Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation

doi:10.1007/s11464-018-0694-z

Front. Math. China

2018, Vol. 13

Issue (3) : 525-534 https://doi.org/10.1007/s11464-018-0694-z

RESEARCH ARTICLE

Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation

Shou-Ting CHEN¹, Wen-Xiu MA^2,^3,⁴(

)

¹. School of Mathematics and Physical Science, Xuzhou Institute of Technology, Xuzhou 221008, China
². Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA
³. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
⁴. International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa

Download: PDF(779 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

A (2+ 1)-dimensional generalized Bogoyavlensky-Konopelchenko equation that possesses a Hirota bilinear form is considered. Starting with its Hirota bilinear form, a class of explicit lump solutions is computed through conducting symbolic computations with Maple, and a few plots of a specific presented lump solution are made to shed light on the characteristics of lumps. The result provides a new example of (2+ 1)-dimensional nonlinear partial differential equations which possess lump solutions.

Keywords Symbolic computation lump solution soliton theory

Corresponding Author(s): Wen-Xiu MA

Issue Date: 11 June 2018

Cite this article:

Shou-Ting CHEN,Wen-Xiu MA. Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation[J]. Front. Math. China, 2018, 13(3): 525-534.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-018-0694-z
https://academic.hep.com.cn/fmc/EN/Y2018/V13/I3/525

