Extended generalized Riccati equation mapping method for the fifth-order Sawada-Kotera equation

Naher, Hasibun; Abdullah, Farah Aini; Mohyud-Din, Syed Tauseef

doi:10.1063/1.4804433

In this article, the generalized Riccati equation mapping together with the basic (G′/G)-expansion method is implemented which is advance mathematical tool to investigate nonlinear partial differential equations. Moreover, the auxiliary equation G′(ϕ) = h + f G(ϕ) + g G²(ϕ) is used with arbitrary constant coefficients and called the generalized Riccati equation. By applying this method, we have constructed abundant traveling wave solutions in a uniform way for the Sawada-Kotera equation. The obtained solutions of this equation have vital and noteworthy explanations for some practical physical phenomena.

I. INTRODUCTION

The rapid developments of nonlinear sciences, a wide range of straightforward and effective methods have been introduced to obtain traveling wave solutions of nonlinear PDEs. Such as, the truncated Painleve expansion method,^1,2 the Hirota's bilinear transformation method,³ the weierstrass elliptic function method,⁴ the inverse scattering method,⁵ the Backlund transformation method,^6–8 the Jacobi elliptic function expansion method,^9–11 the tanh-coth method,^12,13 the direct algebraic method,¹⁴ the Cole-Hopf transformation method,¹⁵ the homotopy perturbation method,^16–19 the Exp-function method,^20–25 the variational iteration method^26–32 and others.^33–41

Recently, Wang et al.⁴² presented a widely used simple and concise method which is called the (G′/G)-expansion method for constructing traveling wave solutions of some nonlinear evolution equations (NLEEs). In addition, in this method the second order linear ordinary differential equation G′′(ϕ) + λG′(ϕ) + μG(ϕ) = 0 is implemented, as an auxiliary equation, where λ and μ are arbitrary constants. Subsequently, many researchers studied many nonlinear PDEs to obtain exact traveling wave solutions via this powerful (G′/G)-expansion method.^43–51

Zhu⁵² implemented the generalized Riccati equation mapping with the extended tanh-function method to investigate the (2+1)-dimensional Boiti-Leon-Pempinelle equation. Moreover, G′(ϕ) = h + f G(ϕ) + g G²(ϕ) is utilized as an auxiliary equation and called generalized Riccati equation, where f, g and h are arbitrary constants. Li et al.⁵³ concerned about the generalized Riccati equation expansion method to study higher-dimensional Jimbo-Miwa equation. In Ref. 54 Bekir and Cevikel investigated nonlinear coupled equation in mathematical physics via the tanh-coth method combined with the Riccati equation while Salas⁵⁵ applied the projective Riccati equation method to obtain some exact solutions for the Caudrey-Dodd-Gibbon equation. In Ref. 56 Guo et al. executed the extended Riccati equation mapping method to study the diffusion-reaction and the mKdV equation with variable coefficient whereas Naher and Abdullah⁵⁷ established traveling wave solutions of the modified Benjamin-Bona-Mahony equation by applying the generalized Riccati equation mapping with the basic (G′/G)-expansion method and so on.

Many researchers constructed traveling wave solutions of the fifth order Sawada-Kotera equation by using different methods. Such as, Liu and Dai⁵⁸ executed the Hirota's bilinear method to obtain exact solutions of the same equation. Feng and Zheng⁵⁹ investigated this equation to establish traveling wave solutions via the (G′/G)-expansion method. In Ref. 60, Wazwaz implemented the extended tanh method for constructing analytical solutions of the same equation. Salas⁶¹ concerned about the projective Riccati equation method to obtain exact solutions of this equation. But, to the best of our knowledge, the fifth-order Sawada-Kotera equation is not investigated to construct exact solutions by applying the generalized Riccati equation mapping together with the(G′/G)-expansion method. The obtained traveling wave solutions in this article are new and have not been found in the previous literature.

The importance of our present work is, in order to construct abundant traveling wave solutions including solitons, periodic and rational solutions, and a Sawada-Kotera equation is considered by applying the extended generalized Riccati equation mapping method.

II. THE EXTENDED GENERALIZED RICCATI EQUATION MAPPING METHOD

Suppose the general nonlinear partial differential equation:

\begin{equation}S(u,\,u_x,\,u_t,\,u_{xx},\,u_{xt},\,u_{tt},...) = 0,\end{equation}

S (u, u_{x}, u_{t}, u_{x x}, u_{x t}, u_{t t}, . . .) = 0,

(1)

where u = u(x, t) is an unknown function, S is a polynomial in u(x, t) and the subscripts indicate the partial derivatives.

