled nonlinear Schrödinger equations-II
,” J. Math. Phys.
57
, 043508
(2016
). 39.
G.
Zhang
, Z.
Yan
, and X. Y.
Wen
, “Three-wave resonant interactions: Multi-dark-dark-dark solitons, breathers, rogue waves, and their interactions and dynamics
,” Physica D
366
, 27
(2018
). 40.
G.
Zhang
and Z.
Yan
, “Three-component nonlinear Schrödinger equations: Modulational instability, Nth-order vector rational and semi-rational rogue waves, and dynamics
,” Commun. Nonlinear Sci. Numer. Simul.
62
, 117
(2018
). 41.
D.
Bilman
and P. D.
Miller
, “A robust inverse scattering transform for the focusing nonlinear Schrödinger equation
,” Comm. Pure Appl. Math.
72
, 1722
(2019
). 42.
D.
Bilman
and R.
Buckingham
, “Large-order asymptotics for multiple-pole solitons of the focusing nonlinear Schrödinger equation
,” J. Nonlinear Sci.
29
, 2185
(2019
). 43.
D.
Bilman
, L.
Ling
, and P. D.
Miller
, “Extreme superposition: Rogue waves of infinite order and the Painlevé-III hierarchy
,” Duke Math. J.
169
, 671
(2020
). 44.
D.
Bilman
and T.
Trogdon
, “On numerical inverse scattering for the Korteweg-de Vries equation with discontinuous step-like data
,” Nonlinearity
33
, 2211
(2020
). 45.
D.
Bilman
, R.
Buckingham
, and D. S.
Wang
, “Far-field asymptotics for multiple-pole solitons in the large-order limit
,” J. Differ. Equ.
297
, 320
(2021
). 46.
M. J.
Ablowitz
, B.
Prinari
, and A.
Trubatch
, Discrete and Continuous Nonlinear Schrödinger Systems
(Cambridge University Press
, Cambridge
, 2004
).47.
G.
Zhang
, L.
Ling
, Z.
Yan
, and V. V.
Konotop
, “Parity-time-symmetric rational vector rogue waves of the n-component nonlinear Schrödinger equation
,” Chaos
31
, 063120
(2021
). 48.
G.
Zhang
, L.
Ling
, and Z.
Yan
, “Multi-component nonlinear Schrödinger equations with nonzero boundary conditions: Higher-order vector Peregrine solitons and asymptotic estimates
,” J. Nonlinear Sci.
31
, 81
(2021
). 49.
W.
Weng
, G.
Zhang
, L.
Wang
, M.
Zhang
, and Z.
Yan
, “Rational vector rogue waves for the
-component Hirota equation with non-zero backgrounds
,” Physica D
427
, 133005
(2021
). 50.
W.
Weng
, G.
Zhang
, M.
Zhang
, Z.
Zhou
, and Z.
Yan
, “Semi-rational vector rogon-soliton solutions and asymptotic analysis for any n-component nonlinear Schrödinger equation with mixed boundary conditions
,” Physica D
432
, 133150
(2022
). © 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
Author(s)
You do not currently have access to this content.
