Md Khorshed Alam

Determination of the rich structural wave dynamic solutions to the caudrey–dodd–gibbon equation and the lax equation

  • Authors Details :  
  • Md. Khorshed Alam,  
  • Md. Dulal Hossain,  
  • M. Ali Akbar,  
  • Khaled A. Gepreel

Journal title : Letters in Mathematical Physics

Publisher : Springer Science and Business Media LLC

Online ISSN : 1573-0530

Journal volume : 111

Journal issue : 4

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This article addresses the implementation of the new generalized (G'∕G)-expansion method to the Caudrey–Dodd–Gibbon (CDG) equation and the Lax equation which are associated with the fifth-order KdV (fKdV) equation. The method works well to derive a variety of standard and functional closed-form wave solutions with distinct physical structures, such as, soliton, kink, periodic soliton, and bell-shaped soliton solutions. The solutions obtained using this method are useful and adequate than other methods. In order to understand the physical aspects and importance of the method, the attained solutions have been simulated graphically. The extracted results definitely establish that the new generalized(G'∕G)-expansion method is an effective mathematical tool to work out new solutions to different types of local nonlinear evolution equations emerging in applied science and engineering, but this method is not effective in solving nonlocal equations.

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DOI : https://doi.org/10.1007/s11005-021-01443-9

