Exact traveling wave solutions of the KP-BBM equation by using the new approach of generalized (G′/G)-expansion method

  • Md Nur Alam1 and

    Affiliated with

    • M Ali Akbar2Email author

      Affiliated with

      SpringerPlus20132:617

      DOI: 10.1186/2193-1801-2-617

      Received: 13 September 2013

      Accepted: 13 November 2013

      Published: 19 November 2013

      Abstract

      The new approach of the generalized (G′/G)-expansion method is an effective and powerful mathematical tool in finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in science, engineering and mathematical physics. In this article, the new approach of the generalized (G′/G)-expansion method is applied to construct traveling wave solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation. The solutions are expressed in terms of the hyperbolic functions, the trigonometric functions and the rational functions. By means of this scheme, we found some new traveling wave solutions of the above mentioned equation.

      New generalized (G′/G)-expansion method KP-BBM equation Exact solutions Nonlinear partial differential equation Homogeneous balance Traveling wave solutions Solitary wave solutions
      35C07 35C08 35P99

      Introduction

      The world around us is inherently nonlinear and NLEEs are widely used as models to describe the complex physical phenomena. The exact traveling wave solutions of NLEEs play a vital role in nonlinear science and engineering. Therefore, investigating traveling wave solutions is becoming increasingly attractive in nonlinear sciences day by day. However, not all equations posed of these models are solvable. As a result, many new techniques have been successfully developed by diverse group of scientists, such as, the rank analysis method (Feng 2000), the complex hyperbolic function method (Zayed et al. 2006; Chow 1995), the generalized Riccati equation method (Yan and Zhang 2001; Porubov 1996), the Jacobi elliptic function method (Chen and Wang 2005; Xu 2006; Yusufoglu and Bekir 2008; Zayed et al. 2004a), the ansatz method (Hu 2001a2001b), the Adomian decomposition method (Wazwaz 2002), the He’s homotopy perturbation method (Ganji and Rafei 2006; Ganji 2006; Ganji et al. 2007), the homogeneous balance method (Wang 19951996; Zayed et al. 2004b), the inverse scattering transform method (Ablowitz and Clarkson 1991), the Darboux transformation method (Matveev and Salle 1991), the Backlund transformation method (Miura 1978), the (G′/G)-expansion method (Wang et al. 2008; Akbar et al. 2012a2012b2012c2012d2012e; Bekir 2008; Zayed 2009; Zhang et al. 2008), the improved (G′/G)-expansion method (Zhang et al. 2010), the modified simple equation method (Jawad et al. 2010; Khan et al. 2013; Zayed and Ibrahim 2012), the Exp-function method (He and Wu 2006; Akbar and Ali 2012; Mohyud-Din et al. 2010), the tanh-function method (Malfliet 1992; Fan 2000), the sine-cosine method (Wazwaz 20042005; Bibi and Mohyud-Din 2013), the first integral method (Feng 2002; Tascan and Bekir 2010) etc.

      Recently, the new generalized (G′/G) expansion method has been initiated by Naher and Abdullah (2013). The significance of the new generalized (G′/G) expansion method is that one can treat the nonlinear problems by essentially linear method. Moreover, it transforms a nonlinear evolution equation to a simple algebraic computation. The merits of the new generalized (G′/G) expansion method over the other methods are that it gives more general solutions with some free parameters and it handles NLEEs in a direct manner with no requirement for initial/boundary conditions or initial trial function at the outset.

      Our aim in this paper is to present an application of the new generalized (G′/G) expansion method to solve the KP-BBM equation by using this method for the first time.

      The rest of the paper is organized as follows: In Section “Description of the new generalized (G′/G)-expansion method”, we give the description of the method. In Section “Application of the method”, we exert this method to the KP-BBM equation. In Section “Discussions”, Discussions are presented. Conclusions are given in Section “Conclusion”.

      Description of the new generalized (G′/G)-expansion method

      Let us consider a general nonlinear PDE in the form
      P u , u x , u t , u x x , u x t , u t t , , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ1_HTML.gif
      (1)

      where u = u(x, t) is an unknown function, P is a polynomial in u(x, t) and its derivatives wherein the highest order derivatives and nonlinear terms are involved and the subscripts are used for the partial derivatives.

