The Modified Benjamin-Bona-Mahony Equation. via the Extended Generalized Riccati Equation. Mapping Method

Applied Mathematical Sciences Vol. 6 0 no. 5495 55 The Modified Benjamin-Bona-Mahony Equation via the Extended Generalized Riccati Equation Mapping Method Hasibun Naher and Farah Aini Abdullah School of Mathematical Sciences Universiti Sains Malaysia 800 Penang Malaysia Abstract - The generalized Riccati equation mapping is extended together with the ( G / G expansion method and is a powerful mathematical tool for solving nonlinear partial differential equations. In this article we construct twenty seven new exact traveling wave solutions including solitons and periodic solutions of the modified Benjamin-Bona-Mahony equation by applying the extended generalized Riccati equation mapping method. In this method G ( p+ rg( + sg ( is implemented as the auxiliary equation where rsand p are arbitrary constants and called the generalized Riccati equation. The obtained solutions are described in four different families including the hyperbolic functions the trigonometric functions and the rational functions. In addition it is worth mentioning that one of newly obtained solutions is identical for a special case with already published result which validates our other solutions. Mathematics Subject Classification: 35K99 35P99 35P05 Keywords: The modified Benjamin-Bona-Mahony equation the generalized Riccati equation the ( G / G -expansion method traveling wave solutions nonlinear evolution equations. Corresponding author: Hasibun Naher e-mail : hasibun06tasauf@gmail.com

5496 H. Naher and F. A. Abdullah. Introduction The investigation of traveling wave solutions of the nonlinear partial differential equations (PDEs plays the most important role in the study of nonlinear physical phenomena which arise in mathematical physics engineering sciences and other technical arena [-43]. In recent times many researchers established various methods to construct exact traveling wave solutions of the nonlinear PDEs such as the inverse scattereing method [] the variational iteration method [3] the Hirota s bilinear transformation method [4] the Jacobi elliptic function expansion method [5] the tanh-coth method [67] the Backlund transformation method [8] the direct algebraic method [9] the Cole-Hopf transformation method [0] the sine-cosine method [] the Exp-function method [-4] the Adomian decomposition method [5] and others [6-]. G / G -expansion Recently ang et al. [3] presented a method called the ( method and they established traveling wave solutions for some nonlinear PDEs. In this method they employed the second order linear ordinary differential equation with constant coefficients G ( θ + λg ( θ + G( θ 0 as an auxiliary equation. Subsequently many researchers implemented this powerful ( G / G -expansion method to investigate different nonlinear PDEs for obtaining exact traveling wave solutions. For example Feng et al. [4] studied the Kolmogorov-Petrovskii- Piskunov equation to construct exact solutions via this method. Naher et al. [5] applied the same method to obtain traveling wave solutions of the higher-order Caudrey-Dodd-Gibbon equation. In Ref. [6] Abazari investigated the Zoomeron equation for establishing solitary wave solutions by using this method. Zayed and Al-Joudi [7] concerned about this method for constructing analytical solutions of some nonlinear partial differential equations whilst Gepreel [8] studied nonlinear PDEs with variable coefficients in mathematical physics by using this method and found exact solutions. Ozis and Aslan [9] applied the ( G / G -expansion method to establish traveling wave solutions for the Kawahara type equations using symbolic computation while Aslan [30] investigated three nonlinear evolution equations to construct exact solutions by applying this method. Naher et al. [3] concerned about the improved ( G / G -expansion method to obtain traveling wave solutions of the higher dimensional nonlinear evolution equation while Naher and Abdullah [3] constructed some new traveling wave solutions of the nonlinear reaction diffusion equation via this method and so on. Zhu [33] introduced the generalized Riccati equation mapping with the extended tanh-function method to investigate the (+-dimensional Boiti-Leon-Pempinelle equation. In this generalized Riccati equation mapping the auxiliary equation G ( p+ rg( + sg ( is used where rs and p are arbitrary constants. Bekir and Cevikel [34] implemented the tanh-coth method combined with the

