A (3 + 1) dimensional Kudryashov–Sinelshchikov equation is investigated in this paper, which describes bubbles in the liquid fluctuations. By virtue of the binary Bell polynomials, the bilinear representation, bilinear Bcklund transformation with associated Lax pair are obtained, respectively. Moreover, utilizing Hirota’s bilinear representation, four new lump solutions are constructed and the interaction phenomenon between lump and periodic solution is thoroughly examined. The work also illustrates the intriguing dynamical behavior with the aid of Maple software, which plots the three-dimensional surface, two-dimensional density, and contour profiles of the solutions constructed in this work in various planes.
REFERENCES
1.
P. M.
Jordan
and C.
Feuillade
, “On the propagation of harmonic acoustic waves in bubbly liquids
,” Int. J. Eng. Sci.
42
(11–12
), 1119
–1128
(2004
).2.
J.
Boussinesq
, “Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond
,” J. Math. Pures Appl.
17
, 55
–108
(1872
).3.
R.
Hirota
, “Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons
,” Phys. Rev. Lett.
27
(18
), 1192
–1194
(1971
).4.
A.-M.
Wazwaz
, “Two reliable methods for solving variants of the KdV equation with compact and noncompact structures
,” Chaos, Solitons Fractals
28
(2
), 454
–462
(2006
).5.
R.
Ishizeki
, M.
Kruczenski
, and S.
Ziama
, “Notes on Euclidean Wilson loops and Riemann theta functions
,” Phys. Rev. D
85
(10
), 106004
(2012
).6.
U.
Al Khawaja
, “A comparative analysis of Painlevé, Lax pair, and similarity transformation methods in obtaining the integrability conditions of nonlinear Schrödinger equations
,” J. Math. Phys.
51
(5
), 053506
(2010
).7.
H. D.
Wahlquist
and F. B.
Estabrook
, “Bäcklund transformation for solutions of the Korteweg-de Vries equation
,” Phys. Rev. Lett.
31
(23
), 1386
–1390
(1973
).8.
X.
Guo
, “On bilinear representations and infinite conservation laws of a nonlinear variable-coefficient equation
,” Appl. Math. Comput.
248
, 531
–535
(2014
).9.
N.
Kudryashov
and D. I.
Sinelshchikov
, “Equation for three–dimensional nonlinear waves in liquid with gas bubbles
,” Phys. Scr.
85
(2
), 025402
(2012
).10.
Y. B.
Chukkol
, M. N. B.
Mohamad
, and M.
Muminov
, “Explicit solutions to the (3 + 1)-dimensional Kudryashov-Sinelshchikov equations in bubbly flow dynamics
,” J. Appl. Math.
2018
, 1
–9
.11.
X.
Tang
and Y.
Chen
, “Lumps, breathers, rogue waves and interaction solutions to a (3 + 1)-dimensional Kudryashov–Sinelshchikov equation
,” Mod. Phys. Lett. B
34
(12
), 2050117
(2020
).12.
M. J.
Ablowitz
and H.
Segur
, “Asymptotic solutions of the Korteweg–de Vries equation
,” Stud. Appl. Math.
57
(1
), 13
–44
(1977
).13.
E.
Kuznetsov
and S.
Turitsyn
, “Two- and three-dimensional solitons in weakly dispersive media
,” Sov. Phys. JETP
55
(5
), 844
–847
(1982
).14.
M. J.
Ablowitz
and H.
Segur
, “On the evolution of packets of water waves
,” J. Fluid Mech.
92
(4
), 691
–715
(1979
).15.
E.
Infeld
and G.
Rowlands
, “Three dimensional stability of Korteweg-de Vries waves and solitons
,” Acta Phys. Pol., A
56
(3
), 329
–332
(1979
).16.
W.-X.
Ma
, “Comment on the 3 + 1 dimensional Kadomtsev–Petviashvili equations
,” Commun. Nonlinear Sci. Numer. Simul.
16
(7
), 2663
–2666
(2011
).17.
W.-X.
Ma
, “Lump solutions to the Kadomtsev–Petviashvili equation
,” Phys. Lett. A
379
(36
), 1975
–1978
(2015
).18.
A.-J.
Zhou
and A.-H.
Chen
, “Exact solutions of the Kudryashov–Sinelshchikov equation in ideal liquid with gas bubbles
,” Phys. Scr.
93
(12
), 125201
(2018
).19.
W.-X.
Ma
, “Binary Darboux transformation of vector nonlocal reverse-time integrable NLS equations
,” Chaos, Solitons Fractals
180
, 114539
(2024
).20.
W.-X.
