Abstract
In this paper, a generalized \((2+1)\)-dimensional nonlinear wave equation is obtained by extending the generalized \((2+1)\)-dimensional Hirota bilinear equation into a more generalized form. The obtained new equation is useful in describing nonlinear wave phenomena in nonlinear optics, shallow water and oceanography. Based on the bilinear method, the N-soliton solutions of the generalized \((2+1)\)-dimensional nonlinear wave equation are obtained. M-lump solutions are investigated by applying the long wave limit to the N-soliton solutions. The propagation orbits and velocities of the M-lump wave are analyzed. The high-order breather waves are obtained by establishing the complex conjugate relations in the parameters of the N-solitons. Furthermore, the interaction hybrid solutions are constructed, which contain hybrid solutions composed of breathers, solitons and lumps. The dynamical behaviors of the hybrid solutions are systematically analyzed via numerical simulations. The obtained results will enrich the study of theory of the nonlinear localized waves.
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References
Manakov, S.V., Zakharov, V.E., Bordag, L.A., Its, A.R., Matveev, V.B.: Two-dimensional solitons of the Kadomtsev–Petviashvili equation and their interaction. Phys. Lett. A 63, 205–206 (1977)
Satsuma, J., Ablowitz, M.J.: Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys. 20, 1496–1503 (1979)
Zhang, Y., Liu, Y.P., Tang, X.Y.: M-lump solutions to a \((3+1)\)-dimenisonal nonlinear evolution equation. Comput. Math. Appl. 76, 592–601 (2018)
Zhang, Y., Liu, Y.P., Tang, X.Y.: M-lump and interactive solutions to a \((3+1)\)-dimensional nonlinear system. Nonlinear Dyn. 93, 2533–2541 (2018)
An, H.L., Feng, D.L., Zhu, H.X.: General M-lump, high-order breather and localized interaction solutions to the \(2+1\)-dimensional Sawada–Kotera equation. Nonlinear Dyn. 98, 1275–1286 (2019)
Tan, W., Dai, Z.D., Yin, Z.Y.: Dynamics of multi-breathers, N-solitons and M-lump solutions in the \((2+1)\)-dimensional KdV equation. Nonlinear Dyn. 96, 1605–1614 (2019)
Manafian, J., Lakestani, M.: N-lump and interaction solutions of localized waves to the \((2+1)\)-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation. J. Geom. Phys. 150, 103598 (2020)
Guo, H.D., Xia, T.C., Hu, B.B.: High-order lumps, high-order breathers and hybrid solutions for an extended \((3+1)\)-dimensional Jimbo–Miwa equation in fluid dynamics. Nonlinear Dyn. (2019). https://doi.org/10.1007/s11071-020-05514-9
Zhang, Z., Yang, X.Y., Li, W.T., Li, B.: Trajectory equation of a lump before and after collision with line, lump, and breather waves for \((2+1)\)-dimensional Kadomtsev–Petviashvili equation. Chin. Phys. B 28, 110201 (2019)
Ma, W.X.: Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Lett. A 379, 2305–2310 (2015)
Zhang, H.Q., Ma, W.X.: Lump solutions to the \((2+1)\)-dimensional Sawada–Kotera equation. Nonlinear Dyn. 87, 1605–1614 (2017)
Ren, B., Ma, W.X., Yu, J.: Characteristics and interactions of solitary and lump waves of a \((2+1)\)-dimensional coupled nonlinear partial differential equation. Nonlinear Dyn. 96, 717–727 (2019)
Ma, W.X., Qin, Z.Y., 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(2), 1217–1222 (2016)
