Abstract
We construct new (\(3+1\))-dimensional Burgers and Sharma–Tasso–Olver-type equations. We determine the dispersion relation for each of the newly derived models. By using the simplified Hirota’s method, we derive multiple-soliton solutions for each equation. We derive a generalized dispersion relation that works for both equations.
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Wazwaz, A.M., El-Tantawy, S.A.: A new integrable (3\(+\)1)-dimensional KdV-like model with its multiple-soliton solutions. Nonlinear Dyn. 83, 15291534 (2016)
Peng, Y.Z.: A new (2\(+\)1)-dimensional KdV equation and its localized structures. Commun. Theor. Phys. 54, 863–865 (2010)
Lu, X., Ma, W.X., Khalique, C.M.: A direct bilinear Backlund transformation of a (2\(+\)1) dimensional Korteweg-de Vries equation. Appl. Math. Lett. 50, 37–42 (2015)
Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)
Khalique, C.M., Biswas, A.: Optical solitons with power law nonlinearity using Lie group analysis. Phys. Lett. A 373, 2047–2049 (2009)
Biswas, A.: Solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coefficients. Nonlinear Dyn. 58(1–2), 345–348 (2009)
Biswas, A.: Solitary waves for power-law regularized long wave equation and R(m, n) equation. Nonlinear Dyn. 59(3), 423–426 (2010)
Biswas, A., Khalique, C.M.: Stationary solitons for nonlinear dispersive Schrodinger’s equation. Nonlinear Dyn. 63(4), 623–626 (2011)
Kolev, B.: Geometric differences between the Burgers and the Camassa–Holm equations. J. Nonlinear Math. Phys. 15(2), 116–132 (2008)
Ma, W.X., Abdeljabbar, A., Asaad, M.G.: Wronskian and Grammian solutions to a (3\(+\)1)-dimensional generalized KP equation. Appl. Math. Comput. 217, 10016–10023 (2011)
El-Tantawy, S.A., Moslem, W.M., Schlickeiser, R.: Ion-acoustic dark solitons collision in an ultracold neutral plasma. Phys. Scr. 90(8), 085606 (2016)
Burgers, J.M.: A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1, 171–199 (1948)
Wazwaz, A.M.: Burgers hierarchy: multiple kink solutions and multiple singular kink solutions. J. Frankl. Inst. 347, 618–626 (2010)
Wazwaz, A.M.: Combined equations of Burgers hierarchy: multiple kink solutions and multiple singular kink solutions. Phys. Scr. 82, 025001 (2010)
Wazwaz, A.M.: New (3\(+\)1)-dimensional evolution equations with Burgers and Sharma–Tasso–Olver equations constituting the main parts. Proc. Roman. Acad. Ser. A 16(1), 32–40 (2015)
Tasso, H.: Coles ansatz and extension of Burgers equation. Report IPP6/142 Ber. MPI fur Plasmaphysik (Garching). (1976)
Sharma, A.S., Tasso, H.: Connection between wave envelope and explicit solution of a nonlinear dispersive equation. Report IPP6/158 Ber. MPI fur Plasmaphysik (Garching). 1–10 (1970)
Olver, P.J.: Evolution equation possessing infinite many symmetries. J. Math. Phys. 18(6), 1212–1215 (1977)
Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theorem. Springer and HEP, Berlin (2009)
Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. 83(1), 591–596 (2015)
Wazwaz, A.M.: A KdV6 hierarchy: integrable members with distinct dispersion relations. Appl. Math. Lett. 45, 86–92 (2015)
Wazwaz, A.M.: New solutions for two integrable cases of a generalized fifth-order nonlinear equation. Mod. Phys. Lett. B 29(14), 1550065 (2015)
Wazwaz, A.M.: New (3+1)-dimensional nonlinear evolution equations with Burgers and Sharma–Tasso–Olver equations constituting the main part. Proc. Roman. Acad. Ser. A 16(1), 32–40 (2015)
Wazwaz, A.M.: New (3+1)-dimensional nonlinear evolution equations with mKdV equation constituting its main part: multiple soliton solutions. Chaos Solitons Fractals 76, 93–97 (2015)
Wazwaz, A.M.: A new integrable (2+1)-dimensional generalized breaking soliton equation: N-soliton solutions and travelling wave solutions. Commun. Theor. Phys. 66, 385–388 (2016)
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Wazwaz, AM., El-Tantawy, S.A. New (3\(\varvec{+}\)1)-dimensional equations of Burgers type and Sharma–Tasso–Olver type: multiple-soliton solutions. Nonlinear Dyn 87, 2457–2461 (2017). https://doi.org/10.1007/s11071-016-3203-5
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DOI: https://doi.org/10.1007/s11071-016-3203-5