On the New Hyperbolic Function Solutions to the (2+1)-Dimensional BKK System

Firat University, Faculty of Science, Department of Mathematics, Elazig, Turkey.

Abstract

In this paper, we successfully implement the powerful sine-Gordon expansion method the (2+1)dimensional BKK system. We succeed in constructing some new analytical hyperbolic function solutions. We check all the analytical solutions by using Wolfram Mathematica 9, and they are indeed verified to be the solutions of (2+1)dimensional BKK system. We also plot the two- and threedimensional surfaces for all the analytical solutions obtained in this paper using the same computer program. We finally, submit a comprehensive conclusion.

Keywords

The sine-Gordon expansion method; the (2+1)dimensional BKK system; hyperbolic function solution

References

[1] H.M. Baskonus and H. Bulut, “Exponential prototype structures for (2+1)-dimensional Boiti-LeonPempinelli systems in mathematical physics”, Waves in Random and Complex Media, 26(2), 189-196, 2016.

[2] H.M. Baskonus, H. Bulut and F.M. Belgacem, “Analytical Solutions for Nonlinear Long-Short Wave Interaction Systems with Highly Complex Structure”, Journal of Computational and Applied Mathematics, 15 pages, 2016.

[3] M.A. Abdou and A.A. Soliman, “Variational Iteration method for Solving Burger’s and Coupled Buger’s Equations”, Journal Computational and Applied Mathematics . 181, 245-251, 2005.

[4] M.A. Noor, K.I. Noor, A. Waheed and E.A. Al-Said, “Some new solitonary solutions of the modified Benjamin-Bona-Mahony equation”, Computer and Mathematics with Applications. 62, 2126-2131, 2011.

[5] E.M.E. Zayed and K.A.E. Alurrfi, “The Modified Extented Tanh-Function Method and its Applications to the Generalized KdV-mKdV Equation with AnyOrder Nonlinear Term”, International Journal of Enviromental Engineering Science and Technology. 1(8), 165-170, 2013.

[6] C. Ye, “New Exact Solutions for the Generalized mKdV Equation with variable coefficients”, Applied Mathathematical Sciences, 5(75), 1484-1490, 2010.

[7] M. Song, S. Li and J. Cao, “New Exact Solutions for the (2+1)-Dimensional Broer-Kaup-Kupershmidt Equations”, Abstract and Applied Analysis, 9 pages, 2010.

[8] M.M. Hossain, H.O. Roshid, M.A.N. Sheikh, “Abundant Exact Traveling Wave Solutions of the (2+1)-Dimensional Couple Broer-Kaup Equations”, Journal for Foundations and Applications of Physics, 3(1), 13 pages, 2016.

[9] E. Yomba, “The Modified Extended Fan SubEquation Method and its Application to the (2+1)Dimensional Broer-Kaup-Kuperschmidt Equations”, Chaos, Solitons and Fractals. 27(1), 187-196, 2006.

[10] Y. Jin-Ping and L. Sen-Yue, “Abundant Coherent Structures of the (2+1)-Dimensional Broer-KaupKupershmidt Equation”, Zeitschrift fur Naturforschung A. 56, 619-625, 2001.

[11] N. Taghizadeh and A. Neirameh, “New Complex Travelling Wave Solutions to the Nonlinear BroerKaup-Kupershmidt System”, Middle-East Journal of Scientific Research. 11(11), 1542-1545, 2012.

[12] F. Jian-Ping and Z. Chun-Long, “New Exact Excitations and Soliton Fission and Fusion for the (2+1)-Dimensional Broer-Kaup-Kupershmidt System”, Chinese Physics B. 14(4), 669-675, 2005.

[13] X.Y. Wen, “N-Soliton Solutions and Localized Structures for the (2+1)-Dimensional Broer-KaupKupershmidt System”, Nonlinear Analysis: RealWorld Applications. 12(6), 3346-3355, 2011.

[14] C.L. Bai and H. Zhao, “A New General Algebraic Method and its Applications to the (2+1)Dimensional Broer-Kaup-Kupershmidt Equations”, Applied Mathematics and Computation. 217(4), 1719-1732, 2010.

[15] X.R. Hu and Y. Chen, “Nonlocal Symmetries, Consistent Riccati Expansion Integrability, and their Applications of the (2+1)-Dimensional Broer-KaupKupershmidt System”, Chin. Phys. B. 24, 090203 2015.

[16] J. Lin and H.M. Li, “Painleve Integrability and Abundant Localized Structures of (2+1)-Dimensional Higher Order Broer-Kaup System”, Z. Naturforsch. 57, 929-936, 2002.

[17] A.G. Davodi, D.D. Ganji and A. Asgari, “Finding General and Explicit Solutions (2+1)-Dimensional Broer-Kaup-Kupershmidt System Nonlinear Equation by Exp-Function Method”, Applied Mathematics and Computation. 217(4), 1415-1420, 2010.

[18] C.L. Chen and S.Y. Lou, “CTE Solvability and Exact Solution to the Broer-Kaup System”, Chin. Phys. Lett. 30(11), 110202, 2013.

[19] C. Yan, “A Simple Transformation for Nonlinear Waves”, Physics Letters A, 22(4), 77-84, 1996. [20] H.M. Baskonus, “New AcousticWave Behaviors to the Davey-Stewartson Equation with Power Nonlinearity Arising in Fluid Dynamics”, Nonlinear Dynamics, 86(1), 177-183, 2016.

[21] Z. Yan and H. Zhang, “New Explicit and exact Travelling Wave Solutions for a System of Variant Boussinesq equations in Mathematical Physics”, Physics Letters A, 252, 291-296, 1999.

[22] E.W. Weisstein, “Concise Encyclopedia of Mathematics”, 2nd edition. New York: CRC Press, 2002.