In plasma physics, the Kadomtsev–Petviashvili I (KPI) equation is a fundamental model for investigating the evolution characteristics of nonlinear waves. For the KPI equation, the constraint method is an effective tool for generating solitonic or rational solutions from the solutions of lower-dimensional integrable systems. In this work, various nonsingular, rational lump solutions of the KPI equation are constructed by employing the vector one-constraint method and the generalized Darboux transformation of the (1 + 1)-dimensional vector Ablowitz–Kaup–Newell–Segur system. Furthermore, we investigate the large-time asymptotic behavior of high-order lumps in detail and discover distinct types of patterns. These lump patterns correspond to the high-order rogue wave patterns of the (1 + 1)-dimensional vector integrable equation and are associated with root structures of generalized Wronskian–Hermite polynomials.
REFERENCES
1.
Y.
Kodama
, KP Solitons and the Grassmannians: Combinatorics and Geometry of Two-Dimensional Wave Patterns
(Springer
, 2017
).2.
B.
Kadomtsev
and V.
Petrviashvili
, “On the stability of solitary waves in weakly dispersing media
,” Sov. Phys. Dokl.
15
(6
), 539
–541
(1970
).3.
K. E.
Lonngren
, “Ion acoustic soliton experiments in a plasma
,” Opt. Quantum Electron.
30
(7–10
), 615
–630
(1998
).4.
U.
Kumar Samanta
, A.
Saha
, and P.
Chatterjee
, “Bifurcations of nonlinear ion acoustic travelling waves in the frame of a Zakharov-Kuznetsov equation in magnetized plasma with a kappa distributed electron
,” Phys. Plasmas
20
(5
), 052111
(2013
).5.
A.
Saha
, N.
Pal
, and P.
Chatterjee
, “Bifurcation and quasiperiodic behaviors of ion acoustic waves in magnetoplasmas with nonthermal electrons featuring tsallis distribution
,” Braz. J. Phys.
45
(3
), 325
–333
(2015
).6.
R.
Saeed
and A.
Mushtaq
, “Ion acoustic waves in pair-ion plasma: Linear and nonlinear analyses
,” Phys. Plasmas
16
(3
), 032307
(2009
).7.
U. N.
Ghosh
, P.
Chatterjee
, and B.
Kaur
, “Study of lump soliton structures in pair-ion plasmas
,” Braz. J. Phys.
53
(2
), 48
(2023
).8.
D. E.
Pelinovsky
, Y. A.
Stepanyants
, and Y. S.
Kivshar
, “Self-focusing of plane dark solitons in nonlinear defocusing media
,” Phys. Rev. E
51
(5
), 5016
–5026
(1995
).9.
F.
Baronio
, S.
Wabnitz
, and Y.
Kodama
, “Optical kerr spatiotemporal dark-lump dynamics of hydrodynamic origin
,” Phys. Rev. Lett.
116
(17
), 173901
(2016
).10.
S.
Tsuchiya
, F.
Dalfovo
, and L.
PitaevskII
, “Solitons in two-dimensional Bose-Einstein condensates
,” Phys. Rev. A
77
(4
), 045601
(2008
).11.
S.
Turitsyn
and G.
Fai’kovich
, “Stability of magnetoelastic solitons and self-focusing of sound in antiferromagnets
,” Sov. Phys. JETP
62
(1
), 146
–152
(1985
).12.
X. W.
Jin
, S. J.
Shen
, Z. Y.
Yang
, and J.
Lin
, “Magnetic lump motion in saturated ferromagnetic films
,” Phys. Rev. E
105
(1
), 014205
(2022
).13.
M.
Ablowitz
and P.
Clarkson
, Solitons, Nonlinear Evolution Equations and Inverse Scattering
(Cambridge University Press
, Cambridge
, 1991
).14.
S.
Chakravarty
and Y.
Kodama
, “Soliton solutions of the KP equation and application to shallow water waves
,” Stud. Appl. Math.
123
(1
), 83
–151
(2009
).15.
M. J.
Ablowitz
, S.
Chakravarty
, A. D.
Trubatch
, and J.
Villarroel
, “A novel class of solutions of the non-stationary Schrödinger and the Kadomtsev-Petviashvili I equations
,” Phys. Lett. A
267
(2–3
), 132
–146
(2000
).16.
M. J.
Ablowitz
and J.