Modified Benjamin-Bona-Mahony equation 5497 Riccati equation to solve nonlinear coupled equation in mathematical physics. In Ref. [35] Guo et al. studied the diffusion-reaction and the mkdv equation with variable coefficient via the extended Riccati equation mapping method while Li et al. [36] used the generalized Riccati equation expansion method to study the (3+-dimensional Jimbo-Miwa equation. Salas [37] utilized the projective Riccati equation method to obtain some exact solutions for the Caudrey-Dodd- Gibbon equation. Li and Dai [38] executed the generalized Riccati equation mapping method to construct traveling wave solutions for the higher dimensional nonlinear evolution equation and so on. Many researchers implemented different methods to obtain traveling wave solutions of the modified Benjamin-Bona-Mahony equation. For example Taghizadeh and Mirzazadeh [39] used modified extended tanh method to establish traveling wave solutions of this equation. Yusufoglu [40] applied the Expfunction method to construct traveling wave solutions of the same equation while Yusufoglu and Bekir [4] investigated this equation to seek exact solutions via the tanh and sine-cosine methods. In Ref. [4] Abbasbandy and Shirzadi executed the first integral method to obtain analytical solutions of the same equation. Gao [43] studied this equation by using the algebraic method to establish exact solutions. Aslan [30] studied the same equation for constructing traveling wave solutions by '/ G'/ G -expansion applying the basic ( G G -expansion method. In the basic ( method the second order linear ordinary differential equation (LODE with constant coefficients is considered as an auxiliary equation. To the best of our knowledge no body studied the modified Benjamin-Bona-Mahony equation to construct exact traveling wave solutions by applying the extended generalized Riccati equation mapping method. In this article we investigate the modified Benjamin-Bona-Mahony equation to construct exact traveling wave solutions including solitons periodic and rational solutions via the extended generalized Riccati equation mapping method.. The extended generalized Riccati equation mapping method Suppose the general nonlinear partial differential equation: Avv v v v v... 0 ( ( t x xt tt xx where v v( x t is an unknown function A is a polynomial in v v( x t and the subscripts indicate the partial derivatives. The most important steps of the generalized Riccati equation mapping together G / G -expansion method [333] are as follows: with the ( Step. Consider the traveling wave variable: v x t p Kx+ Ht ( ( (

5498 H. Naher and F. A. Abdullah Now using Eq. ( Eq. ( is converted into an ordinary differential equation for p ( : ( B p p' p p... 0 (3 where the superscripts stand for the ordinary derivatives with respect to. Step. Eq. (3 integrates term by term one or more times according to possibility yields constant(s of integration. The integral constant(s may be zero for simplicity. Step 3. Suppose that the traveling wave solution of Eq. (3 can be expressed in the form [333]: n j G ' p( aj j 0 G (4 where aj ( j 0... n and an 0 with G G( is the solution of the generalized Riccati equation: G p+ rg+ sg (5 where rsp are arbitrary constants and s 0. Step 4. To decide the positive integer n consider the homogeneous balance between the nonlinear terms and the highest order derivatives appearing in Eq. (3. Step 5. Substitute Eq. (4 along with Eq. (5 into the Eq. (3 then collect all the coefficients with the same order the left hand side of Eq. (3 converts into m m polynomials in G ( and G ( ( m 0.... Then equating each coefficient of the polynomials to zero yield a set of algebraic equations for a j 0... n r s p K and H. j ( Step 6. Solve the system of algebraic equations which are found in Step 5 with the a j 0... n and aid of algebraic software Maple and we obtain values for ( H. Then substitute obtained values in Eq. (4 along with Eq. (5 with the value of n we obtain exact solutions of Eq. (. In the following we have twenty seven solutions including four different families of Eq. (5. Family : hen r 4sp > 0 and rs 0 or sp 0 the solutions of Eq. (5 are: r 4sp r + r 4sp tanh s j

Modified Benjamin-Bona-Mahony equation 5499 r 4sp r + r 4sp coth s ( ( ( ( 3 r+ r 4sp tanh r 4sp ± isech r 4 sp s ( ( ( ( 4 r+ r 4sp coth r 4sp ± csch r 4 sp s r 4sp r 4sp 5 r + r 4 sp tanh + cot h 4s 4 4 ( Q + R ( r 4sp Q r 4sp cosh( r 4sp 6 r s + Qsinh( r 4sp + R ( R Q ( r 4sp + Q r 4sp sinh( r 4sp 7 r s Q cosh( r 4 sp R + where Q and R are two non-zero real constants and satisfies R Q > 0. r 4sp pcosh 8 r 4sp r 4sp r 4spsinh rcosh r 4sp psinh 9 r 4sp r 4sp rsinh r 4spcosh 0 pcosh( r 4sp ( ( psinh( r 4sp ( ( r 4sp sinh r 4sp rcosh r 4sp ± i r 4sp + ± rsinh r 4sp r 4sp cosh r 4sp r 4sp

