The New Exact Solutions of the New Coupled Konno-Oono Equation By Using Extended Simplest Equation Method

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1 Applied Mathematical Sciences, Vol. 12, 2018, no. 6, HIKARI Ltd, The New Exact Solutions of the New Coupled Konno-Oono Equation By Using Extended Simplest Equation Method Montri Torvattanabun, Papraporn Juntakud, Adsadawut Saiyun and Nattawut Khansai Department of Mathematics, Faculty of Science and Technology Loei Rajabhat University, Loei 42000, Thailand Corresponding author Copyright c 2018 Montri Torvattunabun et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The extended simplest equation method is employed to solve new coupled Konno-Oono equation. By means of this scheme, we found new exact solution. The method is straight for ward, concise and effective, it can also be applied to other nonlinear solution in mathematical physics. Mathematics Subject Classification: 35E05, 35D99, 35A20, 35Q51 Keywords: the new coupled Konno-Oono equation, the extended simplest equation method, exact solution 1 Introduction Nowadays solitary wave solution of nonlinear evolution equation play an important vole in various branches of scientific disciplines, such as plasma physics, optical fibers, fluid mechanics, chemical physics, solid state physics and many other scientific applications. The search for the exact solution of nonlinear evolution equation has attracted a lot of attention. In the past years, many powerful methods for finding exact solution have been proposed

2 294 Montri Torvattunabun et al. and applied to the NLEEs, including Modified Kudryashov method [1], Exponential rational function method [2], Exp-function method [3], Improved tanh-coth method [4,5], Generalized Kudryashov method [6], Lie symmetry method [7], F-expansion method [8], Modified(w/g)- expansion method [9], Exp( φ(ξ))-expansion method [10], Sine-cosine functions method [11], Semiinverse variational principle [12] and so on. Another powerful method has been presented by Kudryashov [13,14], this method is called the simplest equation method. Recently, Bilige et al [15,16], extended and improved this method which is called the extended simplest equation method. After wards, several researchers applied this method to obtain new exact solutions for nonlinear PDEs [15-20]. In this paper, we focus on using the extended and improved this method for finding exact solutions of the new coupled Konno-Oono equation [21-24]. In section 2 we briefly describe the extended and improved this method, in section 3 is applied extended simplest equation method to the new coupled Konno-Oono equation. The last section is some conclusions are given. 2 Description of the extended simplest equation method (ESEM) Consider the nonlinear partial differential equation in the variables x and t P (u, u x, u t, u xx, u xt,...) = 0. (1) In order to make better use of the extended simplest equation method, we take its main steps as follows : step 1. By using the traveling wave transformation u(x, t) = u(ξ), ξ = µ(x ct). (2) We reduce Eq. (1) into the ordinary differential equation, namely O(u, µu, cµu, µ 2 u, c 2 µ 2 u,...) = 0, (3) where the prime denotes the derivation with respect to ξ. step 2. We suppose that the solution for Eq.(3) can be expressed by a finite series of the form u(ξ) = N i=0 N 1 a i ( φ φ )i + b j ( φ φ )j ( 1 ), (4) φ where a i, b j (i = 0, 1, 2, 3,..., N, j = 0, 1, 2, 3,..., N 1) are constants and a N b N 1 0. the function φ = φ(ξ) satisfies the second order linear ODE in the from j=0 φ = σ δφ, (5)

3 The new exact solutions of the new coupled Konno-Oono equation 295 where δ and v are constants. Eq.(5) has three types of general solution with double arbitrary parameters as follows : and A 1 cosh δξ + A 2 sinh δξ + σ, δ < 0 δ φ(ξ) = cos δξ + A 2 sin δξ + σ, δ > 0 δ σ 2 ξ2 + ξ + A 2, δ = 0 ( φ φ )2 = (δ 2 δa 2 2 σ2 δ )( 1 φ )2 δ + 2σ φ, δ < 0, (6) (δa δa 2 2 σ2 δ φ )2 δ + 2σ, φ δ > 0, (7) (A 2 1 2σA 2 ) 1 + 2σ, φ 2 φ δ = 0 where and A 2 are arbitrary constants. N is a positive integer that will be determined by balancing the highest order derivative term with the highest order nonlinear term. step 3. By substituting Eq.(4) into Eq.(3) and using the second order linear ODE Eq.(5) and expressions Eq.(7), collecting all terms with the same 1 order of and ( 1 )( φ ) together, the left-hand side of Eq.(3) is converted φ i φ i φ into another polynomial in 1 and ( 1 )( φ ). Equating each coefficient of this φ i φ i φ different power terms to zero yields a set of nonlinear algebraic equations for a i, b j (i = 0, 1, 2, 3,..., N, j = 0, 1, 2, 3,..., N 1), c, µ, δ and v. We can construct a variety of exact solutions of Eq.(1). 3 The new coupled Konno-Oono equation In this section, we will construct some new exact solutions of the new coupled Konno-Oono equation system. This system has the form : [21-24] v t (x, t) + 2u(x, t)u x (x, t) = 0, (8) u xt (x, t) 2v(x, t)u(x, t) = 0. (9) At first, we use the traveling wave transformation formula Eq.(2), we get the following equations cµ 2 u 2uv = 0, (10) cµv + 2µuu = 0. (11)