Villarroel
, “Solutions to the time dependent Schrödinger and the Kadomtsev-Petviashvili equations
,” Phys. Rev. Lett.
78
(4
), 570
–573
(1997
).17.
S.
Chakravarty
and M.
Zowada
, “Classification of KPI lumps
,” J. Phys. A: Math. Theor.
55
(21
), 215701
(2022
).18.
X. M.
Zhu
, D. J.
Zhang
, and D. Y.
Chen
, “Lump solutions of Kadomtsev-Petviashvili I equation in non-uniform media
,” Commun. Theor. Phys.
55
(1
), 13
(2011
).19.
S.
Chakravarty
and M.
Zowada
, “Multi-lump wave patterns of KPI via integer partitions
,” Physica D
446
, 133644
(2023
).20.
J.
Dong
, L.
Ling
, and X.
Zhang
, “Kadomtsev-Petviashvili equation: One-constraint method and lump pattern
,” Physica D
432
, 133152
(2022
).21.
K.
Gorshkov
, D.
PelinovskII
, and Y.
Stepanyants
, “Normal and anomalous scattering, formation and decay of bound states of two-dimensional solitons described by the Kadomtsev-Petviashvili equation
,” J. Exp. Theor. Phys.
77
(2
), 237
–245
(1993
).22.
R. S.
Johnson
and S.
Thompson
, “A solution of the inverse scattering problem for the Kadomtsev-Petviashvili equation by the method of separation of variables
,” Phys. Lett. A
66
(4
), 279
–281
(1978
).23.
S. V.
Manakov
, V. E.
Zakharov
, L. A.
Bordag
, A. R.
Its
, and V. B.
Matveev
, “Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction
,” Phys. Lett. A
63
(3
), 205
–206
(1977
).24.
D.
PelinovskII
and Y.
Stepanyants
, “New multisoliton solutions of the Kadomtsev-Petviashvili equation
,” JETP
57
(1
), 24
–28
(1993
).25.
J.
Satsuma
and M. J.
Ablowitz
, “Two-dimensional lumps in nonlinear dispersive systems
,” J. Math. Phys.
20
(7
), 1496
–1503
(1979
).26.
B.
Yang
and J.
Yang
, “Pattern transformation in higher-order lumps of the Kadomtsev-Petviashvili I equation
,” J. Nonlinear Sci.
32
(4
), 52
(2022
).27.
W. X.
Ma
, “Lump solutions to the Kadomtsev-Petviashvili equation
,” Phys. Lett. A
379
(36
), 1975
–1978
(2015
).28.
C.
Terng
and K.
Uhlenbeck
, “Bäcklund transformations and loop group actions
,” Commun. Pure Appl. Math.
53
(1
), 1
–75
(2000
).29.
P.
Dubard
, P.
Gaillard
, C.
Klein
, and V.
Matveev
, “On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation
,” Eur. Phys. J.: Spec. Top.
185
(1
), 247
–258
(2010
).30.
P.
Dubard
and V. B.
Matveev
, “Multi-rogue waves solutions: From the NLS to the KP-I equation
,” Nonlinearity
26
(12
), R93
–R125
(2013
).31.
S.
Chen
, P.
Grelu
, D.
Mihalache
, and F.
Baronio
, “Families of rational soliton solutions of the Kadomtsev-Petviashvili I equation
,” Rom. Rep. Phys.
68
(4
), 1407
–1424
(2016
).32.
J. H.
Chang
, “Asymptotic analysis of multilump solutions of the Kadomtsev-Petviashvili-I equation
,” Theor. Math. Phys.
195
(2
), 676
–689
(2018
).33.
P.
Gaillard
, “Multiparametric families of solutions of the Kadomtsev-Petviashvili-I equation, the structure of their rational representations, and multi-rogue waves
,” Theor. Math. Phys.
196
(2
), 1174
–1199
(2018
).34.
S.
Novikov
, S.
Manakov
, L.
PitaevskII
, and V.
Zakharov
, Theory of Solitons: The Inverse Scattering Methods
(Springer Science & Business Media
, 1984
).35.
X. H.
Wu
and Y. T.
Gao
, “Generalized Darboux transformation and solitons for the Ablowitz–Ladik equation in an electrical lattice
,” Appl. Math. Lett.
137
, 108476
(2023
).36.
X.
Zhang
, Y. F.
Wang
, and S. X.