The most important steps of the generalized Riccati equation mapping together with the (G′/G)-expansion method^42,52 are as follows:

Step 1. Consider the traveling wave variable:

\begin{equation}u\left( {x,t} \right) = q\left( \phi \right),\quad \phi = x - Qt,\end{equation}

u (x, t) = q (ϕ), ϕ = x - Q t,

(2)

where Q is the wave speed. Now using Eq. (2), Eq. (1) is converted into an ordinary differential equation for q(ϕ):

\begin{equation}M\left( {q,q',q'',q''',...} \right) = 0,\end{equation}

M (q, q^{'}, q^{''}, q^{'''}, . . .) = 0,

(3)

where the superscripts stand for the ordinary derivatives with respect to ϕ.

Step 2. 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:^42,52

\begin{equation}q( \phi ) = \sum\limits_{j = 0}^m {b_j ( {G'/G} )} ^j \end{equation}

q (ϕ) = \sum_{j = 0}^{m} {b_{j} (G^{'} / G)}^{j}

(4)

where b_j (j = 0, 1, 2, ..., m) and b_m ≠ 0, with G = G(ϕ) is the solution of the generalized Riccati equation:

\begin{equation}G'( \phi ) = h + f\,G( \phi ) + g\,G^2 ( \phi )\end{equation}

G^{'} (ϕ) = h + f G (ϕ) + g G^{2} (ϕ)

(5)

where f, g, h are arbitrary constants and g ≠ 0.

Step 4. To decide the positive integer m, 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 polynomials in G ^k(ϕ) and G^−k(ϕ), (k = 0, 1, 2, ...). Then equating each coefficient of the polynomials to zero, yield a set of algebraic equations for b_j (j = 0, 1, 2, ..., m), f, g, h and Q.

Step 6. Solve the system of algebraic equations which are found in Step 5 with the aid of algebraic software Maple to obtain values for b_j (j = 0, 1, 2, ..., m), f, g, h and Q. Then, substitute obtained values in Eq. (4) along with Eq. (5) with the value of n, we obtain exact solutions of Eq. (1).

In the following, we have twenty seven solutions including four different families of Eq. (5).

Family 2.1: When f² − 4gh < 0 and f g ≠ 0 or gh ≠ 0 , the solutions of Eq. (5) are:

\begin{eqnarray*}G_1 &=& \frac{1}{{2g}}\left( { - f + \sqrt {4gh - f^2 } \tan \left( {\frac{{\sqrt {4gh - f^2 } }}{2}\,\phi } \right)} \right),\\G_2 &=& \frac{{ - 1}}{{2g}}\left( {f + \sqrt {4gh - f^2 } \cot \left( {\frac{{\sqrt {4gh - f^2 } }}{2}\,\phi } \right)} \right),\\G_3 &=& \frac{1}{{2g}}\left( { - \,f + \sqrt {4gh - f^2 } \,\left( {\tan \left( {\sqrt {4gh - f^2 } \,\phi } \right) \pm \sec \left( {\sqrt {4gh - f^2 } \,\phi } \right)} \right)} \right),\\G_4 &=& \frac{{ - 1}}{{2g}}\left( {f + \sqrt {4gh - f^2 } \,\left( {\cot \left( {\sqrt {4gh - f^2 } \,\phi } \right) \pm \csc \left( {\sqrt {4gh - f^2 } \,\phi } \right)} \right)} \right),\\G_5 &=& \frac{1}{{4g}}\left( { - 2f + \sqrt {4gh - f^2 } \,\left( {\tan \left( {\frac{{\sqrt {4gh - f^2 } \,}}{4}\phi } \right) - \cot \left( {\frac{{\sqrt {4gh - f^2 } \,}}{4}\,\,\phi } \right)} \right)} \right),\\G_6 &=& \frac{1}{{2g}}\left( { - \,f + \,\,\frac{{\sqrt {\left( {A^2 - B^2 } \right)\left( {4gh - f^2 } \right)} \,\,\, - A\sqrt {4gh - f^2 } \cos \left( {\sqrt {4gh - f^2 } \,\phi } \right)}}{{A\sin \left( {\sqrt {4gh - f^2 } \,\phi } \right) + B}}\,} \right),\\G_7 &=& \frac{1}{{2g}}\left( { - f - \,\,\frac{{\sqrt {\left( {A^2 - B^2 } \right)\left( {4gh - f^2 } \right)} \,\,\, + A\sqrt {4gh - f^2 } \cos \,\left( {\sqrt {4gh - f^2 } \,\phi } \right)}}{{A\sin \left( {\sqrt {4gh - f^2 } \,\phi } \right) + B}}\,} \right),\end{eqnarray*}

\begin{matrix} G_{1} & = & \frac{1}{2 g} (- f + \sqrt{4 g h - f^{2}} \tan (\frac{\sqrt{4 g h - f^{2}}}{2} ϕ)), \\ G_{2} & = & \frac{- 1}{2 g} (f + \sqrt{4 g h - f^{2}} \cot (\frac{\sqrt{4 g h - f^{2}}}{2} ϕ)), \\ G_{3} & = & \frac{1}{2 g} (- f + \sqrt{4 g h - f^{2}} (\tan (\sqrt{4 g h - f^{2}} ϕ) \pm \sec (\sqrt{4 g h - f^{2}} ϕ))), \\ G_{4} & = & \frac{- 1}{2 g} (f + \sqrt{4 g h - f^{2}} (\cot (\sqrt{4 g h - f^{2}} ϕ) \pm \csc (\sqrt{4 g h - f^{2}} ϕ))), \\ G_{5} & = & \frac{1}{4 g} (- 2 f + \sqrt{4 g h - f^{2}} (\tan (\frac{\sqrt{4 g h - f^{2}}}{4} ϕ) - \cot (\frac{\sqrt{4 g h - f^{2}}}{4} ϕ))), \\ G_{6} & = & \frac{1}{2 g} (- f + \frac{\sqrt{(A^{2} - B^{2}) (4 g h - f^{2})} - A \sqrt{4 g h - f^{2}} \cos (\sqrt{4 g h - f^{2}} ϕ)}{A \sin (\sqrt{4 g h - f^{2}} ϕ) + B}), \\ G_{7} & = & \frac{1}{2 g} (- f - \frac{\sqrt{(A^{2} - B^{2}) (4 g h - f^{2})} + A \sqrt{4 g h - f^{2}} \cos (\sqrt{4 g h - f^{2}} ϕ)}{A \sin (\sqrt{4 g h - f^{2}} ϕ) + B}), \end{matrix}

where A and B are two non-zero real constants and satisfies A² − B² > 0.

\begin{eqnarray*}G_8 &=& \frac{{ - 2\,h\cos \left( {\frac{{\sqrt {4gh - f^2 } }}{2}\,\phi } \right)}}{{\sqrt {4gh - f^2 } \sin \left( {\frac{{\sqrt {4gh - f^2 } }}{2}\,\phi } \right) + f\cos \left( {\frac{{\sqrt {4gh - f^2 } }}{2}\phi } \right)}},\\G_9 &=& \frac{{2\,h\sin \left( {\frac{{\sqrt {4gh - f^2 } }}{2}\phi } \right)}}{{ - \,f\sin \left( {\frac{{\sqrt {4gh - f^2 } }}{2}\phi } \right) + \sqrt {4gh - f^2 } \cos \left( {\frac{{\sqrt {4gh - f^2 } }}{2}\phi } \right)}},\\G_{10} &=& \frac{{ - \,2\,h \cos \left( {\sqrt {4gh - f^2 } \,\phi } \right)}}{{\sqrt {4gh - f^2 } \sin \left( {\sqrt {4gh - f^2 } \,\phi } \right) + f\cos \left( {\sqrt {4gh - f^2 } \,\phi } \right) \pm \sqrt {4gh - f^2 } \,}},\\G_{11} &=& \frac{{2\,h\sin \left( {\sqrt {4gh - f^2 } \,\phi } \right)}}{{ - \,f\sin \left( {\sqrt {4gh - f^2 } \phi } \right) + \sqrt {4gh - f^2 } \cos \left( {\sqrt {4gh - f^2 } \,\phi } \right) \pm \sqrt {4gh - f^2 } \,}},\\G_{12}\! &=&\! \frac{{4\,h\sin \left( {\frac{{\sqrt {4gh - f^2 } \,}}{4}\,\phi } \right)\cos \left( {\frac{{\sqrt {4gh - f^2 }}}{4}\phi } \right)}}{{ - 2f\sin \left( {\frac{{\sqrt {4gh - f^2 }}}{4}\,\phi } \right)\cos \left( {\frac{{\sqrt {4gh - f^2 }}}{4}\,\phi } \right)\! +\! 2\sqrt {4gh - f^2 } \cos ^2 \left( {\frac{{\sqrt {4gh - f^2 } \,}}{4}\phi } \right)\! -\! \sqrt {4gh - f^2 } }},\end{eqnarray*}

