Exploring new nonlinear wave solutions to integrable systems has always been an open issue in physics, applied mathematics, and engineering. In this paper, the Maccari system, a two-dimensional analog of nonlinear Schrödinger equation, is investigated. The system is derived from the Kadomtsev–Petviashvili (KP) equation and is widely used in nonlinear optics, plasma physics, and water waves. A large family of semi-rational solutions of the Maccari system are proposed with the KP hierarchy reduction method and Hirota bilinear method. These semi-rational solutions reduce to the breathers of elastic collision and resonant collision under special parameters. In case of resonant collisions between breathers and rational waves, these semi-rational solutions describe lumps fusion into breathers, or lumps fission from breathers, or a mixture of these fusion and fission. The resonant collisions of semi-rational solutions are semi-localized in time (i.e., lumps exist only when t → +∞ or t → −∞), and we also discuss their dynamics and asymptotic behaviors.

1.
N. N.
Akhmediev
,
A.
Ankiewicz
, and
J. M.
Soto-Crespo
, “
Multisoliton solutions of the complex Ginzburg-Landau equation
,”
Phys. Rev. Lett.
79
,
4047
4051
(
1997
).
2.
M.
Tajiri
and
Y.
Watanabe
, “
Breather solutions to the focusing nonlinear Schrödinger equation
,”
Phys. Rev. E
57
,
3510
3519
(
1998
).
3.
T.
Kanna
and
M.
Lakshmanan
, “
Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations
,”
Phys. Rev. Lett.
86
,
5043
5046
(
2001
).
4.
L.
Khaykovich
,
F.
Schreck
,
G.
Ferrari
,
T.
Bourdel
,
J.
Cubizolles
,
L. D.
Carr
,
Y.
Castin
, and
C.
Salomon
, “
Formation of a matter-wave bright soliton
,”
Science
296
,
1290
1293
(
2002
).
5.
Z.
Yan
, “
Periodic, solitary and rational wave solutions of the 3D extended quantum Zakharov–Kuznetsov equation in dense quantum plasmas
,”
J. Nonlinear Sci.
373
,
2432
2437
(
2009
).
6.
H.
Zhao
and
Z.
Zhu
, “
Multisoliton, multipositon, multinegaton, and multiperiodic solutions of a coupled Volterra lattice system and their continuous limits
,”
J. Math. Phys.
52
,
023512
(
2011
).
7.
X.
Geng
,
R.
Li
, and
B.
Xue
, “
A vector general nonlinear Schrödinger equation with (m + n) components
,”
J. Nonlinear Sci.
30
,
991
1013
(
2019
).
8.
B. F.
Feng
,
L.
Ling
, and
D. A.
Takahashi
, “
Multi-breather and high-order rogue waves for the nonlinear Schrödinger equation on the elliptic function background
,”
Stud. Appl. Math.
144
,
46
101
(
2020
).
9.
G.-j.
Dong
and
Z.
Liu
, “
Soliton resulting from the combined effect of higher order dispersion, self-steepening and nonlinearity in an optical fiber
,”
Opt. Commun.
128
,
8
14
(
1996
).
10.
Z.
Li
,
L.
Li
,
H.
Tian
, and
G.
Zhou
, “
New types of solitary wave solutions for the higher order nonlinear Schrödinger equation
,”
Phys. Rev. Lett.
84
,
4096
4099
(
2000
).
11.
L. C.
Zhao
,
S. C.
Li
, and
L.
Ling
, “
W-shaped solitons generated from a weak modulation in the Sasa-Satsuma equation
,”
Phys. Rev. E
93
,
032215
(
2016
).
12.
B. F.
Feng
,
L. M.
Ling
, and
Z. N.
Zhu
, “
A focusing and defocusing semi-discrete complex short-pulse equation and its various soliton solutions
,”
Proc. R. Soc. A
477
(
2247
),
20200853
(
2021
).
13.
P. Y. P.
Chen
and
B. A.
Malomed
, “
Single- and multi-peak solitons in two-component models of metamaterials and photonic crystals
,”
Opt. Commun.
283
,
1598
1606
(
2010
).
14.
G.
Xu
,
A.
Gelash
,
A.
Chabchoub
,
V.
Zakharov
, and
B.
Kibler
, “
Breather wave molecules
,”
Phys. Rev. Lett.
122
,
084101
(
2019
).
15.
B.
Kibler
,
A.
Chabchoub
,
A.
Gelash
,
N.
Akhmediev
, and
V. E.
Zakharov
, “
Superregular breathers in optics and hydrodynamics: Omnipresent modulation instability beyond simple periodicity
,”
Phys. Rev. X
5
,
041026
(
2013
).
16.
C.
Liu
,
Y.