5500 H. Naher and F. A. Abdullah r 4sp r 4sp 4psinh cosh 4 4 r 4sp r 4sp r 4sp rsinh cosh + r 4sp cosh r 4sp 4 4 4 Family : hen r 4sp< 0 and rs 0 or sp 0 the solutions of Eq. (5 are: 4sp r 3 r+ sp r s 4 tan 4sp r 4 r + 4sp r cot s ( ( ( ( 5 r+ 4sp r tan 4sp r ± sec 4 sp r s ( ( ( ( 6 r+ 4sp r cot 4sp r ± csc 4 sp r s 4sp r 4sp r 7 r+ 4sp r tan cot 4s 4 4 ± ( Q R ( 4sp r Q 4sp r cos( 4sp r 8 r s + Q sin( 4 sp r R + ± ( Q R ( 4sp r + Q 4sp r cos( 4sp r 9 r s Q sin( 4 sp r R + where Q and R are two non-zero real constants and satisfies Q R > 0. 0 pcos 4sp r 4sp r 4sp r 4 sin + cos sp r r

Modified Benjamin-Bona-Mahony equation 550 3 4sp r psin 4sp r 4sp r rsin + 4sp r cos pcos( 4sp r ( ( 4sp r sin 4sp r + rcos 4sp r ± 4sp r psin( 4sp r rsin 4sp r + 4sp r cos 4sp r ± 4sp r ( ( 4 4sp r 4sp r 4psin cos 4 4 4sp r 4sp r 4sp r rsin cos + 4sp r cos 4sp r 4 4 4 Family 3: when p 0 and rs 0 the solution Eq. (5 becomes: 5 6 rg s g r r ( + cosh( sinh ( r( cosh( r + sinh ( r ( + cosh( + sinh( s g r r where g is an arbitrary constant. Family 4: when s 0 and p r 0 the solution of Eq. (5 becomes: 7 s + d where d is an arbitrary constant.

550 H. Naher and F. A. Abdullah 3. Applications of the method e construct twenty seven exact traveling wave solutions including solitons periodic and rational solutions of the modified Benjamin-Bona-Mahony equation in this section. 3. The Modified Benjamin-Bona-Mahony equation e consider the modified Benjamin-Bona-Mahony equation with parameters followed by Aslan [30]: ut + αux + βu ux γuxxt 0 (6 where α γ are free parameters and β 0. Now we use the transformation Eq. ( into the Eq. (6 which yields: ( H + Kα p + Kβp p HK γ p 0 (7 Eq. (7 is integrable therefore integrating with respect once yields: Kβ 3 ( H + Kα p+ p HK γ p + C 0 3 (8 where C is an integral constant which is to be determined later. 3 Taking the homogeneous balance between p and p in Eq. (8 we obtain n. Therefore the solution of Eq. (8 is of the form: p( a( G'/ G + a0 a 0. (9 Using Eq. (5 Eq. (9 can be re-written as: p ( a( r+ pg + sg + a 0 (0 where rs and p are free parameters. By substituting Eq. (0 into Eq. (8 collecting all coefficients of G k and k G ( k 0... and setting them equal to zero we obtain a set of algebraic equations for a 0 a r s p C and H (algebraic equations are not shown for simplicity. Solving the system of algebraic equations with the help of algebraic software Maple we obtain 3αγ 3αγ α K a0 mkr a ± K H β γ β γ + + ( + K ( r + 8sp + K ( r + 8sp 4 8K rst 3 C ± αγ αγ + K γ r + 8sp β + K γ r + 8sp ( ( ( ( K γ ( r 8sp