\begin{matrix} G_{8} & = & \frac{- 2 h \cos (\frac{\sqrt{4 g h - f^{2}}}{2} ϕ)}{\sqrt{4 g h - f^{2}} \sin (\frac{\sqrt{4 g h - f^{2}}}{2} ϕ) + f \cos (\frac{\sqrt{4 g h - f^{2}}}{2} ϕ)}, \\ G_{9} & = & \frac{2 h \sin (\frac{\sqrt{4 g h - f^{2}}}{2} ϕ)}{- f \sin (\frac{\sqrt{4 g h - f^{2}}}{2} ϕ) + \sqrt{4 g h - f^{2}} \cos (\frac{\sqrt{4 g h - f^{2}}}{2} ϕ)}, \\ G_{10} & = & \frac{- 2 h \cos (\sqrt{4 g h - f^{2}} ϕ)}{\sqrt{4 g h - f^{2}} \sin (\sqrt{4 g h - f^{2}} ϕ) + f \cos (\sqrt{4 g h - f^{2}} ϕ) \pm \sqrt{4 g h - f^{2}}}, \\ G_{11} & = & \frac{2 h \sin (\sqrt{4 g h - f^{2}} ϕ)}{- f \sin (\sqrt{4 g h - f^{2}} ϕ) + \sqrt{4 g h - f^{2}} \cos (\sqrt{4 g h - f^{2}} ϕ) \pm \sqrt{4 g h - f^{2}}}, \\ G_{12} & = & \frac{4 h \sin (\frac{\sqrt{4 g h - f^{2}}}{4} ϕ) \cos (\frac{\sqrt{4 g h - f^{2}}}{4} ϕ)}{- 2 f \sin (\frac{\sqrt{4 g h - f^{2}}}{4} ϕ) \cos (\frac{\sqrt{4 g h - f^{2}}}{4} ϕ) + 2 \sqrt{4 g h - f^{2}} \cos^{2} (\frac{\sqrt{4 g h - f^{2}}}{4} ϕ) - \sqrt{4 g h - f^{2}}}, \end{matrix}

Family 2.2: When f² − 4gh > 0 and f g ≠ 0 or gh ≠ 0 , the solutions of Eq. (5) are:

\begin{eqnarray*}G_{13} &=& \frac{{ - 1}}{{2g}}\left( {f + \sqrt {f^2 - 4gh} \tanh \left( {\frac{{\sqrt {f^2 - 4gh} }}{2}\,\phi } \right)} \right),\\G_{14} &=& \frac{{ - 1}}{{2g}}\left( {f + \sqrt {f^2 - 4gh} \coth \left( {\frac{{\sqrt {f^2 - 4gh} }}{2}\,\phi } \right)} \right),\\[-4pt]G_{15} &=& \frac{{ - 1}}{{2g}}\left( {f + \sqrt {f^2 - 4gh} \,\left( {\tanh \left( {\sqrt {f^2 - 4gh} \,\phi } \right) \pm i\sec h\left( {\sqrt {f^2 - 4gh} \,\phi } \right)} \right)} \right),\\[-4pt]G_{16} &=& \frac{{ - 1}}{{2g}}\left( {f + \sqrt {f^2 - 4gh} \left( {\coth \left( {\sqrt {f^2 - 4gh} \,\phi } \right) \pm \csc h\left( {\sqrt {f^2 - 4gh} \,\phi } \right)} \right)\,} \right),\\[-4pt]G_{17} &=& \frac{{ - 1}}{{4g}}\left( {2f + \sqrt {f^2 - 4gh} \,\left( {\tanh \left( {\frac{{\sqrt {f^2 - 4gh} }}{4}\,\,\phi } \right) + \cot h\left( {\frac{{\sqrt {f^2 - 4gh} }}{4}\,\,\phi } \right)} \right)} \right),\\[-4pt]G_{18} &=& \frac{1}{{2g}}\left( { - f + \frac{{\sqrt {\left( {A^2 + B^2 } \right)\left( {f^2 - 4gh} \right)} \,\,\, - A\sqrt {f^2 - 4gh} \cosh \left( {\sqrt {f^2 - 4gh} \,\phi } \right)}}{{A\sinh \left( {\sqrt {f^2 - 4gh} \,\phi } \right) + B}}\,} \right),\\G_{19} &=& \frac{1}{{2g}}\left( { - f - \,\,\frac{{\sqrt {\left( {B^2 - A^2 } \right)\left( {f^2 - 4gh} \right)} \,\,\, + A\sqrt {f^2 - 4gh} \sinh \left( {\sqrt {f^2 - 4gh} \,\phi } \right)}}{{A\cosh \left( {\sqrt {f^2 - 4gh} \,\phi } \right) + B}}\,} \right),\end{eqnarray*}

