Abstract
In this paper, we consider an extended KdV equation, which arises in the analysis of several problems in soliton theory. First, we converted the underlying equation into the Hirota bilinear form. Then, using the novel test function method, abundant multi-soliton solutions were obtained. Second, we have performed some distinct methods to extended KdV equation for getting some exact wave solutions. In this regard, Kudryashov’s simplest equation methods were examined. Third, the local conservation laws are deduced by multiplier/homotopy methods. Finally, the graphical simulations of the exact solutions are depicted.
Similar content being viewed by others
References
Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform. Cambridge University Press, Cambridge (1990)
Vakhnenko, V.O., Parkes, E.J., Morrison, A.J.: A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation. Chaos Solitons Fractals 17(4), 683 (2003)
Hirota, R.: Direct method of finding exact solutions of nonlinear evolution equations. In: Bullough, R., Caudrey, P. (eds.) Backlund Transformations. Springer, Berlin (1980)
Malfliet, W., Hereman, W.: The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys. Scr. 54(6), 563–568 (1996)
Wazwaz, A.M.: The tanh method for travelling wave solutions of nonlinear equations. Appl. Math. Comput. 154(3), 713 (2004)
Ma, W.X., Tingwen, H., Yi, Z.: A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr. 82(6), 6065003 (2010)
Ma, W.X., Yuncheng, Y.: Solving the Korteweg–de Vries equation by its bilinear form: Wronskian solutions. Trans. Am. Math. Soc. 357(5), 1753–1778 (2005)
He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30(3), 700–708 (2006)
Feng, Z.: The first-integral method to study the Burgers–Korteweg–de Vries equation. J. Phys. A Math. Gen. 35(2), 343 (2002)
Ma, W.X.: Bilinear equations, Bell polynomials and linear superposition principle. Journal of Physics: Conference Series. vol. 411 No. 1. IOP Publishing (2013)
Marchant, T.R.: Solitary wave interaction for the extended BBM equation. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 456, No. 1994, pp. 433–453. The Royal Society (2000)
Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)
Wazwaz, A.M.: The simplified Hirota’s method for studying three extended higher-order KdV-type equations. J. Ocean Eng. Sci. 1, 181–185 (2016)
Seadawy, A.R.: Fractional solitary wave solutions of the nonlinear higher-order extended KdV equation in a stratified shear flow: Part I. Comput. Math. Appl. 70, 345–352 (2015)
Miao, Q., Wang, Y., Chen, Y., Yang, Y.: PDEBellII: A Maple package for finding bilinear forms, bilinear Bcklund transformations, Lax pairs and conservation laws of the KdV-type equations. Comput. Phys. Commun. 185(1), 357–367 (2014)
Hirota, R.: The Direct Method in Soliton Theory, vol. 155. Cambridge University Press, Cambridge (2004)
Singh, M., Gupta, R.K.: Bäcklund transformations, Lax system, conservation laws and multisoliton solutions for Jimbo–Miwa equation with Bell-polynomials. Commun. Nonlinear Sci. Numer. Simul. 37, 362–373 (2016)
Singh, M., Gupta, R.K.: Exact solutions for nonlinear evolution equations using novel test function. Nonlinear Dyn. 86(2), 1171–1182 (2016)
Kudryashov, N.A.: Exact solitary waves of the Fisher equation. Phys. Lett. A 342, 99–106 (2005)
Kudryashov, N.A.: Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Solitons Fractals 24, 1217–1231 (2005)
Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 2248–2253 (2012)
Adem, A.R., Khalique, C.M.: Symbolic computation of conservation laws and exact solutions of a coupled variable-coefficient modified Korteweg–de Vries system. Comput. Math. Math. Phys. 56(4), 650–660 (2016)
Vitanov, N.K.: Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling wave solutions for a class of PDEs with polynomial nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 15, 2050–2060 (2010)
Yu, J., Wang, D.S., Sun, Y., Suping, Wu: Modified method of simplest equation for obtaining exact solutions of the Zakharov–Kuznetsov equation, the modified Zakharov–Kuznetsov equation, and their generalized forms. Nonlinear Dyn. 85, 2449–2465 (2016)
Adem, A.R., Khalique, C.M.: Exact solutions and conservation laws of a two-dimensional integrable generalization of the Kaup–Kupershmidt equation. J. Appl. Math. 647313, 6 (2013)
Cheviakov, A.F.: GeM software package for computation of symmetries and conservation laws of differential equations. Comput. Phys. Commun. 176, 48–61 (2007)
Cheviakov, A.F.: Symbolic computation of local symmetries of nonlinear and linear partial and ordinary differential equations. Math. Comput. Sci. 4, 203–222 (2010)
Adem, A.R., Khalique, C.M.: Conserved quantities and solutions of a (2+ 1)-dimensional Haragus-Courcelle–Il’ichev model. Comput. Math. Appl. 71(5), 1129–1136 (2016)
Moleleki, L.D., Muatjetjeja, B., Adem, A.R.: Solutions and conservation laws of a (3+1)-dimensional Zakharov–Kuznetsov equation. Nonlinear Dyn. 87, 2187 (2017)
Triki, H., Ak, T., Ekici, M., Sonmezoglu, A., Mirzazadeh, M., Kara, A.H., Aydemir, T.: Some new exact wave solutions and conservation laws of potential Korteweg–de Vries equation. Nonlinear Dyn. (2017). doi:10.1007/s11071-017-3467-4
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yıldırım, Y., Yaşar, E. An extended Korteweg–de Vries equation: multi-soliton solutions and conservation laws. Nonlinear Dyn 90, 1571–1579 (2017). https://doi.org/10.1007/s11071-017-3749-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-017-3749-x