      Step 1

      We combine the real variables x and t by a compound variable ξ:
      u x , t = u ( ξ ) , ξ = x ± V t , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ2_HTML.gif
      (2)
      where V is the speed of the traveling wave. The traveling wave transformation (2) converts Eq. (1) into an ordinary differential equation (ODE) for u = u(ξ):
      Q u , u , u , u , = 0 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ3_HTML.gif
      (3)

      where Q is a polynomial of u and its derivatives and the superscripts indicate the ordinary derivatives with respect to ξ.

      Step 2

      According to possibility Eq. (3) can be integrated term by term one or more times, yields constant(s) of integration. The integral constant may be zero, for simplicity.

      Step 3

      Suppose the traveling wave solution of Eq. (3) can be expressed as follows:
      u ξ = i = 0 N a i d + H i + i = 1 N b i d + H i , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ4_HTML.gif
      (4)
      where either a N or b N may be zero, but both a N and b N could be zero at a time, a i (i = 0, 1, 2,…, N) and b i (i = 1, 2,…, N) and d are arbitrary constants to be determined later and H(ξ) is
      H ξ = G / G http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ5_HTML.gif
      (5)
      where G = G(ξ) satisfies the following auxiliary equation:
      AG G BG G E G 2 C G 2 = 0 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ6_HTML.gif
      (6)

      where the prime stands for derivative with respect to ξ; A, B, C and E are real parameters.

      Step 4

      To determine the positive integer N, taking the homogeneous balance between the highest order nonlinear terms and the derivatives of the highest order appearing in Eq. (3).

      Step 5

      Substitute Eq. (4) and Eq. (6) including Eq. (5) into Eq. (3) with the value of N obtained in Step 4, we obtain polynomials in (d + H) N (N = 0, 1, 2,…) and (d + H)N (N = 0, 1, 2,…). We collect each coefficient of the resulted polynomials and setting them to zero yields a set of algebraic equations for a i (i = 0, 1, 2,…, N) and b i (i = 1, 2,…, N), d and V.

      Step 6

      Suppose that the value of the constants a i (i = 0, 1, 2,…, N), b i (i = 1, 2,…, N), d and V can be found by solving the algebraic equations obtained in Step 5. Since the general solution of Eq. (6) is well known to us, inserting the values of a i (i = 0, 1, 2,…, N), b i (i = 1, 2,…, N), d and V into Eq. (4), we obtain more general type and new exact traveling wave solutions of the nonlinear partial differential equation (1).

      Using the general solution of Eq. (6), we have the following solutions of Eq. (5):

      Family 1

      When B ≠ 0, Ψ = A − C and Ω = B2 + 4E(A − C) > 0,
      H ξ = G G = B 2 Ψ + Ω 2 Ψ C 1 sinh Ω 2 A ξ + C 2 cosh Ω 2 A ξ C 1 cosh Ω 2 A ξ + C 2 sinh Ω 2 A ξ http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ7_HTML.gif
      (7)

      Family 2

      When B ≠ 0, Ψ = A − C and Ω = B2 + 4E(A − C) < 0,
      H ξ = G G = B 2 Ψ + Ω 2 Ψ C 1 sin Ω 2 A ξ + C 2 cos Ω 2 A ξ C 1 cos Ω 2 A ξ + C 2 sin Ω 2 A ξ http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ8_HTML.gif
      (8)

      Family 3

      When B ≠ 0, Ψ = A − C and Ω = B2 + 4E(A − C) = 0,
      H ξ = G G = B 2 Ψ + C 2 C 1 + C 2 ξ http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ9_HTML.gif
      (9)

      Family 4

      When B = 0, Ψ = A − C and ∆ = ΨE > 0,
      H ξ = G G = Ψ C 1 sinh A ξ + C 2 cosh A ξ C 1 cosh A ξ + C 2 sinh A ξ http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ10_HTML.gif
      (10)

      Family 5

      When B = 0, Ψ = A − C and ∆ = ΨE < 0,
      H ξ = G G = Ψ C 1 sin A ξ + C 2 cos A ξ C 1 cos A ξ + C 2 sin A ξ http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ11_HTML.gif
      (11)

