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

Haci Mehmet Baskonus, Hasan Bulut


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.


The sine-Gordon expansion method; generalized-Zakharov equation with complex structure; complex hyperbolic function solution; dark soliton solutions.

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Yang, X.F. Deng Z.C. and Wei Y., A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Advances in Difference Equations, 117, 1-17 (2015). Crossref

Bekir A., Application of the -expansion method for nonlinear evolution equations, Phisics Letters Physics A, 372(19), 3400-3406 (2008). Crossref

Bekir A., Boz A., Exact solutions for nonlinear evolution equations using Exp-function method Physics Letters A, 372(10), 1619–1625 (2008). Crossref

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).

Alofi A. S., Extended Jacobi Elliptic Function Expansion Method for Nonlinear Benjamin-Bona Mahony Equations, International Mathematical Forum, 7(53), 2639–2649 (2012).

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). Crossref

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).

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). Crossref

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).

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).

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).

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: Mathematics-Informatics, 36, 235-243 (2013).

Tuluce Demiray, S., and Bulut, H., New Exact Solutions of the New Hamiltonian Amplitude Equation and Fokas Lenells Equation, Entropy, 17, 6025-6043 (2015). Crossref

Tuluce Demiray, S., and Bulut, H., Generalized Kudryashov method for nonlinear fractional double sinh-Poisson equation, J. Nonlinear Sci. Appl., 9, 1349-1355 (2016).

Taghizadeh N., Mirzazadeh M., and Farahrooz F., Exact Solutions of the Generalized- Zakharov (GZ) Equation by the Infinite Series Method, Applications and Applied Mathematics: An International Journal, 05(2), 621 – 628 (2010).

Yan C., A simple transformation for nonlinear waves, Physics Letters A, 224, 77-84 (1996). Crossref

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). Crossref

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).

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). Crossref

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). Crossref

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). Crossref

Beyer, W. H.,CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press (1987).

Weisstein, E.W. Concise Encyclopedia of Mathematics, 2nd ed.; CRC: New York, NY, USA (2002).

Jin S., Markowich P.A. and Zheng C., Numerical simulation of a generalized Zakharov system, Journal of Computational Physics, 201, 376–395 (2004). Crossref

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). Crossref

Malomed, B., Anderson, D., Lisak, M., Quiroga-Teixeiro, M.L. and Stenflo, L. Dynamics of Solitary Waves in the Zakharov Model Equations. Physical Review E, 55, 962-968 (1997). Crossref

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). Crossref

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). Crossref

Ebadi G., E. V. Krishnan and Biswas A., Solitons and Cnoidal waves of the Klein-Gordon Zakharov Equation in Plasmas, Pramana, 79(2), 185-198 (2012). Crossref

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).

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).

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). Crossref

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). Crossref

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). Crossref

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). Crossref

Biswas A., Dispersion-Managed Solitons in Optical Fibers, Journal of Optics A., 4(1), 84-97 (2002). Crossref

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).

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). Crossref

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).

Johnson S., Biswas A., Breather Dynamics of the Sine-Gordon Equation, Communications in Theoretical Physics, 59(6), 664-670 (2013). Crossref

Johnson S., Biswas A., Topological Soliton Perturbation for Sine-Gordon Equation with Full Nonlinearity, Physics Letters A., 374(34), 3437-3440 (2010). Crossref

Johnson S., Chen F. Biswas A., Mathematical Structure Of Topological Solitons Due To Sine-Gordon Equation, Applied Mathematics and Computation, 217(13), 6372-6378 (2011). Crossref

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). Crossref

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). Crossref

Wang G.W., Xu T., Zedan H.,Abazari R., Triki H., Biswas A., Solitary Waves, Shock Waves And Other Solutions To Nizhniki-Novikov-Veselov Equation, Applied and Computational Mathematics, 14(3), 260-283 (2015).

Erdogan F., Amiraliyev G.M., Fitted finite difference method for singularly perturbed delay differential equations, Numerical Algorithms, 59, 131-145 (2012). Crossref

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). Crossref

Hammouch Z., Mekkaoui T., A Laplace-variational iteration method for solving the homogeneous Smoluchowski coagulation equation, (2010).

Hammouch Z., Multiple solutions of steady MHD flow of dilatant fluids, (2008).

Baskonus H.M., Altan Koç D., Bulut H., Dark and New Travelling Wave Solutions to the Nonlinear Evolution Equation, Optik- International Journal for Light and Electron Optics, 127, 8043-8055 (2016). Crossref

Baskonus H.M., New acoustic wave behaviors to the Davey–Stewartson equation with power-law nonlinearity arising in fluid dynamics, Nonlinear Dynamics, DOI:10.1007/s11071-016-2880-4, 1-7 (2016). Crossref



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