New complex exact travelling wave solutions for the generalized- Zakharov equation with complex structures

In this paper, we apply the sine-Gordon expansion method which is one of the powerful methods to the generalized-Zakharov equation with complex structure. This algorithm yields new complex hyperbolic function solutions to the generalized-Zakharov equation with complex structure. Wolfram Mathematica 9 has been used throughout the paper for plotting two- and three-dimensional surface of travelling wave solutions obtained.

PDF

___

[1] Yang, X.F. Deng Z.C. and Wei Y., A RiccatiBernoulli sub-ODE method for nonlinear partial differential equations and its application, Advances in Difference Equations, 117, 1-17 (2015).
[2] Bekir A., Application of the G G? -expansion method for nonlinear evolution equations, Phisics Letters Physics A, 372(19), 3400-3406 (2008).
[3] Bekir A., Boz A., Exact solutions for nonlinear evolution equations using Exp-function method Physics Letters A, 372(10), 1619-1625 (2008).
[4] Bekir A., Boz A., Exact Solutions for a Class of Nonlinear Partial Differential Equations using Exp-Function Method, International Journal of Nonlinear Sciences and Numerical Simulation, 8(4), 505-512 (2011).
[5] Alofi A. S., Extended Jacobi Elliptic Function Expansion Method for Nonlinear BenjaminBona Mahony Equations, International Mathematical Forum, 7(53), 2639-2649 (2012).
[6] Khan K., Akbar M.A., Exact solutions of the (2+1)-dimensional cubic Klein-Gordon equation and the (3+1)-dimensional Zakharov- Kuznetsov equation using the modified simple equation method, Journal of the Association of Arab Universities for Basic and Applied Sciences, 15, 74-81 (2014).
[7] Zheng B., Application of A Generalized Bernoulli Sub-ODE Method For Finding Traveling Solutions of Some Nonlinear Equations, WSEAS Transactions on Mathematics, 7(11), 618-626 (2012).
[8] Ya L., Li K. and Lin, C. Exp-function method for solving the generalized-Zakharov equation, Appl. Math. Comput, Vol. 205, pp.197-201 (2008).
[9] Salam A., Uddin S. and Dey P., Generalized Bernoulli Sub-ODE Method and its Applications, Annals of Pure and Applied Mathematics, 10(1),1-6 (2015).
[10] Bulut H., Baskonus H.M. and Belgacem F.B.M., The Analytical Solutions of Some Fractional Ordinary Differential Equations By Sumudu Transform Method, Abstract and Applied Analysis, Volume 2013, Article ID 203875, 6 pages, (2013).
[11] Hammouch Z., Mekkaoui T., Traveling-wave solutions of the Generalized Zakharov Equation with time-space fractional derivatives, Mathematics In Engineering, Science And Aerospace,5(4), 489-499 (2014).
[12] Bulut, H., Baskonus, H.M and Tuluce, S., The solutions of homogeneous and nonhomogeneous linear fractional differential equations by variational iteration method, Acta Universitatis Apulensis: MathematicsInformatics, 36, 235-243 (2013).
[13] Tuluce Demiray, S., and Bulut, H., New Exact Solutions of the New Hamiltonian Amplitude Equation and Fokas Lenells Equation, Entropy, 17, 6025-6043 (2015).
[14] Tuluce Demiray, S., and Bulut, H., Generalized Kudryashov method for nonlinear fractional double sinh-Poisson equation, J. Nonlinear Sci. Appl., 9, 1349-1355 (2016).
[15] Taghizadeh N., Mirzazadeh M., and Farahrooz F., Exact Solutions of the GeneralizedZakharov (GZ) Equation by the Infinite Series Method, Applications and Applied Mathematics: An International Journal, 05(2), 621 - 628 (2010).
[16] Yan C., A simple transformation for nonlinear waves, Physics Letters A, 224, 77-84 (1996).
[17] Yan Z., Zhang H., New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics, Physics Letters A, 252, 291-296 (1999).
[18] Zhen-Ya Y., Hong-Oing Z., En-Gui F., New explicit and travelling wave solutions for a class of nonlinear evaluation equations, Acta Physica Sinica, 48(1), 1-5 (1999).
[19] Chen S., Grelu P., Soto-Crespo J. M., Dark- and bright-rogue-wave solutions for media with long-wave-short-wave resonance Phys. Rev. E 89 011201 (2014).
[20] Nistazakis, H. E., Frantzeskakis D. J., Balourdos P S, Tsigopoulos A, Malomed B A, Dynamics of anti-dark and dark solitons in (2+1)-dimensional generalized nonlinear Schrödinger equation, Phys. Lett. A, 278, 68-76 (2000).
[21] Crosta M., Fratalocchi A., Trillo S., Bistability and instability of dark-antidark solitons in the cubic-quintic nonlinear Schrödinger equation, Phys. Rev. A, 84, 063809 (2011).
[22] Beyer, W. H.,CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press (1987).
[23] Weisstein, E.W. Concise Encyclopedia of Mathematics, 2nd ed.; CRC: New York, NY, USA (2002).
[24] Jin S., Markowich P.A. and Zheng C., Numerical simulation of a generalized Zakharov system, Journal of Computational Physics, 201, 376-395 (2004).
[25] Sun Y., Hu H., Zhang J., New Exact Explicit Solutions of the Generalized Zakharov Equation via the First Integral Method, Open Journal of Applied Sciences, 4, 249-257 (2014).