Modified Benjamin-Bona-Mahony equation 5503 where α γ are free parameters and β 0. Family : The soliton and soliton-like solutions of Eq. (6 (when r 4sp> 0 and rs 0 or sp 0 are: Φ Φ sech p a + a0 Φ r +Φ tanh 3αγ where Φ r 4sp Φ r 4sp a m Kr 0 β + K γ r + 8sp a ± K 3αγ ( + K γ ( r + 8sp β ( ( α K and Kx + K γ r + 8sp ( Φ Φ csch p a + a0 Φ r +Φcoth Φ ( sech ( Φ m itanh( Φ sech( Φ p3 a + a0 r+φ( tanh( Φ ± isech( Φ Φ ( csch ( Φ ± coth( Φ csch( Φ p4 a + a0 r+φ( coth( Φ ± csch( Φ Φ Φ Φ sech csch 4 4 p5 a + a0 Φ Φ 8r + 4Φ tanh + coth 4 4 Q( r Q sinh ( Φ r R 4spQ+ 4spRsinh ( Φ Φ ( Q + R cosh ( Φ p6 a + a0 Qsinh Φ + R rqsinh Φ + rr Φ Q + R + QΦcosh Φ ( ( sinh ( 4 4 sinh ( ( cosh ( Qsinh ( Φ + R ( rqsinh Φ + rr+φ Q + R + QΦcosh Φ ( ( ( ( ( Q r Q Φ r R spq+ spr Φ +Φ Q + R Φ p a + a ( ( ( ( 7 0 t. where Q and R are two non-zero real constants and satisfies R Q > 0.

5504 H. Naher and F. A. Abdullah Φ p a a Φ Φ Φ cosh Φsinh r cosh Φ p a + a Φ Φ Φ sinh r sinh +Φcosh Φ + ir sinh ( Φ i4spsinh ( Φ p0 a + a0 Φsinh Φ rcosh Φ + iφcosh Φ 8 + 0 9 0 ( ( ( Φ + r cosh ( Φ 4spcosh ( Φ ( r sinh ( Φ +Φcosh ( Φ +Φ sinh ( Φ p a + a 0 Φ p a + a0 Φ Φ Φ Φ Φ 4sinh cosh r sinh cosh + Φcosh Φ 4 4 4 4 4 Family : The periodic form solutions of Eq. (6 (when r 4sp< 0 and rs 0 or sp 0 are: Π p a a Π Π Π cos r cos +Πsin 3 + 0 where a ± K Π r + 4 sp 3αγ ( + K γ ( r + 8sp β Π 4 sp r a 3αγ m Kr β ( + K γ ( r + 8sp 0 α K and Kx + K γ r + 8sp Π p4 a + a0 Π Π + cos r +Πcot Π ( + sin( Π p5 a + a0 cos Π r cos Π +Πsin Π +Π ( ( ( ( Π sin ( Π ( r ( ( r ( p 6 a + 0 cos Π sin Π +Πcos Π sin Π Π a ( Π p a + a 7 0 Π Π Π Π 4 cos cos r tan cot 4 + +Π 4 4 4 t.

Modified Benjamin-Bona-Mahony equation 5505 ( 4 4 sin( Π + + sin( Π + 4 ( cos( Π ( cos( Π Π ( Q R ( Q + Q cos ( Π QRsin ( Π R r+ QΠcos( Π Q spq spr r Q r R sp Q R r Q R p a + a 8 0 Qsin ( Π + R ( 4 4 sin( Π + + sin( Π 4 ( cos( Π + ( cos( Π Π ( Q R ( Q + Q cos ( Π QRsin ( Π R r + QΠcos( Π Q spq spr r Q r R sp Q R r Q R p a + a 9 0 ( Qsin Π + R where Q and R are two non-zero real constants and satisfies Q R > 0. Π Π Π Π sec sin cos Π + r p0 a + a0 Π Π Π Π 4sp 4sp cos r + r cos + rπsin cos Π Π Π r sin + Π cos p a + a0 Π Π Π Π Π sin r r cos r sin cos 4spcos + + Π p + a a sec( Π ( Π 4spsin ( Π + r sin ( Π ( Πsin ( Π + rcos( Π +Π ( 4sp spcos ( Π r + r cos ( Π +Πr sin ( Π cos( Π + 4spsin ( Π r sin ( Π + rπcos( Π 0 Π ( r sin ( Π + Π cos( Π + Π ( ( ( ( p3 a + a0 sin Π cos Π + cos Π + Πsin Π p 4 ( sp r r sp Π Π Π Π Π Π a csc sec rsin cos + Πcos Π 4 4 4 4 4 4 Π Π Π Π Π Π Π + + Π Π + 4 4 4 4 4 4 4 4 3 4 8r cos 8r cos 8 rcos sin 4r sin cos 6spcos 6spc Π os Π 4 + a0 Family 3: The soliton and soliton-like solutions of Eq. (6 (when p 0 and rs 0 are: r( cosh( r sinh( r p5 a + a0 g + cosh r sinh r ( (