Guan, X., Liu, W.J., Zhou, Q., Biswas, A.: Some lump solutions for a generalized \((3+1)\)-dimensional Kadomtsev–Petviashvili equation. Appl. Math. Comput. 366, 124757 (2020)
Zhao, Z.L., Han, B.: Residual symmetry, Bäcklund transformation and CRE solvability of a \((2+1)\)-dimensional nonlinear system. Nonlinear Dyn. 94, 461–474 (2018)
Wang, C.J.: Lump solution and integrability for the associated Hirota bilinear equation. Nonlinear Dyn. 87, 2635–2642 (2017)
Wang, X.B., Tian, S.F., Yan, H., Zhang, T.T.: On the solitary waves, breather waves and rogue waves to a generalized \((3+1)\)-dimensional Kadomtsev–Petviashvili equation. Comput. Math. Appl. 74, 556–563 (2017)
Osman, M.S., Wazwaz, A.M.: A general bilinear form to generate different wave structures of solitons for a \((3+1)\)-dimensional Boiti–Leon–Manna–Pempinelli equation. Math. Methods Appl. Sci. 74, 6277–6283 (2019)
Clarkson, P.A., Dowie, E.: Rational solutions of the Boussinesq equation and applications to rogue waves. Trans. Math. Appl. (2017). https://doi.org/10.1093/40imatrm/tnx003
Zhaqilao: A symbolic computation approach to constructing rogue waves with a controllable center in the nonlinear systems. Comput. Math. Appl. 75: 3331–3342 (2018)
Zhao, Z.L., He, L.C.: Multiple lump solutions of the \((3+1)\)-dimensional potential Yu–Toda–Sasa–Fukuyama equation. Appl. Math. Lett. 95, 114–121 (2019)
Zhao, Z.L., He, L.C., Gao, Y.B.: Rogue wave and multiple lump solutions of the \((2+1)\)-dimensional Benjamin–Ono equation in fluid mechanics. Complexity 2019, 8249635 (2019)
He, L.C., Zhao, Z.L.: Multiple lump solutions and dynamics of the generalized the generalized \((3+1)\)-dimensional KP equation. Mod. Phys. Lett. B 33, 2050167 (2020)
Liu, Y.Q., Wen, X.Y., Wang, D.S.: The N-soliton solution and localized wave interaction solutions of the \((2+1)\)-dimensional generalized Hirota–Satsuma–Ito equation. Comput. Math. Appl. 77, 947–966 (2019)
Li, W.T., Zhang, Z., Yang, X.Y., Li, B.: High-order breathers, lumps and hybrid solutions to the \((2+1)\)-dimensional fifth-order KdV equation. Int. J. Mod. Phys. B 33, 1950255 (2019)
Cao, Y.L., He, J.S., Mihalache, D.: Families of exact solutions of a new extended \((2+1)\)-dimensional Boussinesq equation. Nonlinear Dyn. 91, 2593–2605 (2018)
Sun, B.N., Wazwaz, A.M.: General high-order breathers and rogue waves in the \((3+1)\)-dimensional KP–Boussinesq equation. Commun. Nonlinear Sci. Numer. Simul. 64, 1–13 (2018)
Yue, Y.F., Huang, L.L., Chen, Y.: N-solitons, breathers, lumps and rogue wave solutions to a \((3+1)\)-dimensional nonlinear evolution equation. Comput. Math. Appl. 75, 2538–2548 (2018)
Deng, G.F., Gao, Y.T., Su, J.J., Ding, C.C.: Multi-breather wave solutions for a generalized \((3+1)\)-dimensional Yu–Toda–Sasa–Fukuyama equation in a two-layer liquid. Appl. Math. Lett. 98, 177–183 (2019)
Zhao, Z.L., Chen, Y., Han, B.: Lump soliton, mixed lump stripe and periodic lump solutions of a \((2+1)\)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation. Mod. Phys. Lett. B 31, 1750157 (2017)
Chen, L., Chen, J.C., Chen, Q.Y.: Mixed lump-soliton solutions to the two-dimensional Toda lattice equation via symbolic computation. Nonlinear Dyn. 96, 1531–1539 (2019)
