Soliton solutions of Hirota equation and Hirota-Maccari system

Year 2016, Volume: 4 Issue: 3, 231 - 238, 30.09.2016

M. M. El-borai H. M. El-owaidy Hamdy M. Ahmed A. H. Arnous

Abstract

In this paper, the trial equation method is presented
to seek the exact solutions of two nonlinear partial differential equations
(NLPDEs), namely, the Hirota equation and the Hirota-Maccari system. The
obtained solutions are solitary, topological, singular solitons and singular
periodic waves. This method is powerful, effective and it can be extended to
many NLPDEs.

Keywords

Solitons , exact solutions

References

• P. Gray and S. Scott, Chemical oscillations and instabilities, Clarendon, Oxford, 1990.
• M. J. Ablowitz and P. A. Clarkson, Soliton, Nonlinear evolution equations and inverse scattering transform, Cambridg University Press, New York,1991.
• V. O. Vakhnenko, E. J. Parkes and A. J. Morrison, A Backlund transformation and the inverse scattering transform method for the generalized Vakhnenko equation, Chaos, Solitons and Fractals, 17 (2003), 683.
• Mahmoud M. El-Borai, Exact solutions for some nonlinear fractional parabolic fractional partial differential equations, Journal of Applied Mathematics and Computation 206 (2008,) 141-153.
• Mahmoud M. El-Borai, Afaf A. Zaghrout and Amal L. Elshaer, Exact solutions for nonlinear partial differential equations by using the extended multiple Riccati equations expansion method, Inter. J. of Research and Reviews in Applied Sciences, 9(3) (2011), 370-381.
• E. M. E. Zayed and A.H. Arnous, DNA dynamics studied using the homogeneous balance method, Chinese Physics Letters, 29(8) (2012), 080203.
• R. Hirota, Direct method of finding exact solutions of nonlinear evolution equations, in: R. Bulloygh, P. Coudrey (Eds.), Bäcklund transformation, Springer, Berlin, (1980).
• M. L. Wang, X. Li, Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos Solit. Fract., 24 (2005), 1257-1268.
• D. S. Wang, H. Q. Zhang, Further improved F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equations. Chaos Solit. Fract., 25 (2005) 601-610.
• M. A. M. Jawad, M. D. Petkovic and A. Biswas, Modified simple equation method for nonlinear evolution equations, Appl. Math. Comput., 217 (2010), 869-877.
• E. M. E. Zayed and A.H. Arnous, Exact solutions of the nonlinear ZK-MEW and the Potential YTSF equations using the modified simple equation method, AIP Conf. Proc., 1479 (2012), 2044-2048.
• M. L. Wang and X. Li , J. Zhang, The (G'/G)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A., 372 (2008) 417-423.
• C.S. Liu, Trial equationmethod to nonlinear evolution equations with rank inhomogeneous: mathematical discussions and its applications, Commun. Theor. Phys. 45 (2006), 219-223.
• M. Mirzazadeh, A. H. Arnous, M. F. Mahmood, E. Zerrad and A. Biswas, Soliton solutions to resonant nonlinear schrödinger’s equation with time-dependent coefficients by trial solution approach, Nonlinear Dynamics, 81(8) (2015), 277-282.
• M. Mirzazadeh, A. H. Arnous and M. Eslami, Dark optical solitons of Biswas-Milovic equation with dual-power law nonlinearity, The European Physical Journal Plus, 130(4) (2015), 1-7.
• S. T. Demiray and Y. Pandir, H. Bulut, All exact travelling wave solutions of Hirota equation and Hirota-Maccari system, Optik, 127(4) (2016), 1848-1859.
• M. Eslami, M. A. Mirzazadeh, A. Neirameh, New exact wave solutions for Hirota equation, Pramana J. of Physics, 84(1) (2015), 3-8.