      Application of the method

      In this section, we will bring to bear the new generalized (G′/G) expansion method to construct new and more general traveling wave solutions of the KP-BBM equation. Let us consider the KP-BBM equation
      u t + u x a u 2 x b u x x t x + k u y y = 0 . http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ12_HTML.gif
      (12)
      Now, we use the wave transformation (2) into the Eq. (12), which yields
      V u + u 2 a u 2 2 a u u + b V u 4 + k u = 0 . http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ13_HTML.gif
      (13)
      Integrating Eq. (13) twice with respect to ξ, we obtain
      P + 1 V + k u + b V u / / 2 a u 2 = 0 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ14_HTML.gif
      (14)

      where P is an integral constant which is to be determined.

      Taking the homogeneous balance between u2 and u″ in Eq. (14), we obtain N = 2. Therefore, the solution of Eq. (14) is of the form:
      u ξ = a 0 + a 1 d + H + a 2 d + H 2 + b 1 d + H 1 + b 2 d + H 2 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ15_HTML.gif
      (15)

      where a0, a1, a2, b1, b2 and d are constants to be determined.

      Substituting Eq. (15) together with Eqs. (5) and (6) into Eq. (14), the left-hand side is converted into polynomials in (d + H) N (N = 0, 1, 2, ……) and (d + H)N(N = 1, 2,…). We collect each coefficient of these resulted polynomials and setting them zero yields a set of simultaneous algebraic equations (for simplicity the equations are not presented here) for a0, a1, a2, b1, b2d, P and V. Solving these algebraic equations with the help of symbolic computation software Maple, we obtain following:

      Set 1

      P = 0, V = V, d = d, a1 = 0, a2 = 0,
      a 0 = 1 2 a A 2 12 bV d 2 Ψ 2 + A 2 k + A 2 V A 2 + 12 bVBdΨ 8 bVEΨ + bV B 2 , b 1 = 6 bV a A 2 2 d 3 Ψ 2 + 3 B d 2 Ψ + 2 EdΨ + B 2 d EB , b 2 = 6 bV a A 2 d 4 Ψ 2 + 2 B d 3 Ψ 2 E d 2 Ψ 2 BdE + B 2 d 2 + E 2 . http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ16_HTML.gif
      (16)

      where Ψ = A − C, V, d, A, B, C, E are free parameters.

      Set 2

      P = 0, V = V, d = d, b1 = 0, b2 = 0,
      a 1 = 6 bV a A 2 2 d Ψ 2 + , a 0 = 1 2 a A 2 ( 12 bV d 2 Ψ 2 + A 2 k + A 2 V A 2 + 12 bVBdΨ 8 bVEΨ + bV B 2 ) , a 2 = 6 bV Ψ 2 a A 2 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ17_HTML.gif
      (17)

      where Ψ = A − C, V, d, A, B, C, E are free parameters.

      Set 3

      P = 0, V = V, d = B 2 Ψ , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_IEq1_HTML.gifa1 = 0, b1 = 0,
      a 0 = 1 2 a A 2 A 2 k A 2 + A 2 V + 8 bVEΨ + 2 bV B 2 , a 2 = 6 bV Ψ 2 a A 2 , a 1 = 0 , V = V , b 2 = 3 bV 8 a A 2 Ψ 2 16 E 2 Ψ 2 + 8 E B 2 Ψ + B 4 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ18_HTML.gif
      (18)

      where Ψ = A − C, V, A, B, C, E are free parameters.