Hu, C.C., Tian, B., Yin, H.M., Zhang, C.R., Zhang, Z.: Dark breather waves, dark lump waves and lump wave-soliton interactions for a \((3+1)\)-dimensional generalized Kadomtsev–Petviashvili equation in a fluid. Comput. Math. Appl. 78, 166–177 (2019)
Lü, J.Q., Bilige, S.D., Temuer, C.L.: The study of lump solution and interaction phenomenon to \((2+1)\)-dimensional generalized fifth-order KdV equation. Nonlinear Dyn. 91, 1669–1676 (2018)
Liu, J.G., He, Y.: Abundant lump and lump-kink solutions for the new \((3+1)\)-dimensional generalized Kadomtsev–Petviashvili equation. Nonlinear Dyn. 92, 1103–1108 (2018)
He, C.H., Tang, Y.N., Ma, J.L.: New interaction solutions for the \((3+1)\)-dimensional Jimbo–Miwa equation. Comput. Math. Appl. 76, 2141–2147 (2019)
Xu, G.Q.: Painlevé analysis, lump-kink solutions and localized excitation solutions for the \((3+1)\)-dimensional Boiti–Leon–Manna–Pempinelli equation. Appl. Math. Lett. 97, 81–87 (2019)
Yan, X.W., Tian, S.F., Dong, M.J., Zou, L.: Bäcklund transformation, rogue wave solutions and interaction phenomena for a \((3+1)\)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation. Nonlinear Dyn. 92, 709–720 (2018)
Wang, J., An, H.L., Li, B.: Non-traveling lump solutions and mixed lump-kink solutions to \((2+1)\)-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation. Mod. Phys. Lett. B 33, 1950262 (2019)
Fang, T., Wang, Y.H.: Lump-stripe interaction solutions to the potential Yu–Toda–Sasa–Fukuyama equation. Anal. Math. Phys. 9, 1481–1495 (2019)
Fang, T., Wang, Y.H.: Interaction solutions for a dimensionally reduced Hirota bilinear equation. Comput. Math. Appl. 76, 1476–1485 (2018)
Tan, W.: Evolution of breathers and interaction between high-order lump solutions and N-solitons (\(N \rightarrow \infty \)) for Breaking Soliton system. Phys. Lett. A 383, 125907 (2019)
Rao, J.G., He, J.S., Mihalache, D., Cheng, Y.: PT-symmetric nonlocal Davey–Stewartson I equation: general lump-soliton solutions on a background of periodic line waves. Appl. Math. Lett. 104, 106246 (2020)
Dong, J.J., Li, B., Yuen, M.W.: Soliton molecules and mixed solutions of the \((2+1)\)-dimensional bidirectional Sawada–Kotera equation. Commun. Theor. Phys. 72, 025002 (2020)
Hua, Y.F., Guo, B.L., Ma, W.X., Lü, X.: Interaction behavior associated with a generalized \((2+1)\)-dimensional Hirota bilinear equation for nonlinear waves. Appl. Math. Model. 74, 184–198 (2019)
Lou, S.Y.: Soliton molecules and asymmetric solitons in fluid systems via velocity resonance. arXiv:1909.03399 (2019)
Zhang, Z., Yang, X.Y., Li, B.: Soliton molecules and novel smooth positons for the complex modified KdV equation. Appl. Math. Lett. 103, 106168 (2020)
Acknowledgements
This work is supported by Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (No. 2019L0531), Shanxi Province Science Foundation for Youths (No. 201901D211274), and the Fund for Shanxi “1331KIRT”.
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Zhao, Z., He, L. M-lump, high-order breather solutions and interaction dynamics of a generalized \((2 + 1)\)-dimensional nonlinear wave equation. Nonlinear Dyn 100, 2753–2765 (2020). https://doi.org/10.1007/s11071-020-05611-9
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DOI: https://doi.org/10.1007/s11071-020-05611-9