Applied Mathematical Sciences, Vol. 12, 2018, no. 6, 293-301 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.8118 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
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)
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 2 1 + 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)
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)
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)
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), 75-85. https://doi.org/10.1016/j.cjph.2017.11.020 [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), 393-397. https://doi.org/10.1016/j.ijleo.2017.08.116 [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), 114-120. https://doi.org/10.1016/j.rinp.2017.11.023
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