1	Ablowitz M J, Clarkson P A. Solitons, Nonlinear Evolution Equations and Inverse Scattering.Cambridge: Cambridge Univ Press, 1991 https://doi.org/10.1017/CBO9780511623998
2	Caudrey P J. Memories of Hirota's method: application to the reduced Maxwell-Bloch system in the early 1970s. Philos Trans R Soc A Math Phys Eng Sci, 2011, 369: 1215–1227
3	Dong H H, Zhang Y, Zhang X E. The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation. Commun Nonlinear Sci Numer Simul, 2016, 36: 354–365 https://doi.org/10.1016/j.cnsns.2015.12.015
4	Dorizzi B, Grammaticos B, Ramani A, Winternitz P. Are all the equations of the Kadomtsev-Petviashvili hierarchy integrable? J Math Phys, 1986, 27: 2848–2852 https://doi.org/10.1063/1.527260
5	Gilson C, Lambert F, Nimmo J, Willox R. On the combinatorics of the Hirota D-operators. Proc R Soc Lond Ser A, 1996, 452: 223–234 https://doi.org/10.1098/rspa.1996.0013
6	Gilson C R, Nimmo J J C. Lump solutions of the BKP equation. Phys Lett A, 1990, 147: 472–476 https://doi.org/10.1016/0375-9601(90)90609-R
7	Harun-Or-Roshid, Ali M Z. Lump solutions to a Jimbo-Miwa like equation. arXiv: 1611.04478
8	Hirota R. The Direct Method in Soliton Theory.New York: Cambridge Univ Press, 2004 https://doi.org/10.1017/CBO9780511543043
9	Ibragimov N H. A new conservation theorem. J Math Anal Appl, 2007, 333: 311–328 https://doi.org/10.1016/j.jmaa.2006.10.078
10	Imai K. Dromion and lump solutions of the Ishimori-I equation. Progr Theoret Phys, 1997, 98: 1013–1023 https://doi.org/10.1143/PTP.98.1013
11	Kaup D J. The lump solutions and the Bäcklund transformation for the threedimensional three-wave resonant interaction. J Math Phys, 1981, 22: 1176–1181 https://doi.org/10.1063/1.525042
12	Kofane T C, Fokou M, Mohamadou A, Yomba E. Lump solutions and interaction phenomenon to the third-order nonlinear evolution equation. Eur Phys J Plus, 2017, 132: 465 https://doi.org/10.1140/epjp/i2017-11747-6
13	Konopelchenko B, Strampp W. The AKNS hierarchy as symmetry constraint of the KP hierarchy. Inverse Problems, 1991, 7: L17–L24 https://doi.org/10.1088/0266-5611/7/2/002
14	Li X Y, Zhao Q L. A new integrable symplectic map by the binary nonlinearization to the super AKNS system. J Geom Phys, 2017, 121: 123–137 https://doi.org/10.1016/j.geomphys.2017.07.010
15	Li X Y, Zhao Q L, Li Y X, Dong H H. Binary Bargmann symmetry constraint associated with 3 × 3 discrete matrix spectral problem. J Nonlinear Sci Appl, 2015, 8(5): 496–506 https://doi.org/10.22436/jnsa.008.05.05
16	Lin F H, Chen S T, Qu Q X, Wang J P, Zhou X W, Lü X. Resonant multiple wave solutions to a new (3+ 1)-dimensional generalized Kadomtsev-Petviashvili equation: Linear superposition principle. Appl Math Lett, 2018, 78: 112–117 https://doi.org/10.1016/j.aml.2017.10.013
17	Lu C N, Fu C, Yang H W. Time-fractional generalized Boussinesq equation for Rossby solitary waves with dissipation effect in stratified uid and conservation laws as well as exact solutions. Appl Math Comput, 2018, 327: 104–116 https://doi.org/10.1016/j.amc.2018.01.018
18	Lü X, Chen S T, Ma W X. Constructing lump solutions to a generalized Kadomtsev-Petviashvili-Boussinesq equation. Nonlinear Dynam, 2016, 86: 523–534 https://doi.org/10.1007/s11071-016-2905-z
19	Lü X, Wang J P, Lin F H, Zhou X W. Lump dynamics of a generalized two-dimensional Boussinesq equation in shallow water. Nonlinear Dynam, 2018, 91: 1249–1259 https://doi.org/10.1007/s11071-017-3942-y
20	Ma W X. Wronskian solutions to integrable equations. Discrete Contin Dyn Syst, 2009, Suppl: 506–515
21	Ma W X. Bilinear equations, Bell polynomials and linear superposition principle. J Phys Conf Ser, 2013, 411: 012021 https://doi.org/10.1088/1742-6596/411/1/012021
22	Ma W X. Lump solutions to the Kadomtsev-Petviashvili equation. Phys Lett A, 2015, 379: 1975–1978 https://doi.org/10.1016/j.physleta.2015.06.061
23	Ma W X. Lump-type solutions to the (3+1)-dimensional Jimbo-Miwa equation. Int J Nonlinear Sci Numer Simul, 2016, 17: 355–359 https://doi.org/10.1515/ijnsns-2015-0050
24	Ma W X. Conservation laws by symmetries and adjoint symmetries. Discrete Contin Dyn Syst Ser S, 2018, 11: 707–721
25	Ma W X, Fan E G. Linear superposition principle applying to Hirota bilinear equations. Comput Math Appl, 2011, 61: 950–959 https://doi.org/10.1016/j.camwa.2010.12.043
26	Ma W X, Qin Q Z, Lü X. Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dynam, 2016, 84: 923–931 https://doi.org/10.1007/s11071-015-2539-6
27	Ma W X, Yong X L, Zhang H Q. Diversity of interaction solutions to the (2+ 1)-dimensional Ito equation. Comput Math Appl, 2018, 75: 289–295 https://doi.org/10.1016/j.camwa.2017.09.013
28	Ma W X, You Y. Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions. Trans Amer Math Soc, 2005, 357: 1753–1778 https://doi.org/10.1090/S0002-9947-04-03726-2
29	Ma W X, Zhou Y. Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J Differential Equations, 2018, 264: 2633–2659 https://doi.org/10.1016/j.jde.2017.10.033
