Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation

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

References

[1] Ablowitz M J, Clarkson P A. Solitons, Nonlinear Evolution Equations and Inverse Scattering.Cambridge: Cambridge Univ Press, 1991

[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

[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

[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

[6] Gilson C R, Nimmo J J C. Lump solutions of the BKP equation. Phys Lett A, 1990, 147: 472–476

[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

[9] Ibragimov N H. A new conservation theorem. J Math Anal Appl, 2007, 333: 311–328

[10] Imai K. Dromion and lump solutions of the Ishimori-I equation. Progr Theoret Phys, 1997, 98: 1013–1023

[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

[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

[13] Konopelchenko B, Strampp W. The AKNS hierarchy as symmetry constraint of the KP hierarchy. Inverse Problems, 1991, 7: L17–L24

[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

[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

[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

[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

[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

[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

[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

[22] Ma W X. Lump solutions to the Kadomtsev-Petviashvili equation. Phys Lett A, 2015, 379: 1975–1978

[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

[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

[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

[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

[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

[29] Ma W X, Zhou Y. Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J Differential Equations, 2018, 264: 2633–2659

[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

[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

[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

[34] Satsuma J, Ablowitz M J. Two-dimensional lumps in nonlinear dispersive systems. J Math Phys, 1979, 20: 1496–1503

[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

[37] Triki H, Jovanoski Z, Biswas A. Shock wave solutions to the Bogoyavlensky-Konopelchenko equation. Indian J Phys, 2014, 88: 71–74

[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

[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

[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

[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

[42] Yang J Y, Ma W X. Lump solutions of the BKP equation by symbolic computation. Internat J Modern Phys B, 2016, 30: 1640028

[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

[44] Yang J Y, Ma W X. Abundant interaction solutions of the KP equation. Nonlinear Dynam, 2017, 89: 1539–1544

[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,

[47] Yu J P, Sun Y L. Study of lump solutions to dimensionally reduced generalized KP equations. Nonlinear Dynam, 2017, 87: 2755–2763

[48] Zhang J B, Ma W X. Mixed lump-kink solutions to the BKP equation. Comput Math Appl, 2017, 74: 591–596

[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

[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

[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

[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

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature