World Applied Sciences Journal 16 (11): 1551-1558, 2012 ISSN 1818-4952 © IDOSI Publications, 2012 Some New Exact Traveling Wave Solutions to the (3+1)-dimensional Kadomtsev-Petviashvili equation 1,2 1 3 M. Ali Akbar, Norhashidah Hj. Mohd. Ali and Sayed Tauseef Mohyud-Din 1 School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia Department of Applied Mathematics, University of Rajshahi, Bangladesh 3 Department of Mathematics, HITEC University Taxila Cantt., Pakistan 2 Abstract: Mathematical modeling of numerous physical phenomena often leads to high-dimensional partial differential equations and thus the higher dimensional nonlinear evolution equations come into further attractive in many branches of physical sciences. In this article, we propose a new technique of the (G′/G)-expansion method combine with the Riccati equation for searching new exact traveling wave solutions of the (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation. Consequently, some new solutions of the KP are successfully obtained in a unified way involving arbitrary parameters. When the parameters take special values, solitary waves are derived from the traveling waves. The obtained solutions are expressed by the hyperbolic, trigonometric and rational functions. The method can be applied to many other nonlinear partial differential equations. PACS numbers: 02.30.Jr 05.45.Yv 02.30.Ik • • Key words : The (G′/G)-expansion method the Riccati equation the traveling wave solutions the (3+1)dimensional Kadomtsev-Petviashvili equation • INTRODUCTION In 1834, John Scott Russell [1] first observed the solitary waves. The significant observation motivated him to conduct experiments to underline his observance and to study these solitary waves. In 1965, Zabusky and Kruskal [2] studied the interactions between the solitary waves and the reappearance of initial states and since the KdV equation was solved by Gardner et al. [3] by the inverse scattering method, finding the solitary wave solutions of Nonlinear Evolution Equations (NLEEs) has turned out to be one of the enthusiastic and greatly lucrative areas of research. The appearance of solitary wave in nature is rather frequent, especially in fluids, plasmas, solid state physics, condensed matter physics, optical fibers, chemical kinematics, electrical circuits, bio-genetics, elastic media etc. Therefore, the researchers conducted a huge amount of research work to investigate the exact traveling wave solutions of the phenomena. Consequently, they established many methods and techniques, such as, the Backlund transformation method [4], the Hirota’s bilinear transformation method [5], the variational iteration method [6], the Adomian decomposition method [7], the tanh-function method [8], the homogeneous balance method [9], the F-expansion method [10], the Jacobi • • elliptic function method [11], the variable separation method [12], the Lie group symmetry method [13], the homotopy analysis method [14, 15], the homotopy perturbation method [16], the first integration method [17], the Exp -function method [18-21], the (G′/G)expansion method [22-31] and so on. It is significant to observe that there exist some fundamental relationships among numerous complex nonlinear partial differential equations and some basic and soluble nonlinear Ordinary Differential Equations (ODEs), such as the sine-Gordon equation, the sinhGordon equation, the Riccati equation, the Weierstrass elliptic equation etc. Therefore, it is natural to use the solutions of these nonlinear ODEs to construct exact solutions of various intricate nonlinear partial differential equations. Based on the relationships