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  • (1). Weiss, J., Tabor, M., Carnevale, G.: The Painleve property for partial differential equations. J. Math. Phys. 24, 522–526 (1983)
  • (2). Ablowitz, M.J., Clarkson, P.A.: Solitons, nonlinear evolution equations and inverse scattering, p. 516. Cambridge University Press, Cambridge (1991)
  • (3). Beals, R., Coifman, R.R.: Scattering and inverse scattering for 1st order system. Commun. Pure. Appl. Math. 37, 39–90 (1984)
  • (4). Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons, p. 120. Springer, Berlin (1991)
  • (5). Cai, H., Jing, S., De-Chen, T., Nian-Nin, H.: Darboux transformation method for solving the Sine-Gordon equation in a laboratory reference. Chin. Phys. Lett. 19(7), 908–911 (2002)
  • (6). Cole, J.D.: On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math. 9, 225–236 (1951)
  • (7). Hopf, E.: The partial differential equation ut+uux=µuxx. Commun. Pure Appl. Math. 3, 201–230 (1950)
  • (8). Liu, S., Fu, Z., Liu, S., Zhao, Q.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A. 289, 69–74 (2001)
  • (9). Alquran, M., Jarrah, A.: Jacobi elliptic function solutions for a two-mode KdV equation. J. King Saud. Uni-Sci. (2017). https://doi.org/10.1016/j.jksus.2017.06.010(inpress)
  • (10). Hirota, R.: Exact solution of the Korteweg-de-Vries equation for multiple collisions of solutions. Phys. Rev. Lett. 27, 1192–1194 (1971)
  • (11). Wazwaz, A.M.: Multiple-soliton solutions and multiple-singular soliton solutions for two higher-dimensional shallow water wave equations. Appl. Math. Comput. 211, 495–510 (2009)
  • (12). Mimura, M.R.: Bäcklund Transformation. Springer, Berlin, Germany (1978)
  • (13). Sayed, S.M.: The Bäcklund transformations, exact solutions, and conservation laws for the compound modified Korteweg-de Vries-Sine-Gordon equations which describe pseudo-spherical surfaces. J. Appl. Math. 7, 613065 (2013)
  • (14). Raslan, K.R., EL-Danaf, T.S., Alia, K.K.: New exact solution of coupled general equal width wave equation using sine-cosine function method. J. Egypt. Math. Soc. 25(3), 350–354 (2017)
  • (15). Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60, 650–654 (1992)
  • (16). Abdelkawy, M.A., Bhrawy, A.H., Zerrad, E., Biswas, A.: Application of tanh method to complex coupled nonlinear evolution equations. Acta Phys. Pol. A. 129, 278–283 (2016)
  • (17). Wazwaz, A.M.: Analytic study of the fifth order integrable nonlinear evolution equations by using the tanh method. Appl. Math. Comput. 174, 289–299 (2006)
  • (18). Islam, M.A., Akbar, M.A., Khan, K.: The improved F-expansion method and its application to the MEE circular rod equation and the ZKBBM equation. Cogent. Math. 4, 1378530 (2017)
  • (19). Akbar, M.A., Akinyemi, L., Yao, S.W., Jhangeer, A., Rezazadeh, H., Khater, M.M.A., Ahmad, H., Inc, M.: Soliton solutions to the Boussinesq equation through sine-Gordon method and Kudryashov method. Results Phys. 25, 1–10 (2021)
  • (20). Ravi, L.K., Ray, S.S., Sahoo, S.: New exact solutions of coupled Boussinesq-Burgers equations by Exp-function method. J. Ocean Engg. Sci. 2(1), 34–46 (2017)
  • (21). Kadkhoda, N., Jafari, H.: Analytical solutions of the Gerdjikov-Ivanov equation by using exp(-??)-expansion method. Optik Int. J. Light Electron Opt. 139, 72–76 (2017)
  • (22). Biswas, A., Yildirim, Y., Yasar, E., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical solitons for Lakshmanan-Porsezian-Daniel model by modified simple equation method. Optik 160, 24–32 (2018)
  • (23). Yassin, O., Alquran, M.: Constructing new solutions for some types of two-mode nonlinear equations. Appl. Math. Inf. Sci. 12(2), 361–367 (2018)
  • (24). Naher, H., Abdullah, F.A., Akbar, M.A.: The -expansion method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon equation. Math. Prob. Engg. 11, 218216 (2011)
  • (25). Hafez, M.G.: New traveling wave solutions of the (1+1)-dimensional cubic nonlinear Schrodinger equation using novel (G’/G)-expansion method. Beni-Seuf Univ. J. Appl. Sci. 5, 109–118 (2016)
  • (26). Hassaballa, A., Elzaki, T.M.: Applications of the improved (G’/G)-expansion method for solve Burgers-Fishers equation. J. Comput. Theor. Nanosci. 14(10), 4664–4668 (2017)
  • (27). Naher, H., Abdullah, F.A.: New approach of -expansion method and new approach of generalized -expansion method for nonlinear evolution equation. AIP Adv. 3(3), 032116 (2013)
  • (28). Naher, H., Abdullah, F.A.: New generalized -expansion method to the Zhiber-Shabat equation and Liouville equation. J. Phys. Conf. Series. 890, 012018 (2017)
  • (29). Miah, M.M., Ali, H.M.S., Akbar, M.A., Wazwaz, A.M.: Some applications of the -expansion method to find new exact solutions of NLEEs. Eur. Phys. J. Plus. 132(6), 252 (2017)
  • (30). Caudrey, P.J., Dodd, R.K., Gibbon, J.D.: A new hierarchy of Korteweg-de Vries equations. Proc. Roy. Soc. Lond. A 351, 407–422 (1976)
  • (31). Dodd, R.K., Gibbon, J.D.: The prolongation structure of higher order Korteweg-de Vries equations. Proc. Roy. Soc. Lond. A 358, 287–300 (1977)
  • (32). Weiss, J.: On classes of integrable systems and the Painleve´ property. J. Math. Phys. 25(1), 13–14 (1984)
  • (33). Hénon, M., Heiles, C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73–79 (1964)
  • (34). Wazwaz, A.M.: Abundant solitons solutions for several forms of the fifth-order KdV equation by using the tanh method. Appl. Math. Comput. 182, 283–300 (2006)
  • (35). Wazwaz, A.M.: Solitons and periodic solutions for the fifth-order KdV equation. Appl. Math. Lett. 19, 1162–1167 (2006)
  • (36). 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 (2006)
  • (37). Wazwaz, A.M.: Multiple-soliton solutions for the fifth order Caudrey-Dodd-Gibbon (CDG) equation. Appl. Math. Comput. 197, 719–724 (2008)
  • (38). Wazwaz, A.M.: N-soliton solutions for the combined KdV-CDG equation and the KdV-Lax equation. Appl. Math. Comput. 203, 402–407 (2008)
  • (39). Wazwaz, A.M.: Multiple soliton solutions for (2+1)-dimensional Sawada-Kotera and Caudrey-Dodd-Gibbon equations. Math. Meth. Appl. Sci. 34, 1580–1586 (2011)
  • (40). Bilige, S., Chaolu, T.: An extended simplest equation method and its application to several forms of the fifth-order KdV equation. Appl. Math. Comput. 216, 3146–3153 (2010)
  • (41). Salas, A.: Some exact solutions for the Caudrey-Dodd-Gibbon equation. Math. Phys. arXiv: 0805.2969v2 (2008)
  • (42). Xu, Y.G., Zhou, X.W., Yao, L.: Solving the fifth order Caudrey-Dodd-Gibbon (CDG) equation using the exp-function method. Appl. Math. Comput. 206, 70–73 (2008)
  • (43). Go´mez, C.A., Salas, A.H.: The generalized tanh-coth method to special types of the fifth-order KdV equation. Appl. Math. Comput. 203, 873–880 (2008)
  • (44). Jin, L.: Application of the variational iteration method for solving the fifth order Caudrey-Dodd-Gibbon equation. Int. Math. Forum 5(66), 3259–3265 (2010)
  • (45). Biswas, A., Ebadi, G., Triki, H., Yildirim, A., Yousefzadeh, N.: Topological soliton and other exact solutions to KdV-Caudrey-Dodd-Gibbon equation. Results. Math. 63, 687–703 (2013)
  • (46). Abbasbandy, S., Zakaria, F.S.: Soliton solutions for the fifth-order KdV equation with the homotopy analysis method. Nonlin. Dyn. 51, 83–87 (2008)
  • (47). Gomez, C.A.S.: Special forms of the fifth-order KdV equation with new periodic and soliton solutions. Appl. Math. Comput. 189, 1066–1077 (2007)
  • (48). Huang, X., Ling, L.: Soliton solutions for the nonlocal nonlinear Schrödinger equation. Eur. Phys. J. Plus 131(148), 1–11 (2016)
  • (49). Chen, J., Yan, Q.: Bright soliton solutions to a nonlocal nonlinear Schrödinger equation of reverse-time type. Nonlinear Dyn. 100, 2807–2816 (2020)
  • (50). Wang, Y.Y., Dai, C.Q., Xu, Y.Q., Zheng, J., Fan, Y.: Dynamics of nonlocal and localized spatiotemporal solitons for a partially nonlocal nonlinear Schrödinger equation. Nonlinear Dyn. 92, 1261–1269 (2018)



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