      For set 1, substituting Eq. (16) into Eq. (15), along with Eq. (7) and simplifying, yields following traveling wave solutions (if C1 = 0 but C2 ≠ 0; C2 = 0 but C1 ≠ 0) respectively:
      u 1 1 ξ = a 0 + b 1 d + B 2 Ψ + Ω 2 Ψ coth Ω 2 A ξ 1 + b 2 d + B 2 Ψ + Ω 2 Ψ coth Ω 2 A ξ 2 , u 1 2 ξ = a 0 + b 1 d + B 2 Ψ + Ω 2 Ψ tanh Ω 2 A ξ 1 + b 2 d + B 2 Ψ + Ω 2 Ψ tanh Ω 2 A ξ 2 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equa_HTML.gif
      Substituting Eq. (16) into Eq. (15), along with Eq. (8) and simplifying, the exact solutions become (if C1 = 0 but C2 ≠ 0; C2 = 0 but C1 ≠ 0):
      u 1 3 ξ = a 0 + b 1 d + B 2 Ψ + Ω 2 Ψ cot Ω 2 A ξ 1 + b 2 d + B 2 Ψ + Ω 2 Ψ cot Ω 2 A ξ 2 , u 1 4 ξ = a 0 + b 1 d + B 2 Ψ Ω 2 Ψ tan Ω 2 A ξ 1 + b 2 d + B 2 Ψ Ω 2 Ψ tan Ω 2 A ξ 2 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equb_HTML.gif
      Substituting Eq. (16) into Eq. (15), together with Eq. (9) and simplifying, the obtained solution becomes:
      u 1 5 ξ = a 0 + b 1 d + B 2 Ψ + C 2 C 1 + C 2 ξ 1 + b 2 ( d + B 2 Ψ + C 2 C 1 + C 2 ξ ) 2 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equc_HTML.gif
      Substituting Eq. (16) into Eq. (15), along with Eq. (10) and simplifying, we obtain following traveling wave solutions (if C1 = 0 but C2 ≠ 0; C2 = 0 but C1 ≠ 0):
      u 1 6 ξ = a 0 + b 1 d + Ψ coth A ξ 1 + b 2 d + Ψ coth A ξ 2 , u 1 7 ξ = a 0 + b 1 d + Ψ tanh A ξ 1 + b 2 d + Ψ tanh A ξ 2 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equd_HTML.gif
      Substituting Eq. (16) into Eq. (15), together with Eq. (11) and simplifying, the obtained exact solutions become (if C1 = 0 but C2 ≠ 0; C2 = 0 but C1 ≠ 0) respectively:
      u 1 8 ξ = a 0 + b 1 d + Ψ cot A ξ 1 + b 2 d + Ψ cot A ξ 2 , u 1 9 ξ = a 0 + b 1 d Ψ tan A ξ 1 + b 2 d Ψ tan A ξ 2 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Eque_HTML.gif

      where ξ = x − Vt.

      Similarly, for set 2, substituting Eq. (17) into Eq. (15), along with Eq. (7) and simplifying, the traveling wave solutions become (if C1 = 0 but C2 ≠ 0; C2 = 0 but C1 ≠ 0) respectively:
      u 2 1 ξ = 1 2 a A 2 ( bV ( B 2 8 ) + A 2 ( k V + 1 ) + 3 bVΩ coth 2 ( Ω 2 A ξ ) ) , u 2 2 ξ = 1 2 a A 2 ( bV ( B 2 8 ) + A 2 ( k V + 1 ) + 3 bVΩ tanh 2 ( Ω 2 A ξ ) ) , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equf_HTML.gif
      Substituting Eq. (17) into Eq. (15), along with Eq. (8) and simplifying, yields exact solutions (if C1 = 0 but C2 ≠ 0; C2 = 0 but C1 ≠ 0) respectively:
      u 2 3 ξ = 1 2 a A 2 ( bV ( B 2 8 ) + A 2 ( k V + 1 ) 3 bVΩ cot 2 ( Ω 2 A ξ ) ) , u 2 4 ξ = 1 2 a A 2 ( bV ( B 2 8 ) + A 2 ( k V + 1 ) 3 bVΩ tan 2 ( Ω 2 A ξ ) ) , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equg_HTML.gif
      Substituting Eq. (17) into Eq. (15), along with Eq. (9) and simplifying, our obtained solution becomes:
      u 2 5 ξ = 1 2 a A 2 ( A 2 k V + 1 2 bV B 2 + 4 + 12 bV Ψ 2 C 2 C 1 + C 2 ξ 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equh_HTML.gif
      Substituting Eq. (17) into Eq. (15), together with Eq. (10) and simplifying, yields following traveling wave solutions (if C1 = 0 but C2 ≠ 0; C2 = 0 but C1 ≠ 0) respectively:
      u 2 6 ξ = 1 2 a A 2 ( bV ( B 2 8 ) + A 2 ( k V + 1 ) 12 bV ( B coth ( A ξ ) coth 2 ( A ξ ) ) ) , u 2 7 ξ = 1 2 a A 2 ( bV ( B 2 8 ) + A 2 ( k V + 1 ) 12 bV ( B tanh ( A ξ ) tanh 2 ( A ξ ) ) ) , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equi_HTML.gif
      Substituting Eq. (17) into Eq. (15), along with Eq. (11) and simplifying, our exact solutions become (if C1 = 0 but C2 ≠ 0; C2 = 0 but C1 ≠ 0) respectively:
      u 2 8 ξ = 1 2 a A 2 ( bV ( B 2 8 ) + A 2 ( k V + 1 ) 12 bV ( iB cot ( A ξ ) + cot 2 ( A ξ ) ) ) , u 2 9 ξ = 1 2 a A 2 ( bV ( B 2 8 ) + A 2 ( k V + 1 ) + 12 bV ( iB tan ( A ξ ) tan 2 ( A ξ ) ) , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equj_HTML.gif

