On The Existence Conditions for New Kinds of Solutions to (3+1)-Dimensional mKDV and mBBM Equations

Year 2021, Volume: 25 Issue: 1, 141 - 149, 01.02.2021

Hami Gündoğdu , Ömer Faruk Gözükızıl

Abstract

In this work, we consider (3+1) dimensional nonlinear partial differential equations, namely modified KdV and Benjamin-Bona-Mahony equations. Different types of solutions to these equations are derived by Jacobi elliptic sine function expansion method. Besides that, we introduce new types of solutions for two more modified forms of given equations. The gained solutions include exact, singular, periodic, and kink solutions. It is stated that some conditions related to the coefficients provide us with the existence of the gained solutions.

Keywords

Elliptic sine function method, (3+1)-Dimensional mKdV equation, (3+1)-Dimensional mBBM equation

References

• Referans1 A. Wazwaz, “The extended tanh method for new solitons solutions for many forms of the fifth order KdV equations,” Appl. Math. Comput. Vol. 184, pp.1002-1014, 2007.
• Referans2 A. Wazwaz, “New travelling wave solutions of different physical structures to generalized BBM equation,” Phys. Lett. A. Vol. 355, pp.358-362, 2006.
• Referans3 A. Wazwaz and M. A. Helal, “Non-linear variants of the BBM equation with compact and noncompact physical structures,” Chaos. Solitons. Fractals. Vol. 26, pp.767-776, 2005.
• Referans4 S. Elwakil, K. El-Labany, A. Zahran, and R. Sabry, “Modified extended tanh-function method for solving nonlinear partial differential equations,” Phys. Lett. A. Vol. 299, pp.179-88, 2002.
• Referans5 C. T. Yan, “A simple transformation for nonlinear waves,” Phys. Lett. A. Vol. 224, pp. 77-84, 1996.
• Referans6 T. Gao, and B. Tian, “Generalized hyperbolic-function method with computerized symbolic computation to construct the solitonic solutions to nonlinear equations of mathematical physics,” Comput. Phys. Commun. Vol. 133, pp. 158-164, 2001.
• Referans7 M. Wang, X. Li, and J. Zhang, “The (G’/G)- expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Phys. Lett. A. Vol. 372, pp. 417-423, 2008.
• Referans8 J. He and L. Zhang, “Generalized solitary solution and compacton-like solution of the Jaulent-Miodek equations using the Exp-function method,” Phys. Lett. A. Vol. 371, pp. 1044-1047, 2008.
• Referans9 H. Gündogdu and Ö. F. Gözükızıl, “Solving Benjamin-Bona-Mahony equation by usingˇ the sn-ns method and the tanh-coth method,” Math. Morav., Vol. 21, pp. 95-103, 2017.
• Referans10 D. J. Korteweg and G. De Vries, “On the change of long waves advancing in a rectangular canal and a new type of long stationary wave,” Phil. Mag., Vol. 39, pp. 422-443, 1835.
• Referans11 T. B. Benjamin, J. L. Bona, and J. J. Mahony, “Model equations for long waves in nonlinear dispersive systems,” Philos. Trans. R. Soc. London. Ser. A., Vol. 272, pp. 47-48, 1972.
• Referans12 W. Hereman, “Shallow water waves and solitarywaves, Ecnyclopedis of Complexity and Systems Science,” Springer Verlag, Heibelberg, Germany, 2009.
• Referans13 A. Wazwaz, “Exact soliton and kink solutions for new (3+1)-dimensional nonlinear modified equations of wave propagation,” Open. Eng., Vol. 7, pp. 169-174, 2017.
• Referans14 S. K. Liu, Z. T. Fu, and S. D. Liu, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Phys. Lett. A., Vol. 289, pp. 69-74, 2001.
• Referans15 H. Gündogdu and Ö. F. Gözükızıl, “On different kinds of solutions to a simplified modified form of Camassa-Holm equation” J. Appl. Math. & Comput. Mech. Vol. 18, pp. 31-40, 2019.
• Referans16 H. S. Alvaro, “Solving nonlinear partial differential equations by the sn-ns method,” Abstr. Appl. Analysis., Vol. 25, pp. 1-25, 2012.
• Referans17 H. Zhang, “Extended Jacobi elliptic function expansion method and its applications,” Commun. Non. Sci. Numer. Simul., Vol. 12, pp. 627-635, 2007.
• Referans18 H. Gündogdu and Ö. F. Gözükızıl, “On The New Type Of Solutions To Benney-Luke Equation,’’ Bol. Soc. Paran. Mat. doi:10.5269/bspm.41244.