4 296 Montri Torvattunabun et al. Integrating Eq.(11), we get v = 1 c (u2 + p), (12) where p is constant of integration. Putting Eq.(12) into Eq.(10), we obtain c 2 µ 2 u + 2u 3 + 2pu = 0. (13) Balancing the terms u of Eq.(13) is the from and u 3 in Eq.(13), yield N = 1. Suppose the solutions u(ξ) = a 0 + a 1 ( φ φ ) + b 0( 1 φ ) (14) where a 0, a 1 and b 0 are constants to be determined later and function φ = φ(ξ) satisfies the the linear second order ODE Eq.(5).By substituting Eq.(14) into Eq.(13) and using Eqs.(5) and (7), the left-hand sind of Eq.(13) become a polynomial in 1 and ( 1 )( φ ). Equating each coefficient of this polynomial to φ i φ i φ zero, yields a system of algebraic equations for a 0, a 1, b 0, δ, µ and v. After solving these algebraic equations with help of symbolic computation, we obtain the following set of nontrivial solutions : If δ = 0, then we get the trivial solution. So, this case is rejected. If δ < 0, we obtain p a 0 = 0, a 1 = ± δ, b A 2 1δ 2 p A 2 2δ 2 p pσ 2 0 = ±, c = ± 2 p δ µ δ (15) substituting Eq.(15) into Eq.(14) and the general solutions Eq.(7) of Eq.(5) into Eq.(14) and making use of Eq.(12). We construct new exact solutions of the new coupled Konno-Oono equation as follows: u 1,2 (x, t) = ± A p 1 δ sinh(ψ 1)( δ) 3 2 +A p 2 δ cosh(ψ 1)( δ 3 +Ω v 1 (x, t) = µ 2 p δ cosh(ψ 1 )δ+a 2 sinh(ψ 1 )δ+σ (16) p δ sinh(ψ 1)( δ) 3 2 +A 2 p δ cosh(ψ 1)( δ) 3 2 +Ω cosh(ψ 1 )δ+a 2 sinh(ψ 1 )δ+σ (17) u 2,2 (x, t) = ± A p 1 δ sinh(ψ 1)( δ) 3 2 +A p 2 δ cosh(ψ 1)( δ 3 Ω v 2 (x, t) = µ 2 p δ cosh(ψ 1 )δ+a 2 sinh(ψ 1 )δ+σ (18) p δ sinh(ψ 1( δ) 3 2 +A 2 p δ cosh(ψ 1)( δ) 3 2 Ω cosh(ψ 1 )δ+a 2 sinh(ψ 1 )δ+σ (19)