of complex nonlinear partial differential equations and ODEs, a number of methods, such as, the Riccati equation expansion method [32, 33], the projective Riccati equation method [34, 35], the algebraic method [36], the sinh-Gordon equation expansio n method [37], the generalized F-expansion method [38, 39] etc. have been developed. In the present article, we make use of the Riccati equation with the (G′/G)-expansion method for obtaining some new exact traveling wave solutions to Corresponding Author: M. Ali Akbar, School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia 1551 World Appl. Sci. J., 16 (11): 1551-1558, 2012 Family 1: When h 1 and h 2 have same sign and h 1 h 2 ≠ 0, the solutions of Eq. (5) are: the (3+1)-dimensional Kadomtsev-Petviashvili (KP). The Riccati equation has not been used by anybody before to solve the KP equation by (G′/G)-expansion method. 1  h1 h2 tan h2  = G1 DESCRIPTION OF THE (G′/G)-EXPANSION METHOD WITH THE RICCATI EQUATION ( ) h1 h2 η   ( ) 1 G2 = −  h1 h2 cot h1 h 2 η   h2  Suppose the general nonlinear partial differential equation ( ( ) )) ( 1  = G3 h1 h2 tan 2 h 1 h 2 η ± sec 2 h 1 h 2 η   h 2  Φ ( u , ut , u x, u y, u z, u tt, u xx , ) = 0 (1) 1 ( ( ) )) ( G 4 =−  h1 h 2 cot 2 h1 h2 η ± csc 2 h1 h2 η  where u = u (x,y,z,t) is an unknown function, Φ is a  h2  polynomial in u (x,y,z,t) and its partial derivatives in which the highest order partial derivatives and the  1  1  1   nonlinear terms are ni volved. The main steps of the = G5 h1 h 2 η  − cot  h1 h 2 η     h1 h 2  tan  2 h2   2  2   (G′/G)-expansion method combined with the Riccati  equation is as follows: Step 1: The travelling wave variable ansatz u(x,y,z,t) = v( η),η = x + y + z − V t (2) where V is the speed of the traveling wave, allows us to convert the Eq. (1) into an ODE: ψ ( v , v′ , v′ ,) = 0  2 2 h1 h 2  (M − N ) + Msin 2 h 1 h 2 η h2  Mcos 2 h 1 h 2 η + N  G9 = n )   )  )   ( −h1 cos 2 h1 h 2 η ( ) ) h1 h2 sin 2 h 1 h 2 η ± h1 h 2 ( h1 sin 2 h 1 h 2 η ( ) ) h1 h 2 cos 2 h 1 h 2 η ± h1 h 2 (4) G 10 where G = G(η) satisfies the Riccati equation, G′ = h1 + h 2 G 2 ,h 2 ≠ 0 ( ( G8 = Step 3: Suppose the traveling wave solution of Eq. (3) can be expressed by a polynomial in (G′/G) as follows: ∑ G7 = ( where M and N are two non-zero real constants and satisfies the condition M 2 -N2 >0. Step 2: If Eq. (3) is integrable, integrate term by term one or more times, yields constant(s) of integration.  G′  α n   ,αm ≠ 0 v(η)= G n =0 )   2 2 h1 h 2  (M − N ) − Mcos 2 h 1 h 2 η h2  Msin 2 h1 h2 η + N  (3) where the superscripts stands for the ordinary derivatives with respect to η. m ( G6 = 1 1 2h1 sin  h1 h 2 η  cos  h1 h 2 η  2 2     =  2 1 2 h1 h 2 cos  h1 h 2 η  − h1 h 2 2  (5) where αn = (n = 0,1,2,…,m), h1 and h2 are arbitrary constants to be determined later. The Riccati Eq. (5) plays important role in manipulating nonlinear equations to get exact solutions by the (G′/G)-expansion method. It has the following twenty one exact solutions [40]. Family 2: When h 1 and h2 possess opposite sign and h 1 h 2 ≠0, the solutions of Eq. (5) are: 1552 ( − h1 h 2 η   ( − h1 h 2 η   1 G 11 = −  − h1 h2 tanh h2  1 G 12 = −  − h1 h2 coth h2  ) ) World Appl. Sci. J., 16 (11): 1551-1558, 2012 ( ( ) ( )) ( ( ) ( )) G13 =− 1  − h1 h 2 tanh 2 − h1 h 2 η ± isech 2 − h1 h2 η   h 2  G 14 =− 1  − h1 h 2 coth 2 − h1 h 2 η ± csch 2 − h1 h 2 η   h 2  G15 =−  1   1  1   − h1 h2 η  + coth  − h1 h 2 η     − h1 h 2  tanh  2h2  2 2       G16 = G17 G 19 = G 20 ( ) h1 cosh 2 − h1 h 2 η ( ) − h1 h2 sinh 2 − h1 h 2 η ± i − h1 h 2 ( h1 sinh 2 − h1 h 2 η ( ) ) − h1 h 2 cosh 2 − h1 h2 η ± −h1 h2 1  1  2h1 sinh  −h1 h 2 