[26]Malomed, B., Anderson, D., Lisak, M., QuirogaTeixeiro, M.L. and Stenflo, L. Dynamics of Solitary Waves in the Zakharov Model Equations. Physical Review E, 55, 962-968 (1997).
[27] Biswas A., Zerrad E., Gwanmesia J. and Khouri R., 1-Soliton Solution Of The Generalized Zakharov Equation In Plasmas By He's Variational Principle, Applied Mathematics and Computation, 215(12), 4462-4466 (2010).
[28] Suarez P., and Biswas A., Exact 1-Soliton Solution Of The Zakharov Equation In Plasmas With Power Law Nonlinearity, Applied Mathematics and Computation, 217(17), 7372- 7375 (2011).
[29] Ebadi G., E. V. Krishnan and Biswas A., Solitons and Cnoidal waves of the KleinGordon Zakharov Equation in Plasmas, Pramana, 79(2), 185-198 (2012).
[30] Morris R., Kara A.H. and Biswas A., Soliton Solution And Conservation Laws of The Zakharov Equation in Plasmas With Power Law Nonlinearity, Nonlinear Analysis: Modelling and Control, 18(2), 153-159 (2013).
[31] Bouthina S. A., Zerrad E., Biswas A., Kinks And Domain Walls Of The Zakharov Equation In Plasmas, Proceedings of the Romanian Academy, Series A., 14(4), 281-286 (2013).
[32] Song M., Bouthina A., Zerrad E. and Biswas A., Domain Wall And Bifurcation Analysis Of The Klein-Gordon Zakharov Equation In (1+2)- Dimensions With Power Law Nonlinearity, Chaos, 23(3), 033115 (2013).
[33] M. Eslami, Vajargah B. F., Mirzazadeh M.,Biswas A., Application Of First Integral Method To Fractional Partial Differential Equations, Indian Journal of Physics, 88(2), 177-184 (2014).
[34] Sassaman R., Heidari A., Biswas A., Topological And Non-Topological Solitons Of Non-Linear Klein-Gordon Equations By He's Semi-Inverse Variational Principle, Journal of the Franklin Institute, 347(7), 1148-1157 (2010).
[35] Biswas A., Milovic D. and Ranasinghe A., Solitary Waves of Boussinesq Equation in A Power Law Media, Communications in Nonlinear Science and Numerical Simulation, 14(11), 3738-3742 (2009).
[36] Biswas A., Dispersion-Managed Solitons in Optical Fibers, Journal of Optics A., 4(1), 84- 97 (2002).
[37] Ganaini S.E., Mirzazadeh M. and Biswas A., Solitons And Other Solutions to Long-Short Wave Resonance Equation, Applied and Computational Mathematics, 14(3), 248-259 (2015).
[38] Fabian A.L., Kohl R. and Biswas A., Perturbation Of Topological Solitons Due To Sine-Gordon Equation And Its Type, Communications in Nonlinear Science and Numerical Simulation, 14(4), 1227-1244 (2009).
[39] Suarez P., Johnson S.,Biswas A., Chebychev Split-Step Scheme for The Sine-Gordon Equation In (2+1)-Dimensions, International Journal of Nonlinear Sciences and Numerical Simulation, 14(1), 69-75 (2013).
[40] Johnson S., Biswas A., Breather Dynamics of the Sine-Gordon Equation, Communications in Theoretical Physics, 59(6), 664-670 (2013).
[41] Johnson S., Biswas A., Topological Soliton Perturbation for Sine-Gordon Equation with Full Nonlinearity, Physics Letters A., 374(34), 3437-3440 (2010).
[42] Johnson S., Chen F. Biswas A., Mathematical Structure Of Topological Solitons Due To SineGordon Equation, Applied Mathematics and Computation, 217(13), 6372-6378 (2011).
[43] Johnson S., Suarez P.,Biswas A., New Exact Solutions For The Sine-Gordon Equation In (2+1)-Dimensions, Computational Mathematics and Mathematical Physics, 52(1), 98-104 (2012).
[44] Xu Y., Savescu M., Khan K.R., Mahmood M.F., Biswas A., and Belic M., Soliton Propagation Through Nanoscale Waveguides In Optical Metamaterials, Optics and Laser Technology, 77, 177-186 (2016).
[45] Wang G.W., Xu T., Zedan H.,Abazari R., Triki H., Biswas A., Solitary Waves, Shock Waves And Other Solutions To Nizhniki-NovikovVeselov Equation, Applied and Computational Mathematics, 14(3), 260-283 (2015).
[46] Erdogan F., Amiraliyev G.M., Fitted finite difference method for singularly perturbed delay differential equations, Numerical Algorithms, 59, 131-145 (2012).
[47] Amiraliyev G.M., Erdogan F., A finite difference scheme for a class of singularly perturbed initial value problems for delay differential equations, Numerical Algorithms, 52, 663-675 (2009).
[48] Hammouch Z., Mekkaoui T., A Laplacevariational iteration method for solving the homogeneous Smoluchowski coagulation equation, https://hal.archives-ouvertes.fr/hal- 00592481/document (2010).
[49] Hammouch Z., Multiple solutions of steady MHD flow of dilatant fluids, http://arxiv.org/pdf/0802.1851.pdf (2008).
[50] Baskonus H.M., Altan Koç D., Bulut H., Dark and New Travelling Wave Solutions to the Nonlinear Evolution Equation, OptikInternational Journal for Light and Electron Optics, 127, 8043-8055 (2016).
[51] Baskonus H.M., New acoustic wave behaviors to the Davey-Stewartson equation with powerlaw nonlinearity arising in fluid dynamics, Nonlinear Dynamics, DOI:10.1007/s11071- 016-2880-4, 1-7 (2016).