      where ξ = x − Vt.

      Similarly, For set 3, substituting Eq. (18) into Eq. (15), together with Eq. (7) and simplifying, yields following traveling wave solutions (if C1 = 0 but C2 ≠ 0; C2 = 0 but C1 ≠ 0) respectively:
      u 3 1 ξ = a 0 + 3 bVΩ 2 a A 2 coth 2 ( Ω 2 A ξ ) + 4 b 2 Ψ 2 Ω tanh 2 ( Ω 2 A ξ ) , u 3 2 ξ = a 0 + 3 bVΩ 2 a A 2 tanh 2 ( Ω 2 A ξ ) + 4 b 2 Ψ 2 Ω coth 2 ( Ω 2 A ξ ) , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equk_HTML.gif
      Substituting Eq. (18) into Eq. (15), along with Eq. (8) and simplifying, we obtain following solutions (if C1 = 0 but C2 ≠ 0; C2 = 0 but C1 ≠ 0) respectively:
      u 3 3 ξ = a 0 3 bVΩ 2 a A 2 cot 2 ( Ω 2 A ξ ) 4 b 2 Ψ 2 Ω tan 2 ( Ω 2 A ξ ) u 3 4 ξ = a 0 3 bVΩ 2 a A 2 tan 2 ( Ω 2 A ξ ) 4 b 2 Ψ 2 Ω cot 2 ( Ω 2 A ξ ) http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equl_HTML.gif
      Substituting Eq. (18) into Eq. (15), along with Eq. (9) and simplifying, our obtained solution becomes:
      u 3 5 ξ = a 0 + 6 bV Ψ 2 a A 2 C 2 C 1 + C 2 ξ 2 + b 2 C 2 C 1 + C 2 ξ 2 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equm_HTML.gif
      Substituting Eq. (18) into Eq. (15), along with Eq. (10) and simplifying, yields following exact traveling wave solutions (if C1 = 0 but C2 ≠ 0; C2 = 0 but C1 ≠ 0) respectively:
      u 3 6 ξ = a 0 + 6 bV Ψ 2 a A 2 B 2 Ψ + Ψ coth A ξ 2 + b 2 B 2 Ψ + Ψ coth A ξ 2 , u 3 7 ξ = a 0 + 6 bV Ψ 2 a A 2 B 2 Ψ + Ψ tanh A ξ 2 + b 2 B 2 Ψ + Ψ tanh A ξ 2 http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equn_HTML.gif
      Substituting Eq. (18) into Eq. (15), along with Eq. (11) and simplifying, the obtained solutions become (if C1 = 0 but C2 ≠ 0; C2 = 0 but C1 ≠ 0) respectively:
      u 3 8 ξ = a 0 + 6 bV Ψ 2 a A 2 B 2 Ψ + Ψ cot A ξ 2 + b 2 B 2 Ψ + Ψ cot A ξ 2 u 3 9 ξ = a 0 + 6 bV Ψ 2 a A 2 B 2 Ψ Ψ tan A ξ 2 + b 2 B 2 Ψ Ψ tan A ξ 2 http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equo_HTML.gif

      where ξ = x − Vt.