η  cosh  − h1 h 2 η  2 2     =  2 1 2 −h1 h2 cosh  − h1 h 2 η  − −h1 h 2 2  Family 3: When Eq. (5) is: h 2 ≠0 but h 1 =0, the solution of G 21 = − ( ( )  2 2 − h1 h2  N − M + Msinh 2 − h1 h2 η = −  h2 Mcosh 2 − h1 h 2 η + N  where M and N are two non-zero real constants and satisfies the condition N 2 -M 2 >0. G 18 = (  2 2 −h1 h 2  (M + N ) − Mcosh 2 − h1 h 2 η h2  Msinh 2 − h1 h 2 η + N  1 h2 η + d where d is an arbitrary constant. The above solutions help to generate various traveling wave solutions, including solitary, periodic and rational solutions, in elementary functions. Step 4: To determine the positive integer m, put Eq. (4) along with Eq. (5) into Eq. (3) and consider the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in Eq. (3). ( ) )    )    obtain polynomials in Gi and Gi (i = 0,1,2,3…). Setting each coefficient of the resulted polynomial to zero, yields a set of algebraic equations for αn , h 1 , h 2 and V. Step 6: Suppose the value of the constants αn h1 , h 2 and V can be obtained by solving the set of algebraic equations obtained in step 5. Since the general solutions of Eq. (5) are known (arranged in step 3), substituting αn h 1 , h 2 and V into Eq. (4), we obtain new exact traveling wave solutions of the nonlinear evolution Eq. (1). APPLICATION OF THE METHOD In this section, we apply the proposed approach of the (G′/G)-expansion method to construct new exact traveling wave solutions to the Kadomtsev-Petviashvili (KP) equation which is an important nonlinear equation in mathematical physics. Let us consider the (3+1)-dimensional KP equation, (u t + 6 u u x + u x x x ) x + 3u y y + 3u zz = 0 (6) We investigate solutions the KP equation by the method described in section 2. Utilizing the traveling wave variable ansatz organized in Eq. (2), we obtain ( − Vv′ + 6 v v′ + v′′′)′ + 6 v′ = 0 (7) Eq. (7) is integrable, therefore, integrating twice, we obtain Step 5: Substituting Eq. (4) together with Eq. (5) into Eq. (3) along with the value of m obtained in step 4, we (6 − V)v + 3v 2 + v ′′+ C = 0 1553 (8) World Appl. Sci. J., 16 (11): 1551-1558, 2012 where C is a constant of integration. According to step 3, the solution of Eq. (8) can be expressed by a polynomial in (G′/G) as follows: v(η) = α 0 + α1 (G ′/ G ) + α 2 (G ′/ G ) 2 of algebraic equations (we will omit to display them for simplicity) for α0 , α1 , α2 , h 1 , h 2 , V and C. Solving the over-determined set of algebraic equations by using the symbolic computation software, such as Maple, we obtain (9) +  + α m (G′/ G )m , α m ≠ 0 α2 = 2,α 1 = 0,α 0 = α 0 where αn , (n = 0,1,2,…,m) are constants to be determined and G = G(η) satisfies the Riccati Eq. (5). Considering the homogeneous balance between the highest order derivative ν″ and the nonlinear term ν2 we obtain m = 2. Therefore, solution Eq. (9) become v(η) = α0 + α1( G′ / G ) + α 2 (G′/ G )2 , α2 ≠ 0 V = 6 − 1 6 h1h 2 − 6 α0 and C = 3α 02 + 1 6 h1 h2 where α0 , h 1 and h 2 are arbitrary constants. Now on the basis of the solutions of the Riccati Eq. (5), we obtain the following cluster of traveling wave solutions of Eq. (6). (10) Cluster 1: When h 1 and h 2 have same sign and h 1 h 2 ≠0, the periodic form solutions of Eq. (6) are, By means of Eq. (5), Eq. (10) can be rewritten as, v(η) = α0 + α1(h1 G−1 + h 2 G) + α 2 (h1 G −1 + h 2 G) 2 u1 =α 0 + 8h1 h2 csc 2(2 h 1 h 2 η) (11) where η = x+y+z-(6-16h 1 h 2 -6α0 )t and α0 , h1 , h2 are arbitrary constants. Substituting Eq. (11) into Eq. (8), the left hand side of the equation is converted into polynomials in Gi and G-i , (i = 0,1,2,…). Setting each coefficient of these polynomials to zero, we obtain an