Bilinear Bäcklund transformation, soliton and periodic wave solutions for a $$\varvec{(3 + 1)}$$ ( 3 + 1 ) -dimensional variable-coefficient generalized shallow water wave equation

Hirota, R.: The Direct Method in Soliton Theory. Springer, Berlin (1980)

Chai, J., Tian, B., Xie, X.Y.: Conservation laws, bilinear Bäcklund transformations and solitons for a nonautonomous nonlinear Schrödinger equation with external potentials. Commun. Nonlinear Sci. Numer. Simul. 39, 472–480 (2016)

Lan, Z.Z., Gao, Y.T., Yang, J.W.: Bäcklund transformation and Lax pair for a (2 + 1)-dimensional Broer-Kaup-Kupershmidt system in the shallow water of uniform depth. Commun. Nonlinear Sci. Numer. Simul. 44, 360–372 (2017)

Li, H.M., Tian, B., Xie, X.Y.: Soliton and rogue-wave solutions for a (2 + 1)-dimensional fourth-order nonlinear Schrödinger equation in a Heisenberg ferromagnetic spin chain. Commun. Nonlinear Dyn. 86, 369–380 (2016)

Wang, L., Zhang, J.H., Wang, Z.Q.: Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation. Phys. Rev. E 93, 6448–56 (2012)

Wang, L., Zhu, Y.J., Qi, F.H.: Modulational instability, higher-order localized wave structures, and nonlinear wave interactions for a nonautonomous Lenells-Fokas equation in inhomogeneous fibers. Chaos 25, 142 (2015)

Guo, R., Zhao, H.H., Wang, Y.: A higher-order coupled nonlinear Schrödinger system: solitons, breathers, and rogue wave solutions. Nonlinear Dyn. 83, 2475–2484 (2016)

Zhao, H.H., Zhao, X.J., Hao, H.Q.: Breather-to-soliton conversions and nonlinear wave interactions in a coupled Hirota system. Appl. Math. Lett. 61, 8–12 (2016)

Guo, R., Hao, H.Q.: Breathers and multi-soliton solutions for the higher-order generalized nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 18, 2426C2435 (2013)

Hirota, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations And Inverse Scattering Transform. Cambridge University Press, London (1990)

El-Sabbagh, M.F., Ahmad, A.T.: Nonclassical symmetries for nonlinear partial differential equations via compatibility. Commun. Theor. Phys. 56, 611–616 (2011)

Bai, C.J., Zhao, H., Xu, H.Y.: New traveling wave solutions for a class of nonlinear evolution equations. Int. J. Mod. Phys. B 25, 319–327 (2011)

Savescu, M., Bi-Irawy, A.H., Hilal, E.M.: Optical solitons in birefringent fibers with four-wave mixing for Kerr law nonlinearity. Optoelectron. Adv. Mat. 9, 10–13 (2015)

Mirzazadeh, M., Eslami, M., Zerrad, E.: Optical solitons in nonlinear directional couplers by sineCcosine function method and Bernoullis equation approach. Nonlinear Dyn. 81, 1–17 (2015)

Louaked, M., Hanich, L.: TVD scheme for the shallow water equations. J. Hydraul. Res 36, 363–378 (1998)

Jimbo, M., Miwa, T.: Solitons and infinite dimensional Lie algebras. Publ. Res. I. Math. Sci. 19, 943–1001 (1983)

Chen, Y., Liu, R.: Some new nonlinear wave solutions for two (3 + 1)-dimensional equations. Appl. Math. Comput. 260, 397–411 (2015)

Zeng, Z.F., Liu, J.G., Nie, B.: Multiple-soliton solutions, soliton-type solutions and rational solutions for the (3 + 1)-dimensional generalized shallow water equation in oceans, estuaries and impoundments. Nonlinear Dyn. 86, 667–675 (2016)

Tian, B., Gao, Y.T.: Beyond travelling waves: a new algorithm for solving nonlinear evolution equations. Comput. Phys. Commun. 95, 139–142 (1996)

Zayed, E.M.E.: Traveling wave solutions for higher dimensional nonlinear evolution equations using the $$(\frac{G^{\prime }}{G})$$ ( G ′ G ) -expansion method. J. Appl. Math. Inform. 28, 383–395 (2010)

Tang, Y.N., Ma, W.X., Xu, W.: Grammian and Pfaffian solutions as well as Pfaffianization for a (3 + 1)-dimensional generalized shallow water equation. Chin. Phys. B 21, 85–91 (2012)

Wu, J., Xing, X., Geng, X.: Generalized bilinear differential operators application in a (3 + 1)-dimensional generalized shallow water equation. Adv. Math. Phys. 4, 1–9 (2015)

Tang, Y.N., Ma, W.X., Xu, W.: Wronskian determinant solutions of the (3 + 1)-dimensional Jimbo-Miwa equation. Appl. Math. Comput. 21, 8722–8730 (2011)

Darvishi, M.T., Najafi, M.: Some complexiton type solutions of the (3 + 1)-dimensional Jimbo-Miwa equation. Int. J. Comput. Math. Sci. 1, 25–27 (2012)

Tang, X.Y., Liang, Z.F.: Variable separation solutions for the (3 + 1)-dimensional Jimbo-Miwa equation. Phys. Lett. A 6, 398–402 (2006)

Turgut, Öziş, Aslan, İsmail: Exact and explicit solutions to the (3 + 1)-dimensional Jimbo-Miwa equation via the exp-function method. Phys. Lett. A 47, 7011–7015 (2008)

Ma, S.H., Fang, J.P., Zheng, C.L.: New exact solutions for the (3 + 1)-dimensional Jimbo-Miwa system. Chaos Soliton Fract. 3, 1352–1355 (2009)

Li, Z., Dai, Z.: Abundant new exact solutions for the (3 + 1)-dimensional Jimbo-Miwa equation. J. Math. Anal. Appl. 2, 587–590 (2010)

Hong, W., Oh, K.S.: New solitonic solutions to a (3 + 1)-dimensional Jimbo-Miwa equation. Comput. Math. Appl. 5, 29–31 (2000)

Bell, E.T.: Exponential polynomials. Ann. Math. 35, 258–277 (1934)

Lambert, F., Loris, I., Springael, J.: On a direct bilinearization method: Kaup’s higher-order water wave equation as a modified nonlocal Boussinesq equation. J. Phys. A 28, 5325–5334 (2007)

Fan, E., Chow, K.W.: Darboux covariant Lax pairs and infinite conservation laws of the (2+ 1)-dimensional breaking soliton equation. J. Math. Phys. 2, 023504–10 (2011)

Hirota, R.: Exact envelope soliton solutions of a nonlinear wave equation. J. Math. Phys. 7, 805–809 (1973)

Hirota, R., Ohta, Y.: Hierarchies of coupled soliton equations I. J. Phys. Soc. 3, 798–809 (1991)

Tian, S., Zhang, H.: Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations. J. Math. Anal. Appl. 2, 585–608 (2010)

Tian, S.F., Zhang, H.Q.: Riemann theta functions periodic wave solutions and rational characteristics for the (1 + 1)-dimensional and (2 + 1)-dimensional Ito equation. Chaos Soliton Fract. 47, 27–41 (2013)

Tian, F.S., Ma, P.L.: On the quasi-periodic wave solutions and asymptotic analysis to a(3 + 1)-dimensional generalized Kadomtsev-Petviashvili equation. Commun. Theor. Phys. 8, 245–258 (2014)