5 The new exact solutions of the new coupled Konno-Oono equation 297 u 3,2 (x, t) = ± A p 1 δ sinh(ψ 1)( δ) 3 2 +A p 2 δ cosh(ψ 1)( δ 3 +Ω v 3 (x, t) = µ 2 p δ cosh(ψ 1 )δ+a 2 sinh(ψ 1 )δ+σ (20) p δ sinh(ψ 1)( δ) 3 2 +A 2 p δ cosh(ψ 1)( δ) 3 2 +Ω cosh(ψ 1 )δ+a 2 sinh(ψ 1 )δ+σ (21) u 4,2 (x, t) = ± A p 1 δ sinh(ψ 1)( δ) 3 2 +A p 2 δ cosh(ψ 1)( δ 3 Ω v 4 (x, t) = µ 2 p δ cosh(ψ 1 )δ+a 2 sinh(ψ 1 )δ+σ (22) p δ sinh(ψ 1)( δ) 3 2 +A 2 p δ cosh(ψ 1)( δ) 3 2 Ω cosh(ψ 1 )δ+a 2 sinh(ψ 1 )δ+σ (23) Where ψ 1 = δµ( 2 µ Ω = p δ t + x), Ψ 1 = δµ( 2 p µ δ A 2 1δ 2 p A 2 2δ 2 p pσ 2 t + x), (24) If δ > 0,we obtain p a 0 = 0, a 1 = ± δ, b A 2 1δ 2 p + A 2 2δ 2 p pσ 2 0 = ±, c = ± 2 p δ µ δ (25) substituting Eq.(25) into Eq.(14) and the general solutions Eq.(7) of Eq.(5) into Eq.(14) and making use of Eq.(12). We construct new exact solutions of the new coupled Konno-Oono equation as follows: u 5,2 (x, t) = ± A p 1 δ sin(ψ 2)(δ) 3 2 A p 2 δ cos(ψ 2)(δ) 3 2 +Θ v 5 (x, t) = µ 2 p δ cos(ψ 2 )δ+a 2 sin(ψ 2 )δ+σ (26) p δ sin(ψ 2)(δ) 3 2 A 2 p δ cos(ψ 2)(δ) 3 2 +Θ cos(ψ 2 )δ+a 2 sin(ψ 2 )δ+σ (27) u 6,2 (x, t) = ± A p 1 δ sin(ψ 2)(δ) 3 2 A p 2 δ cos(ψ 2)(δ) 3 2 Θ v 6 (x, t) = µ 2 p δ cos(ψ 2 )δ+a 2 sin(ψ 2 )δ+σ (28) p δ sin(ψ 2)(δ) 3 2 A 2 p δ cos(ψ 2)(δ) 3 2 Θ cos(ψ 2 )δ+a 2 sin(ψ 2 )δ+σ (29) u 7,2 (x, t) = ± A p 1 δ sin(ψ 2)(δ) 3 2 A p 2 δ cos(ψ 2)(δ) 3 2 +Θ v 7 (x, t) = µ 2 p δ cos(ψ 2 )δ+a 2 sin(ψ 2 )δ+σ (30) p δ sin(ψ 2)(δ) 3 2 A 2 p δ cos(ψ 2)(δ) 3 2 +Θ cos(ψ 2 )δ+a 2 sin(ψ 2 )δ+σ (31)

6 298 Montri Torvattunabun et al. Where u 8,2 (x, t) = ± A p 1 δ sin(ψ 2)(δ) 3 2 A p 2 δ cos(ψ 2)(δ) 3 2 Θ v 8 (x, t) = µ 2 p δ ψ 2 = δµ( 2 µ Θ = cos(ψ 2 )δ+a 2 sin(ψ 2 )δ+σ (32) p δ sin(ψ 2)(δ) 3 2 A 2 p δ cos(ψ 2)(δ) 3 2 Θ cos(ψ 2 )δ+a 2 sin(ψ 2 )δ+σ p δ t + x), Ψ 2 = δµ( 2 p µ δ A 2 1δ 2 p + A 2 2δ 2 p pσ 2 (33) t + x), (34) 4 Conclusions This paper obtained new exact solutions of the new coupled Konno-Oono equation. The extended simplest equation method is efficient and practically well suited for use in finding more exact solutions. To the best of our knowledge, symbolic computation systems played a crucial role in the computations. We have checked our solutions by putting them back into the original equation. Acknowledgements. The authors would like to the Department of Mathematics, Faculty of Science and Technology, Loei Rajabhat University and the Department of Mathematics, Faculty of Applied Science, King Mongkuts University of Technology North Bangkok. References [1] D. Kumer, A. R. Seadawy, A. K. Joardar, Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology, Chinese Journal of Physics, 56 (2018), [2] M. Kaplan, K.Hosseini, Investigation of exact solutions for the Tzitzica type equations in nonlinear optics, Optik- International Journal for Light and Electron Optics, 154 (2018), [3] Rahmatullah, R. Ellahi, S.T. Mohyud-din,U. Khan, Exact traveling wave solutions of fractional order Boussinesq-like equations by applying Expfunction method, Results in Physics, 8 (2018),