over-determined set { u3 =α 0 + 8h1 h2 sec 2 (2 h 1 h 2 η  2 h h M M + Nsin(2 h h η )− M 2 − N2 cos(2 h h η) 1 2 1 2 1 2  u 6 =α 0 + 2 2  Msin(2 h 1h 2 η) + N Mcos(2 h1h2 η )− M − N2  { { }{ }  2 h h M M + Ncos(2 h h η )+ M 2 − N 2 sin(2 h h η) 1 2 1 2 1 2  u 7 =α 0 + 2  Mcos(2 h 1h 2 η) + N Msin(2 h1 h 2 η) + M 2 − N 2  { (12) }{ } }  2   }  2   where M and N are two non-zero real constants satisfies the condition M 2 -N2 >0. u10  h1h 2 =α0 + 2   2sin(( h h η)/2)cos(( h h η)/2 ) 2cos 2 (( h h η) / 2 ) − 1 1 2 1 2 1 2  { } 2     The solutions corresponding to G2 , G4 , G5 and G9 are identical to the solution u 1 and the solution corresponding to G8 is identical to the solution u 3 . Cluster 2: When h 1 and h 2 possess opposite sign and h1 h 2 ≠0, the soliton and soliton-like solutions of Eqs. (6) are, u11 =α 0 − 8 h1 h2 csch 2 (2 − h1h 2 η) where η = x + y + z −(6 −16h1 h 2 −6 α0 ) t and α0 , h 1 , h 2 are arbitrary constants. 1554 World Appl. Sci. J., 16 (11): 1551-1558, 2012 u13 =α 0 + 8 h1h 2 sech 2(2 − h1h 2 η) { u16  2 −h h M M − Nsinh(2 − h h η) − M 2 + N 2 cosh(2 −h h η) 1 2 1 2 1 2  =α0 + 2  Msinh(2 − h1h 2 η) + N Mcosh(2 − h1h 2 η) − M 2 + N 2   u17  2 −h h M M + Ncosh(2 −h h η) − N 2 − M 2 sinh(2 − h h η) 1 2 1 2 1 2  =α 0 + 2  2  Mcosh(2 − h1 h2 η) + N Msinh(2 − h 1h 2 η) + N − M 2  { }{ { { } }{ } }      2 }  2   where M and N are two non-zero real constants and satisfies the condition M 2 -N2 >0. u 20  − h1h 2 =α0 + 2   2sinh(( −h h η)/2)cosh(( − h h η)/2 ) 2cosh2 (( − h h η)/2 ) − 1 1 2 1 2 1 2  { Fig. 1: Periodic solution corresponding to u 1 for α0 = 1, h 1 = 2, h 2 = 1 } 2     Fig. 2: Periodic solution corresponding to u 3 for α0 = 3, h 1 = 2, h 2 = 2 The solutions corresponding to G12 , G14 , G15 and G19 are identical to the solution u11 and the solution corresponding to G18 is identical to the solution u 13 . Cluster 3: When h1 =0 but h 2 ≠0, the solution of Eq. (6) is,  h2  u21 =α 0 + 2   h2 η + d  2 where d is an arbitrary constant. Because of the arbitrary constants α0 , h1 , h2 and V, in the above obtained solutions, the physical quantity u might possess physically significant rich structures. Fig. 3: Periodic solution corresponding to u 6 for α0 = 5, h 1 = 5, r = 5, M = 2 and N = 1 1555 World Appl. Sci. J., 16 (11): 1551-1558, 2012 GRAPHICAL REPRESENTATIONS Graph is an influential tool for communication and it illustrates clearly the solutions of the problems. We consider the evolutions of the soliton, periodic and rational-like solutions u1 , u 3 , u6 , u 11 , u 17 and u21 along x = 0 and y = 0. The graphs readily have shown the periodic and solitary wave forms of the solutions. CONCLUSION Fig. 4: Soliton solution corresponding to u 11 for α0 = 10, h 1 = -2, h 2 = 2 The (G′/G)-expansion method is an advance mathematical tool for investigating exact solutions of nonlinear partial differential equations associated with complex physical phenomena wherein, in general the second order linear ordinary differential equation is employed as an auxiliary equation. But, in this article, we utilize the Riccati equation as an auxiliary equation; as a result, some new explicit solutions of the Kadomtsev-Petviashvili equation are obtained in a unified way. The obtained exact solutions might be important and significant in the field of water waves of long wavelength with weakly nonlinear restoring forces and frequency dispersion. The algorithm presented in this article is effective and more powerful than the original (G′/G)-expansion method and it can be applied for other kind of nonlinear evolution equations in mathematical physics. 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