Ren
,
Z.-Y.
Yang
, and
W.-L.
Yang
, “
Superregular breathers in a complex modified Korteweg-de Vries system
,”
Chaos
27
,
083120
(
2017
).
17.
V. E.
Zakharov
and
A. A.
Gelash
, “
Nonlinear stage of modulation instability
,”
Phys. Rev. Lett.
111
,
054101
(
2013
).
18.
B.
Kibler
,
J.
Fatome
,
C.
Finot
,
G.
Millot
,
G.
Genty
,
B.
Wetzel
,
N.
Akhmediev
,
F.
Dias
, and
J. M.
Dudley
, “
Observation of Kuznetsov-Ma soliton dynamics in optical fiber
,”
Sci. Rep.
2
,
463
(
2012
).
19.
A.
Chabchoub
,
B.
Kibler
,
J. M.
Dudley
, and
N.
Akhmediev
, “
Hydrodynamics of periodic breathers
,”
Philos. Trans. R. Soc. A
372
,
20140005
(
2014
).
20.
C.
Bao
,
J. A.
Jaramillo-Villegas
,
Y.
Xuan
,
D. E.
Leaird
,
M.
Qi
, and
A. M.
Weiner
, “
Observation of Fermi-Pasta-Ulam recurrence induced by breather solitons in an optical microresonator
,”
Phys. Rev. Lett.
117
,
163901
(
2016
).
21.
J.
Rao
,
Y.
Cheng
, and
J.
He
, “
Rational and semirational solutions of the nonlocal Davey–Stewartson equations
,”
Stud. Appl. Math.
139
,
568
598
(
2017
).
22.
Y.
Cao
,
J.
Rao
,
D.
Mihalache
, and
J.
He
, “
Semi-rational solutions for the (2+1)-dimensional nonlocal Fokas system
,”
Appl. Math. Lett.
80
,
27
34
(
2018
).
23.
J.
Rao
,
Y.
Zhang
,
A. S.
Fokas
, and
J.
He
, “
Rogue waves of the nonlocal Davey–Stewartson I equation
,”
Nonlinearity
31
,
4090
4107
(
2018
).
24.
Y. L.
Cao
,
Y.
Cheng
,
B. A.
Malomed
, and
J. S.
He
, “
Rogue waves and lumps on the non-zero background in the PT-symmetric nonlocal Maccari system
,”
Stud. Appl. Math.
147
,
694
723
(
2021
).
25.
L.
Draper
, “‘
Freak’ ocean waves
,”
Weather
21
,
2
4
(
1966
).
26.
E. I.
El-Awady
and
W. M.
Moslem
, “
On a plasma having nonextensive electrons and positrons: Rogue and solitary wave propagation
,”
Phys. Plasmas
18
,
082306
(
2011
).
27.
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
).
28.
Yu. V.
Bludov
,
V. V.
Konotop
, and
N.
Akhmediev
, “
Matter rogue waves
,”
Phys. Rev. A
80
,
033610
(
2009
).
29.
Y. V.
Bludov
,
V. V.
Konotop
, and
N.
Akhmediev
, “
Vector rogue waves in binary mixtures of Bose-Einstein condensates
,”
Eur. Phys. J.: Spec. Top.
185
,
169
180
(
2010
).
30.
L.
Stenflo
and
M.
Marklund
, “
Rogue waves in the atmosphere
,”
Plasma Phys.
76
,
293
295
(
2010
).
31.
Y.
Ohta
and
J.
Yang
, “
Dynamics of rogue waves in the Davey–Stewartson II equation
,”
J. Phys. A: Math. Theor.
46
,
105202
(
2013
).
32.
Y.
Ohta
and
J.
Yang
, “
Rogue waves in the Davey-Stewartson I equation
,”
Phys. Rev. E
86
,
036604
(
2012
).
33.
Y.
Jiang
,
J.
Rao
,
D.
Mihalache
,
J.
He
, and
Y.
Cheng
, “
Rogue breathers and rogue lumps on a background of dark line solitons for the Maccari system
,”
Commun. Nonlinear Sci. Numer. Simul.
102
,
105943
(
2021
).
34.
J.
Rao
,
J.
He
, and
B. A.
Malomed
, “
Resonant collisions between lumps and periodic solitons in the Kadomtsev–Petviashvili I equation
,”
J. Math. Phys.
63
,
013510
(
2022
).
35.
J.
Rao
,
J.
He
, and
Y.
Cheng
, “
The Davey–Stewartson I equation: Doubly localized two-dimensional rogue lumps on the background of homoclinic orbits or constant
,”
Lett. Math. Phys.
112
,
75
(
2022
).
36.
J.
Rao
,
A. S.
Fokas
, and
J.