      Remark

      The solutions obtained in this article have been checked by putting them back into the original equation and found correct.

      Discussions

      The advantages and validity of the method over the basic (G′/G)-expansion method have been discussed in the following.

      Advantages

      The crucial advantage of the new generalized (G′/G)-expansion method over the basic (G′/G)-expansion method is that the method provides more general and large amount of new exact traveling wave solutions with several free parameters. The exact solutions have its great importance to expose the inner mechanism of the complex physical phenomena. Apart from the physical application, the close-form solutions of nonlinear evolution equations assist the numerical solvers to compare the accuracy of their results and help them in the stability analysis.

      Validity

      Feng and Zheng (2010) investigated the well-established KP-BBM equation to obtain exact solutions via the basic (G′/G)-expansion method and achieved only three solutions (A.1),(A.2),(A.3) (see Appendix). On the other hand, twenty seven solutions are constructed of this equation by applying the new approach of generalized (G′/G)-expansion method. They used the linear ordinary differential equation as an auxiliary equation and traveling wave solutions presented in the form u ξ = i = 0 m a i G / G i , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_IEq2_HTML.gif where a m  ≠ 0. It is noteworthy to point out that some of our solutions are coincided with the solutions obtained by Feng and Zheng (2010) if the parameters are taken particular values, which validate our solutions.

      Conclusion

      The new generalized (G′/G)-expansion method established by Naher and Abdullah has successfully been implemented to construct new and more general exact traveling wave solutions of the KP-BBM equation. The method offers solutions with free parameters that might be important to explain some complex physical phenomena. Comparing the currently proposed method with other methods, such as (G′/G)-expansion method, the Exp-function method and the modified simple equation method, we might conclude that the exact solutions to Eq. (12) can be investigated by simple and systematic way. This study shows that the new generalized (G′/G)-expansion method is quite efficient and practically well suited to be used in finding exact solutions of NLEEs. Also, we observe that the new generalized (G′/G)-expansion method is straightforward and can be applied to many other nonlinear evolution equations.

      Appendix

      Feng and Zheng’s solutions (Feng and Zheng 2010)

      Feng and Zheng (2010) established exact solutions of the well-known the KP-BBM equation by using the basic (G′/G)-expansion method which are as follows:

      When λ2 − 4 μ > 0,
      u 1 = a 0 a 2 4 λ 2 + a 2 4 λ 2 4 μ C 1 sinh 1 2 λ 2 4 μ ξ + C 2 cosh 1 2 λ 2 4 μ ξ C 1 cosh 1 2 λ 2 4 μ ξ + C 2 sinh 1 2 λ 2 4 μ ξ 2 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ19_HTML.gif
      (A.1)

      where ξ = x + y 1 6 a a 2 b t http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_IEq3_HTML.gif and C1, C2 are arbitrary constants.

      When λ2 − 4 μ < 0,
      u 2 = a 0 a 2 4 λ 2 + a 2 4 4 μ λ 2 C 1 sin 1 2 4 μ λ 2 ξ + C 2 cos 1 2 4 μ λ 2 ξ C 1 cos 1 2 4 μ λ 2 ξ + C 2 sin 1 2 4 μ λ 2 ξ 2 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ20_HTML.gif
      (A.2)

      where ξ = x + y 1 6 a a 2 b t http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_IEq4_HTML.gif and C1, C2 are arbitrary constants.

      When λ2 − 4 μ = 0,
      u 5 = 1 4 λ 2 a 2 + a 2 C 2 2 C 1 + C 2 ξ 2 + a 0 , http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_Equ21_HTML.gif
      (A.3)

      where ξ = x + y 1 6 a a 2 b t http://static-content.springer.com/image/art%3A10.1186%2F2193-1801-2-617/MediaObjects/40064_2013_Article_666_IEq5_HTML.gif and C1, C2 are arbitrary constants.

