Application of Extended Transformed Rational Function Method to Some (3+1) Dimensional Nonlinear Evolution Equations

The transformed rational function method can be considered as unification of the tanh type methods, the homogeneous balance method, the mapping method, the exp-function method and the F-expansion type methods. In this paper, we present complexiton solutions of (3+1) dimensional Korteweg-de Vries (KdV) equation and a new (3+1) dimensional generalized Kadomtsev-Petviashvili equation by using extended transformed rational function method which provides very useful and effective way to obtain complexiton solutions of nonlinear evolution equations.

Genişletilmiş Dönüştürülmüş Rasyonel Fonksiyon Metodunun Bazı (3+1) Boyutlu Lineer Olmayan Oluşum Denklemlerine Uygulanması

Dönüştürülmüş rasyonel fonksiyon metodu; tanh tipi metodlar, homojen denge metodu, resmetme metodu, üstel fonksiyon metodu ve F-açılım tipi metodların birleşimi olarak düşünülebilir. Biz bu çalışmada, lineer olmayan oluşum denklemlerinin kompleksiton çözümlerinin elde edilmesinde kullanışlı ve etkili bir yol olan genişletilmiş dönüştürülmüş rasyonel fonksiyon metodunu kullanarak (3+1) boyutlu KdV ve yeni (3+1) boyutlu genelleştirilmiş Kadomtsev-Petviashvili denklemlerinin kompleksiton çözümlerini elde edeceğiz.

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Matveev, VB., Salle, MA. 1980. Darboux transformation and solutions. Springer, Berlin.

Wang, ML. 1995. Solitary wave solutions for variant Boussinesq equations. Phys. Lett. A, 199: 169-172.

Zhou, Y., Wang, ML., Wang, YM., 2003. Periodic wave solutions to a coupled KdV equations with variable coefficients. Phys. Lett. A, 308: 31-36.

Parkes, EJ., Duffy BR. 1996. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput. Phys. Commun., 98: 288-300.

Ma, WX. 1993. Travelling wave solutions to a seventh order generalized KdV equation. Phys. Lett. A, 180: 221-224.

Fuchssteiner, B., Carillo, S. 1992. A new class of nonlinear partial differential equations solvable by quadratures. in: B. Fuchssteiner, W.A.J. Luxemburg (Eds.). Analysis and Geometry, BJ Wissenschaftsverlag, Mannheim; pp. 73-85. Ma, WX., Fuchssteiner, B. 1996. Explicit and exact solutions to a Kolmogorov-Petrovshii-Piskunov equation. Int. J. Nonlinear Mech., 31: 329-338.

Fan, EG. 2000. Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A, 77: 212-218.

Wazwaz, AM. 2006. New solitary wave solutions to the Kuramoto-Sivashinsky and the Kawahara equations. Appl. Math. Comput., 182: 1642-1650.

Wazwaz, AM. 2007. The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. Comput., 188: 1467-1475.

Lü, X., Chen, ST., Ma, WX. 2016. Constructing lump solutions to a generalized Kadomtsev-Petviashvili-Boussinesq equation. Nonlinear Dynam., 86: 523-534.

Lü, X., Ma, WX. 2016. Study of lump Dynamics based on a dimensionally reduced Hirota bilinear equation. Nonlinear Dynam., 85: 1217-1222.

Gao, LN., Zhao, XY., Zi, YY., Yu, J., Lü, X. 2016. Resonant behavior of multiple wave solutions to a Hirota bilinear equation. Comput. Math. Appl., 72: 1225-1229.

Lü, X., Ma, WX., Zhou, Y., Khalique, CM. 2016. Rational solutions to an extended Kadomtsev-Petviashvili-like equation with symbolic computation. Comput. Math. Appl., 71: 1560-1567. Lü, X., Ma, WX., Chen, ST., Khalique, CM. 2016. A note on rational solutions to a Hirota-Satsuma-like equation. Appl. Math. Lett., 58: 13-18.

Lü, X., Ma, WX., Yu, J., Khalique, CM. 2016. Solitary waves with the Madelung fluid description: A generalized derivative nonlinear Schrödinger equation. Commun. Nonlinear Sci., 31: 40-46.

Lü, X., Lin, F. 2016. Soliton excitations and shape-changing collisions in alpha helical proteins with interspine coupling at higher order. Commun. Nonlinear Sci., 32: 241-261.

Ma, WX., Lee, JH. 2009. A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation. Chaos Soliton. Fract., 41: 1356-1363.

Zhang, H., Ma, WX. 2014. Extended transformed rational function method and applications to complexiton solutions. Appl. Math. Comput., 230: 509-515. Ma, WX. 2002. Complexiton solutions to the Kortweg-de Vries equation. Phys. Lett. A, 301: 35-44.

Ma, WX. 2005. Complexiton solutions to integrable equations. Nonlinear Anal., 63: e2461-e2471.

Wazwaz, AM., El-Tantawy, SA. 2016. A new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation. Nonlinear Dynam., 84: 1107-1112.

Yang, XD., Ruan, HY. 2013. HBFGen: A maple package to construct the Hirota bilinear form for nonlinear equations. Appl. Math. Comput., 219: 8018-8025.