He
, “
Doubly localized two-dimensional rogue waves in the Davey–Stewartson I equation
,”
J. Nonlinear Sci.
31
,
67
(
2021
).
37.
J.
Rao
,
J.
He
, and
D.
Mihalache
, “
Doubly localized rogue waves on a background of dark solitons for the Fokas system
,”
Appl. Math. Lett.
121
,
107435
(
2021
).
38.
J. W.
Miles
, “
Obliquely interacting solitary waves
,”
J. Fluid Mech.
79
,
157
169
(
1977
).
39.
J. W.
Miles
, “
Resonantly interacting solitary waves
,”
J. Fluid Mech.
79
,
171
179
(
1977
).
40.
N. C.
Freeman
, “
Soliton interaction in two dimensions
,”
Adv. Appl. Mech.
20
,
1
37
(
1980
).
41.
A. S.
Fokas
and
A. K.
Pogrebkov
, “
Inverse scattering transform for the KPI equation on the background of a one-line soliton
,”
Nonlinearity
16
,
771
783
(
2003
).
42.
J.-Y.
Yang
and
W.-X.
Ma
, “
Abundant interaction solutions of the KP equation
,”
Nonlinear Dyn.
89
,
1539
1544
(
2017
).
43.
X.
Zhang
,
Y.
Chen
, and
X.
Tang
, “
Rogue wave and a pair of resonance stripe solitons to KP equation
,”
Comput. Math. Appl.
76
,
1938
1949
(
2018
).
44.
Y.
Cao
,
J.
He
,
Y.
Cheng
, and
D.
Mihalache
, “
Reduction in the (4 + 1)-dimensional Fokas equation and their solutions
,”
Nonlinear Dyn.
99
,
3013
3028
(
2020
).
45.
J. G.
Rao
,
T.
Kanna
, and
J. S.
He
, “
A study on resonant collision in the two-dimensional multi-component long-wave–short-wave resonance system
,”
Proc. R. Soc. A
478
,
20210777
(
2022
).
46.
D. H.
Peregrine
, “
Water waves, nonlinear Schrödinger equation and their solutions
,”
ANZIAM J.
25
,
16
43
(
1983
).
47.
C.
Rogers
,
K. W.
Chow
, and
R.
Conte
, “
On a capillarity model and the Davey-Stewartson I system: Quasi-doubly periodic wave patterns
,”
Nuovo Cimento B
122
,
105
111
(
2007
).
48.
A. S.
Fokas
, “
The Davey-Stewartson equation on the half-plane
,”
Commun. Math. Phys.
289
,
957
993
(
2009
).
49.
A. S.
Fokas
,
D. E.
Pelinovsky
, and
C.
Sulem
, “
Interaction of lumps with a line soliton for the DSII equation
,”
Physica D
152
,
189
198
(
2001
).
50.
L.
Guo
,
L.
Chen
,
D.
Mihalache
, and
J.
He
, “
Dynamics of soliton interaction solutions of the Davey-Stewartson I equation
,”
Phys. Rev. E
105
,
014218
(
2022
).
51.
A.
Maccari
, “
The Kadomtsev–Petviashvili equation as a source of integrable model equations
,”
J. Math. Phys.
37
,
6207
(
1996
).
52.
A.
Issasfa
and
J.
Lin
, “
N-soliton and rogue wave solutions of (2 + 1)-dimensional integrable system with Lax pair
,”
Int. J. Mod. Phys. B
33
,
1950317
(
2019
).
53.
R.
Hirota
,
The Direct Method in Soliton Theory
(
Cambridge University Press
,
2004
).
54.
M.
Jimbo
and
T.
Miwa
,
Solitons and Infinite Dimensional Lie Algebras
, Publications of the Research Institute for Mathematical Sciences Vol. 19 (
Kyoto University
,
1983
), pp.
943
1001
.
55.
E.
Date
,
M.
Kashiwara
,
M.
Jimbo
, and
T.
Miwa
, “
Transformation groups for soliton equations
,” in
Nonlinear Integrable Systems-Classical Theory and Quantum Theory
, edited by
M.
Jimbo
and
T.
Miwa
(
World Scientific
,
Singapore
, 1983) pp. 39–119.
56.
Y.
Ohta
,
D.-S.
Wang
, and
J.
Yang
, “
General N-dark–dark solitons in the coupled nonlinear Schrödinger equations
,”
Stud. Appl. Math.
127
,
345
371
(
2011
).
57.
M.
Tajiri
and
T.
Arai
, “
Quasi-line soliton interactions of the Davey–Stewartson I equation: On the existence of long-range interaction between two quasi-line solitons through a periodic soliton
,”
J. Phys. A: Math. Theor.
44
,
235204
(
2011
).
You do not currently have access to this content.