      Declarations

      Acknowledgements

      The authors would like to express their sincere thanks to the anonymous referees for their valuable comments and suggestions. The authors also would like to convey their appreciation to Md. Tarik-ul Islam, Professor, Department of English, University of Rajshahi, Bangladesh for his assistance in editing the English language.

      Authors’ Affiliations

      (1)
      Department of Mathematics, Pabna University of Science and Technology
      (2)
      Department of Applied Mathematics, University of Rajshahi

      References

      1. Ablowitz MJ, Clarkson PA: Soliton, nonlinear evolution equations and inverse scattering. New York: Cambridge University Press; 1991.View Article
      2. Akbar MA, Ali NHM: New solitary and periodic solutions of nonlinear evolution equation by Exp-function method. World Appl Sci J 2012, 17(12):1603-1610.
      3. Akbar MA, Ali NHM, Zayed EME: Abundant exact traveling wave solutions of the generalized Bretherton equation via ( G ′/ G )-expansion method. Commun Theor Phys 2012, 57: 173-178.View Article
      4. Akbar MA, Ali NHM, Zayed EME M: A generalized and improved ( G ′/ G )-expansion method for nonlinear evolution equations. Math Prob Engr 2012, vol. 2012: 22 pages. doi: 10.1155/2012/459879
      5. Akbar MA, Ali NHM, Mohyud-Din ST: The alternative ( G ′/ G )-expansion method with generalized Riccati equation: application to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equation. Int J Phys Sci 2012, 7(5):743-752.
      6. Akbar MA, Ali NHM, Mohyud-Din ST: Some new exact traveling wave solutions to the (3 + 1)-dimensional Kadomtsev-Petviashvili equation. World Appl Sci J 2012, 16(11):1551-1558.
      7. Akbar MA, Ali NHM, Mohyud-Din ST: Further exact traveling wave solutions to the (2 + 1)-dimensional Boussinesq and Kadomtsev-Petviashvili equation. J Comput Anal Appl 2012, 15(3):557-571.
      8. Bekir A: Application of the ( G ′/ G )-expansion method for nonlinear evolution equations. Phys Lett A 2008, 372: 3400-3406.View Article
      9. Bibi S, Mohyud-Din ST: Traveling wave solutions of KdVs using sine-cosine method. J Assn Arab Univ Basic Appl Sci 2013. doi: org/10.1016/j.jaubas.2013.03.006 (in press)
      10. Chen Y, Wang Q: Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to (1 + 1)-dimensional dispersive long wave equation. Chaos Soliton Fract 2005, 24: 745-757.View Article
      11. Chow KW: A class of exact periodic solutions of nonlinear envelope equation. J Math Phys 1995, 36: 4125-4137.View Article
      12. Fan E: Extended tanh-function method and its application to nonlinear equations. Phys Lett A 2000, 277(4–5):212-218.View Article
      13. Feng X: Exploratory approach to explicit solution of nonlinear evolutions equations. Int J Theo Phys 2000, 39: 207-222.View Article
      14. Feng Z: The first-integral method to study the Burgers-Korteweg-de Vries equation. J Phys A 2002, 35(2):343-349.View Article
      15. Feng Q, Zheng B: Traveling wave solutions for three nonlinear equations by ( G '/ G )-expansion method. WSEAS Trans Computers 2010, 9(3):1109-2750.
      16. Ganji DD: The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer. Phys Lett A 2006, 355: 137-141.View Article
      17. Ganji DD, Rafei M: Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method. Phys Lett A 2006, 356: 131-137.View Article
      18. Ganji DD, Afrouzi GA, Talarposhti RA: Application of variational iteration method and homotopy perturbation method for nonlinear heat diffusion and heat transfer equations. Phys Lett A 2007, 368: 450-457.View Article
      19. He JH, Wu XH: Exp-function method for nonlinear wave equations. Chaos Soliton Fract 2006, 30: 700-708.View Article
      20. Hu JL: A new method for finding exact traveling wave solutions to nonlinear partial differential equations. Phys Lett A 2001, 286: 175-179.View Article
      21. Hu JL: Explicit solutions to three nonlinear physical models. Phys Lett A 2001, 287: 81-89.View Article