30	Ma W X, Zhou Y, Dougherty R. Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations. Internat J Modern Phys B, 2016, 30: 1640018 https://doi.org/10.1142/S021797921640018X
31	Manakov S V, Zakharov V E, Bordag L A, Matveev V B. Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction. Phys Lett A, 1977, 63: 205–206 https://doi.org/10.1016/0375-9601(77)90875-1
32	Novikov S, Manakov S V, Pitaevskii L P, Zakharov V E. Theory of Solitons—The Inverse Scattering Method.New York: Consultants Bureau, 1984
33	Ray S S. On conservation laws by Lie symmetry analysis for (2+ 1)-dimensional Bogoyavlensky-Konopelchenko equation in wave propagation. Comput Math Appl, 2017, 74: 1158–1165 https://doi.org/10.1016/j.camwa.2017.06.007
34	Satsuma J, Ablowitz M J. Two-dimensional lumps in nonlinear dispersive systems. J Math Phys, 1979, 20: 1496–1503 https://doi.org/10.1063/1.524208
35	Tan W, Dai H P, Dai Z D, Zhong W Y. Emergence and space-time structure of lump solution to the (2+1)-dimensional generalized KP equation. Pramana—J Phys, 2017, 89: 77
36	Tang Y N, Tao S Q, Qing G. Lump solitons and the interaction phenomena of them for two classes of nonlinear evolution equations. Comput Math Appl, 2016, 72: 2334–2342 https://doi.org/10.1016/j.camwa.2016.08.027
37	Triki H, Jovanoski Z, Biswas A. Shock wave solutions to the Bogoyavlensky-Konopelchenko equation. Indian J Phys, 2014, 88: 71–74 https://doi.org/10.1007/s12648-013-0380-7
38	Ünsal Ö, Ma W X. Linear superposition principle of hyperbolic and trigonometric function solutions to generalized bilinear equations. Comput Math Appl, 2016, 71: 1242–1247 https://doi.org/10.1016/j.camwa.2016.02.006
39	Wazwaz A-M, El-Tantawy S A. New (3+1)-dimensional equations of Burgers type and Sharma-Tasso-Olver type: multiple-soliton solutions. Nonlinear Dynam, 2017, 87(4): 2457–2461 https://doi.org/10.1007/s11071-016-3203-5
40	Xu Z H, Chen H L, Dai Z D. Rogue wave for the (2+ 1)-dimensional Kadomtsev-Petviashvili equation. Appl Math Lett, 2014, 37: 34–38 https://doi.org/10.1016/j.aml.2014.05.005
41	Yang H W, Chen X, Guo M, Chen Y D. A new ZKCBO equation for three-dimensional algebraic Rossby solitary waves and its solution as well as fission property. Nonlinear Dynam, 2018, 91: 2019–2032 https://doi.org/10.1007/s11071-017-4000-5
42	Yang J Y, Ma W X. Lump solutions of the BKP equation by symbolic computation. Internat J Modern Phys B, 2016, 30: 1640028 https://doi.org/10.1142/S0217979216400282
43	Yang J Y, Ma W X. Abundant lump-type solutions of the Jimbo-Miwa equation in (3+ 1)-dimensions. Comput Math Appl, 2017, 73: 220–225 https://doi.org/10.1016/j.camwa.2016.11.007
44	Yang J Y, Ma W X. Abundant interaction solutions of the KP equation. Nonlinear Dynam, 2017, 89: 1539–1544 https://doi.org/10.1007/s11071-017-3533-y
45	Yang J Y, Ma W X, Qin Z Y. Mixed lump-soliton solutions of the BKP equation. East Asian J Appl Math, 2017
46	Yang J Y, Ma WX, Qin Z Y. Lump and lump-soliton solutions to the (2+1)-dimensional Ito equation. Anal Math Phys, 2017, https://doi.org/10.1007/s13324-017-0181-9
47	Yu J P, Sun Y L. Study of lump solutions to dimensionally reduced generalized KP equations. Nonlinear Dynam, 2017, 87: 2755–2763 https://doi.org/10.1007/s11071-016-3225-z
48	Zhang J B, Ma W X. Mixed lump-kink solutions to the BKP equation. Comput Math Appl, 2017, 74: 591–596 https://doi.org/10.1016/j.camwa.2017.05.010
49	Zhang Y, Dong H H, Zhang X E, Yang H W. Rational solutions and lump solutions to the generalized (3+ 1)-dimensional shallow water-like equation. Comput Math Appl, 2017, 73: 246–252 https://doi.org/10.1016/j.camwa.2016.11.009
50	Zhang Y, Sun S L, Dong H H. Hybrid solutions of (3+ 1)-dimensional Jimbo-Miwa equation. Math Probl Eng, 2017, 2017: Article ID 5453941
51	Zhao H Q, Ma W X. Mixed lump-kink solutions to the KP equation. Comput Math Appl, 2017, 74: 1399–1405 https://doi.org/10.1016/j.camwa.2017.06.034
52	Zhao Q L, Li X Y. A Bargmann system and the involutive solutions associated with a new 4-order lattice hierarchy. Anal Math Phys, 2016, 6: 237–254 https://doi.org/10.1007/s13324-015-0116-2
53	Zheng H C, Ma W X, Gu X. Hirota bilinear equations with linear subspaces of hyperbolic and trigonometric function solutions. Appl Math Comput, 2013, 220: 226–234 https://doi.org/10.1016/j.amc.2013.06.019

[1]	Sumayah BATWA, Wen-Xiu MA. Lump solutions to a generalized Hietarinta-type equation via symbolic computation[J]. Front. Math. China, 2020, 15(3): 435-450.
[2]	Hui WANG, Shoufu TIAN, Tiantian ZHANG, Yi CHEN. Lump wave and hybrid solutions of a generalized (3+ 1)-dimensional nonlinear wave equation in liquid with gas bubbles[J]. Front. Math. China, 2019, 14(3): 631-643.
[3]	Wen-Xiu MA. Interaction solutions to Hirota-Satsuma-Ito equation in (2+ 1)-dimensions[J]. Front. Math. China, 2019, 14(3): 619-629.

Viewed

Full text

Abstract

Cited

Shared

Discussed