7 The new exact solutions of the new coupled Konno-Oono equation 299 [4] M. Torvattanabun, J. Simmapim, D. Saennuad, T. Somaumchan, The improved generalized tanh-coth method applied to sixth-order solitary wave equation, Journal of Mathematics, 2017 (2017), [5] M. Torvattanabun, S. Koonprasert, Exact traveling wave solutions to the Zakharov-Kuznetsov-Benjamin-Bona-Mahony nonlinear evolution equation using the VIM combined with the improved generalized tanh-coth method, Applied Mathematical Sciences, 11 (2017), no. 64, [6] K. A. Gepreel, T. A. Nofal, A. A. Alesmari, Exact solutions for nonlinear integro-partial differential equations using the generalized Kudryashov method, Journal Egyptian Math. Society, 25 (2017), no. 4, [7] A.R. Adem, Symbolic computation on exact solutions of a coupled KadomtsevPetviashvili equation: Lie symmetry analysis and extended tanh method, Computers and Mathematics with Applications, 74 (2017), no. 8, [8] M. S. Islam, K. Khan, M. A. Akbar, Application of the improved F- expansion method with Riccati equation to find the exact solution of the nonlinear evolution equations, Journal of the Egyptian Mathematical Society, 25 (2017), no. 1, [9] K. A. Gepreel, Exact solutions for nonlinear integral member of KadomtsevPetviashvili hierarchy differential equations using the modified (w/g)- expansion method, Computers and Mathematics with Applications, 72 (2016), no. 9, [10] M. N. Alam, C. Tunc, An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predatorprey system, Alexandria Engineering Journal, 55 (2016), no. 2, [11] K. P. Raslan,T. S. EL-Danaf, K. K. Ali, New exact solutions of coupled generalized regularized long wave equations, Journal of the Egyptian Mathematical Society, 25 (2017), no. 4, [12] A. Biswas, Q. Zhou, H. Triki, M. Z. Ullah, M. Asma, S. P. Moshokoa, M. Belic, Resonant optical solitons with parabolic and dual-power laws by semi-inverse variational principle, Journal of Modern Optics, 65 (2018), no. 2,

8 300 Montri Torvattunabun et al. [13] N. A. Kudryashov, Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos Solitons and Fractals, 24 (2005), [14] N. A. Kudryashov, Exact solitary waves of the Fisher equation, Physics Letters A, 342 (2005), [15] S. Bilige, T.Chaolu, X. Wang, Application of the extended simplest equation method to the coupled Schrodinger-Boussinesq equation, Applied Mathematics and Computation, 224 (2013), [16] S. Bilige,T. Chaolub, An extended simplest equation method and its application to several forms of the fifth-order KdV equation, Applied Mathematics and Computation, 216 (2010), [17] M. O. Al-Amr, S. El-Ganaini, New exact traveling wave solutions of the (4+1)-dimensional Fokas equation, Computers and Mathematics with Applications, 74 (2017), [18] S. El-Ganaini, M. A. Akbar, Conservation Laws and Multiple Simplest Equation Methods to an Extended Quantum Zakharov-Kuznetsov Equation, Nonlinear Analysis and Differential Equations, 4 (2016), no. 9, [19] Y. M. Zhao, New Exact Solutions for a Higher-Order Wave Equation of KdV Type Using the Multiple Simplest Equation Method, Journal of Applied Mathematics, 2014 (2014), [20] Y. J. Yang, L. Yao, New Exact Solutions of the Modified Benjamin-Bona- Mahony Equation, International Conference on Artificial Intelligence and Computer Science, 2016 (2016), [21] K. Konno, H. Oono, New coupled integrable dispersionless equations, Journal of Physical Society of Japan, 63 (1994), [22] M. D. Bashar, G. Mondal, K. Khan, A. Bekir, Traveling wave solutions of new coupled Konno-Oono equation, New Trends in Mathematical Sciences, 4 (2016),

9 The new exact solutions of the new coupled Konno-Oono equation 301 [23] G. Yel, H. M. Baskonus, H. Bulut, Novel archetypes of new coupled KonnoOono equation by using sine-gordon expansion method, Opt. Quant Electron, 49 (2017), [24] Z. F. Kocak, H. Bulut, D. A. Koc, H. Mehmet, Prototype traveling wave solutions of new coupled Konno-Oono equation, Optik - International Journal for Light and Electron Optics, 127 (2016), Received: February 6, 2018; Published: February 28, 2018