Abstract
In this paper, the linear superposition principle and Hirota bilinear equations are simultaneously employed to handle two new (3+1)-dimensional Jimbo–Miwa equations. The corresponding resonant multi-soliton solutions and the related wave numbers are formally established, which are totally different from the previously reported ones. Moreover, the extracted N-soliton waves and dispersion relations have distinct physical structures compared to solutions obtained by Wazwaz. Finally, five graphical representations are portrayed by taking definite values to free parameters which demonstrates various versions of traveling solitary waves. The results show the proposed approach provides enough freedom to construct multi-soliton waves that may be related to a large variety of real physical phenomena and, moreover, enriches the solution structure.
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References
Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theory. Springer and HEP, Berlin (2009)
Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)
Lin, F.H., et al.: Observation of interaction phenomena for two dimensionally reduced nonlinear models. Nonlinear Dyn. 94, 2643–2654 (2018)
Yu, J.P., Sun, Y.L.: Study of lump solutions to dimensionally reduced generalized KP equations. Nonlinear Dyn. 87(4), 2755–2763 (2017)
Li, Z.B.: New multi-soliton solutions for the (2+1)-dimensional Kadomtsev–Petviashvili equation. Commun. Theor. Phys. 49(3), 585 (2008)
Najafi, M., Jamshidi, A.: Multiple soliton solutions of (2+1)-dimensional potential Kadomtsev–Petviashvili equation. Int. J. Math. Comput. Phys. Electr. Comput. Eng. 5(12), 1964–1967 (2011)
Wazwaz, A.M.: Two B-type Kadomtsev–Petviashvili equations of (2+1) and (3+1) dimensions: multiple soliton solutions, rational solutions and periodic solutions. Comput. Fluids 86, 357–362 (2013)
Wazwaz, A.M.: Multiple kink solutions for two coupled integrable (2+1)-dimensional systems. Appl. Math. Lett. 58, 1–6 (2016)
Wazwaz, A.M., El-Tantawy, S.A.: Solving the (3+1)-dimensional KP-Boussinesq and BKP-Boussinesq equations by the simplified Hirota’s method. Nonlinear Dyn. 88(4), 3017–3021 (2017)
Alsayyed, O., et al.: Multi-soliton solutions of the BBM equation arisen in shallow water. J. Nonlinear Sci. Appl 9(4), 1807–1814 (2016)
Gao, X.Y.: Bäcklund transformation and shock-wave-type solutions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation in fluid mechanics. Ocean Eng. 96, 245–247 (2015)
Ablowitz, M.J., Musslimani, Z.H.: Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity 29(3), 915 (2016)
Ji, J.L., Zhu, Z.N.: On a nonlocal modified Korteweg-de Vries equation: Integrability, Darboux transformation and soliton solutions. Commun. Nonlinear Sci. Numer. Simul. 42, 699–708 (2017)
Darvishi, M., et al.: Exact propagating multi-anti-kink soliton solutions of a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation. Nonlinear Dyn. 83(3), 1453–1462 (2016)
Ma, W.X., Huang, T., Zhang, Y.: A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr. 82(6), 065003 (2010)
Ma, W.X., Fan, E.: Linear superposition principle applying to Hirota bilinear equations. Comput. Math. Appl. 61(4), 950–959 (2011)
Ma, W.Z., et al.: Hirota bilinear equations with linear subspaces of solutions. Appl. Math. Comput. 218(13), 7174–7183 (2012)
Zayed, E.M., Al-Nowehy, A.G.: The multiple exp-function method and the linear superposition principle for solving the (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff equation. Zeitschrift für Naturforschung A 70(9), 775–779 (2015)
Wazwaz, A.M.: Multiple-soliton solutions for extended (3+1)-dimensional Jimbo–Miwa equations. Appl. Math. Lett. 64, 21–26 (2017)
Wazwaz, A.M.: Multiple-soliton solutions for the Calogero–Bogoyavlenskii–Schiff, Jimbo–Miwa and YTSF equations. Appl. Math. Comput. 203(2), 592–597 (2008)
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We are most grateful to the anonymous referees for the help in improving the original manuscript. And it is gratefully acknowledged that this work was supported by the Ministry of National Defense, R.O.C.
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Kuo, CK., Ghanbari, B. Resonant multi-soliton solutions to new (3+1)-dimensional Jimbo–Miwa equations by applying the linear superposition principle. Nonlinear Dyn 96, 459–464 (2019). https://doi.org/10.1007/s11071-019-04799-9
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DOI: https://doi.org/10.1007/s11071-019-04799-9