Ma
, “Type (λ*, λ) reduced nonlocal integrable AKNS equations and their soliton solutions
,” Appl. Numer. Math.
199
, 105
(2024
).21.
W.-X.
Ma
, “General solution to a nonlocal linear differential equation of first-order
,” Qual. Theory Dyn. Syst.
23
(4
), 177
(2024
).22.
C.
Gilson
, F.
Lambert
, J.
Nimmo
, and R.
Willox
, “On the combinatorics of the Hirota D-operators
,” Proc. R. Soc. London, Ser. A
452
(1945
), 223
–234
(1996
).23.
F.
Lambert
and J.
Springael
, “Soliton equations and simple combinatorics
,” Acta Appl. Math.
102
(2–3
), 147
–178
(2008
).24.
N.
Liu
and F.
Ding
, “Lax pair, Bäcklund transformation and conservation laws for the (2 + 1)-dimensional extended shallow water wave equation
,” Comput. Fluids
89
, 153
–156
(2014
).25.
W.-X.
Ma
, “Bilinear equations, Bell polynomials and linear superposition principle
,” J. Phys.: Conf. Ser.
411
, 012021
(2013
).26.
J.-M.
Tu
, S.-F.
Tian
, M.-J.
Xu
, X.-Q.
Song
, and T.-T.
Zhang
, “Bäcklund transformation, infinite conservation laws and periodic wave solutions of a generalized (3 + 1)-dimensional nonlinear wave in liquid with gas bubbles
,” Nonlinear Dyn.
83
(3
), 1199
–1215
(2015
).27.
M.-J.
Xu
, S.-F.
Tian
, J.-M.
Tu
, and T.-T.
Zhang
, “Bäcklund transformation, infinite conservation laws and periodic wave solutions to a generalized (2 + 1)-dimensional Boussinesq equation
,” Nonlinear Anal.: Real World Appl.
31
, 388
–408
(2016
).28.
K.
Joseph
and A. V.
Murthy
, “Hopf-Cole transformation to some systems of partial differential equations
,” Nonlinear Differ. Equations Appl.
8
(2
), 173
–193
(2001
).29.
A.
Gorguis
, “A comparison between Cole–Hopf transformation and the decomposition method for solving Burgers’ equations
,” Appl. Math. Comput.
173
(1
), 126
–136
(2006
).30.
X.
Lü
, Y.-F.
Hua
, S.-J.
Chen
, and X.-F.
Tang
, “Integrability characteristics of a novel (2 + 1)-dimensional nonlinear model: Painlevé analysis, soliton solutions, Bäcklund transformation, Lax pair and infinitely many conservation laws
,” Commun. Nonlinear Sci. Numer. Simul.
95
, 105612
(2021
).31.
C.
Chen
, X.
Zhang
, and Z.
Liu
, “A high-order compact finite difference scheme and precise integration method based on modified Hopf-Cole transformation for numerical simulation of n-dimensional Burgers’ system
,” Appl. Math. Comput.
372
, 125009
(2020
).32.
R.
Hirota
, Direct Methods in Soliton Theory
(Springer-Verlag
, Berlin
, 2004
).33.
X.-B.
Hu
, C.-X.
Li
, J. J. C.
Nimmo
, and G.-F.
Yu
, “An integrable symmetric (2 + 1)-dimensional Lotka–Volterra equation and a family of its solutions
,” J. Phys. A: Math. Gen.
38
(1
), 195
–204
(2005
).34.
Q. P.
Liu
, X.-B.
Hu
, and M.-X.
Zhang
, “Supersymmetric modified Korteweg–de Vries equation: Bilinear approach
,” Nonlinearity
18
(4
), 1597
–1603
(2005
).35.
W.
Ma
, R.
Zhou
, and L.
Gao
, “Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in (2 + 1) dimensions
,” Mod. Phys. Lett. A
24
(21
), 1677
–1688
(2009
).36.
E.
Fan
, “The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials
,” Phys. Lett. A
375
(3
), 493
–497
(2011
).37.
Y.
Tian
and J.-G.
Liu
, “Study on dynamical behavior of multiple lump solutions and interaction between solitons and lump wave
,” Nonlinear Dyn.
104
(2
), 1507
–1517
(2021
).38.
B.
Ren
, J.
Lin
, and Z.-M.
Lou
, “A new nonlinear equation with lump-soliton, lump-periodic, and lump-periodic-soliton solutions
,” Complexity
2019
(1
), 4072754
.39.
E. T.
Bell
, “Exponential polynomials
,” Ann. Math.
35
(2
), 258
–277
(1934
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
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