      22. Jawad AJM, Petkovic MD, Biswas A: Modified simple equation method for nonlinear evolution equations. Appl Math Comput 2010, 217: 869-877.View Article
      23. Khan K, Akbar MA, Ali NHM: The modified simple equation method for exact and solitary wave solutions of nonlinear evolution equation: the GZK-BBM equation and right handed non-commutative Burger equations. ISRN Math Phys Vol 2013, 2013(2013):Article ID 146704. 5 pages
      24. Malfliet W: Solitary wave solutions of nonlinear wave equations. Am J Phys 1992, 60(7):650-654.View Article
      25. Matveev VB, Salle MA: Darboux transformation and solitons. Berlin: Springer; 1991.View Article
      26. Miura MR: Backlund transformation. Berlin: Springer; 1978.
      27. Mohyud-din ST, Noor MA, Noor KI: Exp-function method for traveling wave solutions of modified Zakharov–Kuznetsov equation. J King Saud Univ 2010, 22: 213-216.View Article
      28. Naher H, Abdullah FA: New approach of ( G ′/ G )-expansion method and new approach of generalized ( G '/ G )-expansion method for nonlinear evolution equation. AIP Adv 2013, 3: 032116. doi: 10. 1063/1.4794947View Article
      29. Porubov AV: Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid layer. Phys Lett A 1996, 221: 391-394.View Article
      30. Tascan F, Bekir A: Applications of the first integral method to nonlinear evolution equations. Chin Phys B 2010, 19(8):080201.View Article
      31. Wang M: Solitary wave solutions for variant Boussinesq equations. Phys Lett A 1995, 199: 169-172.View Article
      32. Wang ML: Exact solutions for a compound KdV-Burgers equation. Phys Lett A 1996, 213: 279-287.View Article
      33. Wang M, Li X, Zhang J: The ( G ′/ G )-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics. Phys Lett A 2008, 372: 417-423.View Article
      34. Wazwaz AM: Partial Differential equations: Method and Applications. London: Taylor and Francis; 2002.
      35. Wazwaz AM: A sine-cosine method for handing nonlinear wave equations. Math Comput Model 2004, 40: 499-508.View Article
      36. Wazwaz AM: The tanh and the sine-cosine methods for the complex modified KdV and the generalized KdV equations. Comput Math Appl 2005, 49(7–8):1101-1112.View Article
      37. Xu G: An elliptic equation method and its applications in nonlinear evolution equations. Chaos Soliton Fract 2006, 29: 942-947.View Article
      38. Yan Z, Zhang H: New explicit solitary wave solutions and periodic wave solutions for Whitham Broer-Kaup equation in shallow water. Phys Lett A 2001, 285(5–6):355-362.View Article
      39. Yusufoglu E, Bekir A: Exact solution of coupled nonlinear evolution equations. Chaos Soliton Fract 2008, 37: 842-848.View Article
      40. Zayed EME: The ( G ′/ G )-expansion method and its applications to some nonlinear evolution equations in the mathematical physics. J Appl Math Comput 2009, 30: 89-103.View Article
      41. Zayed EME, Ibrahim SAH: Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method. Chin Phys Lett 2012, 29(06):060201.View Article
      42. Zayed EME, Zedan HA, Gepreel KA: On the solitary wave solutions for nonlinear Euler equations. Appl Anal 2004, 83: 1101-1132.View Article
      43. Zayed EME, Zedan HA, Gepreel KA: On the solitary wave solutions for nonlinear Hirota-Sasuma coupled KDV equations. Chaos Soliton Fract 2004, 22: 285-303.View Article
      44. Zayed EME, Abourabia AM, Gepreel KA, Horbaty MM: On the rational solitary wave solutions for the nonlinear Hirota Satsuma coupled KdV system. Appl Anal 2006, 85: 751-768.View Article
      45. Zhang S, Tong J, Wang W: A generalized ( G ′/ G )-expansion method for the mKdV equation with variable coefficients. Phys Lett A 2008, 372: 2254-2257.View Article
      46. Zhang J, Jiang F, Zhao X: An improved ( G ′/ G )-expansion method for solving nonlinear evolution equations. Int J Com Math 2010, 87(8):1716-1725.View Article

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