5506 H. Naher and F. A. Abdullah rg p a + a 6 0 g + cosh( r + sinh( r where g is an arbitrary constant a ± K 3αγ ( + K γ ( r + 8sp β a 3αγ m Kr β ( + K γ ( r + 8sp 0 α K and Kx + K γ r + 8sp ( t. Family 4: The rational function solution (when s 0 and p r 0 is: as p7 s + d 3αγ where d is an arbitrary constant a ± K and β + K γ r + 8sp α K Kx + K γ r + 8sp ( t. ( ( 4. Results and discussion It is important to point out that our solution p 7 is identical for a special case with u x t in subsection 3.3 of section 3 of Aslan [30]. If we substitute ( 56 C3 C 0 λ 4 0 α γ β 6 and u56 x t of Aslan [40] and becomes u56 ( x t m. x t Also if we substitute p ( 7 u56 x t and K α γ p r 0 s β 6 d 0 in our solution p 7 and becomes u56 ( x t m. x t Beyond this we obtain new traveling wave solutions p to p 6 and to the best of our knowledge which have not been reported in the previous literature. Furthermore the graphical demonstrations of some obtained solutions are shown in figure to figure in the following subsection. 4. Graphical illustrations of the solutions k in solution ( Some of our obtained traveling wave solutions are represented in the following figures with the aid of commercial software Maple:

Modified Benjamin-Bona-Mahony equation 5507 Fig. : Periodic solution for Fig. : Periodic solution for Fig. 3: Periodic solution for r s p 3 α r 3 s 4 p 5 r s p 3 α β γ K 3 α β 3 γ K 5 β 3 γ 5 K 8 Fig. 4: Solitons solution for Fig. 5: Solitons solution for Fig. 6: Solitons solution for r 0 s 7 p 0 α β 0.5 r 0 s p 0 α r 0 s 7 p 0 α 4 γ K 5 g 5 β γ K g 0 3 β 3 γ 4 K 5 g 7 Fig. 7: Periodic solutions for Fig. 8: Solitons solution for Fig. 9: Solitons solution for r 3 s 5 p 7 α 4 r 5 s p 7 α 4 r 0 s 5 p 0 α 3 β 3 γ 3 K 9 β 4 γ 0.5 K β 3 γ K 3 g Fig. 0: Solitons solution for Fig. : Periodic solution for Fig. : Periodic solution for r 0 s 5 p 0 α γ β 3 K 9 g 4 r 5 s 3 p 5 r 5 s p 4 α 4 β 3 γ 4 K 9 α 3 β 3 γ 4 K 4

5508 H. Naher and F. A. Abdullah 5. Conclusions By applying the extended generalized Riccati equation mapping method abundant new exact traveling wave solutions of the modified Benjamin-Bona-Mahony equation are constructed in this article. In this method the auxiliary equation G ( p+ rg( + sg ( is used with constant coefficients instead of the second order linear ordinary differential equation with constant coefficients. The obtained solutions disclose noteworthy properties of the shapes in that the solutions come as solitons and periodic solutions. Further it is imperative to mention out that one of our solutions are identical with the solutions available in the literature and some are new. e hope that this straightforward method can be more successfully used to investigate nonlinear evolution equations which arise in mathematical physics engineering sciences and other technical arena. References [] M. J. Ablowitz and P. A. Clarkson Solitons nonlinear evolution equations and inverse scattering transform Cambridge Univ. Press Cambridge 99. [] J. H. He Variational iteration method for delay differential equations Communication in Nonlinear Science and Numerical Simulation (4(997 35-36. [3]. Zhang Solitary solutions and singular periodic solutions of the Drinfeld- Sokolov-ilson equation by variational approach Applied Mathematical Sciences 5(38(0 887-894. [4] R. Hirota Exact solution of the KdV equation for multiple collisions of solutions Physics Review Letters 7(97 9-94. [5] S. Liu Z. Fu S. Liu and Q. Zhao Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations Physics Letters A 89(00 69 74. [6]. Malfliet Solitary wave solutions of nonlinear wave equations Am. J. Phys. 60(99 650 654. [7] A. M. azwaz The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations Applied Mathematics and Computation 88(007 467-475.

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