Yang
, “Hybrid structures of the rogue waves and breather-like waves for the higher-order coupled nonlinear Schrödinger equations
,” Chaos, Solitons Fractals
180
, 114563
(2024
).37.
L.
Ling
and H.
Su
, “Rogue waves and their patterns for the coupled Fokas–Lenells equations
,” Physica D
461
, 134111
(2024
).38.
H.
Lin
and L.
Ling
, “Rogue wave pattern of multi-component derivative nonlinear Schrödinger equations
,” Chaos
34
, 043126
(2024
).39.
L.
Ling
and H.
Yang
, “The determinant representation of Ward soliton solutions and its dynamical behaviors
,” Nonlinear Dyn.
112
, 7417
–7432
(2024
).40.
41.
Y.
Cheng
and Y. S.
Li
, “The constraint of the Kadomtsev-Petviashvili equation and its special solutions
,” Phys. Lett. A
157
(1
), 22
–26
(1991
).42.
Y.
Cheng
, “Constraints of the Kadomtsev-Petviashvili hierarchy
,” J. Math. Phys.
33
(11
), 3774
–3782
(1992
).43.
Y.
Cheng
and Y. S.
Li
, “Constraints of the 2 + 1 dimensional integrable soliton systems
,” J. Phys. A: Math. Gen.
25
(2
), 419
–431
(1992
).44.
Y. J.
Zhang
and Y.
Cheng
, “Solutions for the vector k-constrained KP hierarchy
,” J. Math. Phys.
35
(11
), 5869
–5884
(1994
).45.
Y.
Kodama
, “Young diagrams and N-soliton solutions of the KP equation
,” J. Phys. A: Math. Gen.
37
(46
), 11169
–11190
(2004
).46.
Y.
Ohta
, J.
Satsuma
, D.
Takahashi
, and T.
Tokihiro
, “An elementary introduction to Sato theory
,” Prog. Theor. Phys. Suppl.
94
, 210
–241
(1988
).47.
B.
Konopelchenko
and W.
Strampp
, “The AKNS hierarchy as symmetry constraint of the KP hierarchy
,” Inverse Probl.
7
(2
), L17
–L24
(1991
).48.
G.
Han
, X.
Li
, Q.
Zhao
, and C.
Li
, “Interaction structures of multi localized waves within the Kadomtsev–Petviashvili I equation
,” Physica D
446
, 133671
(2023
).49.
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
(5
), 81
(2021
).50.
L.
Ling
, L.
Zhao
, and B.
Guo
, “Darboux transformation and multi-dark soliton for N-component nonlinear Schrödinger equations
,” Nonlinearity
28
(9
), 3243
(2015
).51.
B.
Yang
and J.
Yang
, “Universal rogue wave patterns associated with the Yablonskii-Vorob’ev polynomial hierarchy
,” Physica D
425
, 132958
(2021
).52.
B.
Yang
and J.
Yang
, “Rogue wave patterns in the nonlinear Schrödinger equation
,” Physica D
419
, 132850
(2021
).53.
B.
Yang
and J.
Yang
, “Rogue waves in (2 + 1)-dimensional three-wave resonant interactions
,” Physica D
432
, 133160
(2022
).54.
B.
Yang
and J.
Yang
, “Rogue wave patterns associated with Okamoto polynomial hierarchies
,” Stud. Appl. Math.
151
, 60
–115
(2023
).55.
G.
Zhang
, P.
Huang
, B. F.
Feng
, and C.
Wu
, “Rogue waves and their patterns in the vector nonlinear Schrödinger equation
,” J. Nonlinear Sci.
33
(6
), 116
(2023
).56.
N.
Bonneux
, C.
Dunning
, and M.
Stevens
, “Coefficients of Wronskian Hermite polynomials
,” Stud. Appl. Math.
144
(3
), 245
–288
(2020
).57.
G.
Felder
, A. D.
Hemery
, and A. P.
Veselov
, “Zeros of Wronskians of Hermite polynomials and Young diagrams
,” Physica D
241
(23–24
), 2131
–2137
(2012
).58.
M.
Garcia-Ferrero
and D.
Gomez-Ullate
, “Oscillation theorems for the Wronskian of an arbitrary sequence of eigenfunctions of Schrödinger’s equation
,” Lett. Math. Phys.
105
(4
), 551
–573
(2015
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
Author(s)
You do not currently have access to this content.
