Abstract
In this paper, we obtained a kind of lump solutions of (\(2+1\))-dimensional bSK equation with the assistance of Mathematica by using the Hirota bilinear method. These lump solutions contain a set of six free parameters. Some contour plots with different determinant values are sequentially made to show that the corresponding lump solutions tend to become zero when the determinant approaches zero. Particularly, the lump solutions with specific values of the included parameters are plotted, as illustrative examples.
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Gilson, C.R., Nimmo, J.J.C.: Lump solutions of the BKP equation. Phys. Lett. A 147, 472–476 (1990)
Imai, K.: Dromion and lump solutions of the Ishimori-I equation. Prog. Theor. Phys. 98, 1013–1023 (1997)
Minzoni, A.A., Smyth, N.F.: Evolution of lump solutions for the KP equation. Wave Motion 24, 291–305 (1996)
Quintero, J.R.: Existence and analyticity of lump solutions for generalized Benney–Luke equations. Rev. Colomb. Mat. 36, 71–95 (2012)
Yu, J.P., Sun, Y.L.: Study of lump solutions to dimensionally reduced generalized KP equations. Nonlinear Dyn. 87, 2755–2763 (2017)
Wang, Y., Chen, M.D., Li, X., et al.: Some interaction solutions of a reduced generalised (\(3+1\))-dimensional shallow water wave equation for lump solutions and a pair of resonance solitons. Z. Naturforsch. A 72, 419–424 (2017)
Ma, H.C., Deng, A.P.: Lump solution of (\(2+1\))-dimensional Boussinesq equation. Commun. Theor. Phys. 65, 546–552 (2016)
Tang, Y.N., Tao, S.Q., Guan, Q.: Lump solitons and the interaction phenomena of them for two classes of nonlinear evolution equations. Comput. Math. Appl. 72, 2334–2342 (2016)
Wang, C.J.: Spatiotemporal deformation of lump solution to (\(2+1\))-dimensional KdV equation. Nonlinear Dyn. 84, 697–702 (2016)
Lü, Z.S., Chen, Y.N.: Construction of rogue wave and lump solutions for nonlinear evolution equations. Eur. Phys. J. B 88, 187 (2015)
Wang, C.J.: Lump solution and integrability for the associated Hirota bilinear equation. Nonlinear Dyn. 87, 2635–2642 (2017)
Ma, W.X.: Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Lett. A 379, 1975–1978 (2015)
Ma, W.X., Qin, Z., Lü, X.: Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn. 84, 923–931 (2016)
Lü, X., Ma, W.X.: Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation. Nonlinear Dyn. 85, 1217–1222 (2016)
Lü, X., Chen, S.T., Ma, W.X.: Constructing lump solutions to a generalized Kadomtsev–Petviashvili–Boussinesq equation. Nonlinear Dyn. 86, 1–12 (2016)
Ma, W.X., Zhou, Y., Dougherty, R.: Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations. Int. J. Mod. Phys. B 30, 1640018 (2016)
Yang, J.Y., Ma, W.X.: Lump solutions to the BKP equation by symbolic computation. Int. J. Mod. Phys. B 30, 1640028 (2016)
Zhang, H.Q., Ma, W.X.: Lump solutions to the (\(2+1\))-dimensional Sawada–Kotera equation. Nonlinear Dyn. 87, 2305–2310 (2017)
Yang, J.Y., Ma, W.X., Qin, Z.Y.: Lump and lump-soliton solutions to the (\(2+1\))-dimensional Ito equation. Anal. Math. Phys. (2017). doi:10.1007/s13324-017-0181-9
Ramani, A.: Inverse scattering, ordinary differential equations of Painlev\(\acute{e}\) type and Hirota’s bilinear formalism. Ann. N. Y. Acad. Sci. 373, 54–67 (1981)
Dye, J.M., Parkera, A.: On bidirectional fifth-order nonlinear evolution equations, Lax pairs, and directionally dependent solitary waves. J. Math. Phys. 42, 2567 (2001)
Wazwaz, A.M.: The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations. Appl. Math. Comput. 184, 1002–1014 (2007)
Dye, J.M., Parker, A.: A bidirectional Kaup–Kupershmidt equation and directionally dependent solitons. J. Math. Phys. 43, 4921–4949 (2002)
Ma, Y.L., Geng, X.G.: Darboux and B\(\ddot{a}\)cklund transformations of the bidirectional Sawada–Kotera equation. Appl. Math. Comput. 218, 6963–6965 (2012)
Lai, X.J., Cai, X.O.: Adomian decomposition method for approximating the solutions of the bidirectional Sawada–Kotera Equation. Z. Naturforsch. 65a, 658–664 (2010)
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This work is supported by the National Natural Science Foundation of China (11661060,11571008).
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Lv, J., Bilige, S. Lump solutions of a (\(2+1\))-dimensional bSK equation. Nonlinear Dyn 90, 2119–2124 (2017). https://doi.org/10.1007/s11071-017-3788-3
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DOI: https://doi.org/10.1007/s11071-017-3788-3