\begin{matrix} G_{13} & = & \frac{- 1}{2 g} (f + \sqrt{f^{2} - 4 g h} \tanh (\frac{\sqrt{f^{2} - 4 g h}}{2} ϕ)), \\ G_{14} & = & \frac{- 1}{2 g} (f + \sqrt{f^{2} - 4 g h} \coth (\frac{\sqrt{f^{2} - 4 g h}}{2} ϕ)), \\ G_{15} & = & \frac{- 1}{2 g} (f + \sqrt{f^{2} - 4 g h} (\tanh (\sqrt{f^{2} - 4 g h} ϕ) \pm i \sec h (\sqrt{f^{2} - 4 g h} ϕ))), \\ G_{16} & = & \frac{- 1}{2 g} (f + \sqrt{f^{2} - 4 g h} (\coth (\sqrt{f^{2} - 4 g h} ϕ) \pm \csc h (\sqrt{f^{2} - 4 g h} ϕ))), \\ G_{17} & = & \frac{- 1}{4 g} (2 f + \sqrt{f^{2} - 4 g h} (\tanh (\frac{\sqrt{f^{2} - 4 g h}}{4} ϕ) + \cot h (\frac{\sqrt{f^{2} - 4 g h}}{4} ϕ))), \\ G_{18} & = & \frac{1}{2 g} (- f + \frac{\sqrt{(A^{2} + B^{2}) (f^{2} - 4 g h)} - A \sqrt{f^{2} - 4 g h} \cosh (\sqrt{f^{2} - 4 g h} ϕ)}{A \sinh (\sqrt{f^{2} - 4 g h} ϕ) + B}), \\ G_{19} & = & \frac{1}{2 g} (- f - \frac{\sqrt{(B^{2} - A^{2}) (f^{2} - 4 g h)} + A \sqrt{f^{2} - 4 g h} \sinh (\sqrt{f^{2} - 4 g h} ϕ)}{A \cosh (\sqrt{f^{2} - 4 g h} ϕ) + B}), \end{matrix}

where A and B are two non-zero real constants and satisfies A² − B² < 0.

\begin{eqnarray*}G_{20} &=& \frac{{2\,h\cosh \left( {\frac{{\sqrt {f^2 - 4gh} }}{2}\,\,\phi } \right)}}{{\sqrt {f^2 - 4gh} \sinh \left( {\frac{{\sqrt {f^2 - 4gh} }}{2}\,\,\phi } \right) - f\cosh \left( {\frac{{\sqrt {f^2 - 4gh} }}{2}\,\,\phi } \right)}},\\[-4pt]G_{21} &=& \frac{{ - 2\,h\sinh \left( {\frac{{\sqrt {f^2 - 4gh} }}{2}\,\,\phi } \right)}}{{f\sinh \left( {\frac{{\sqrt {f^2 - 4gh} }}{2}\,\,\phi } \right) - \sqrt {f^2 - 4gh} \cosh \left( {\frac{{\sqrt {f^2 - 4gh} }}{2}\,\,\phi } \right)}},\\[-4pt]G_{22} &=& \frac{{2\,h\cosh \left( {\sqrt {f^2 - 4gh} \,\phi } \right)}}{{\sqrt {f^2 - 4gh} \sinh \,\left( {\sqrt {f^2 - 4gh} \,\phi } \right) - r\cosh \left( {\sqrt {f^2 - 4gh} \,\phi } \right)\,\,\, \pm \,\,i\sqrt {f^2 - 4gh} \,}},\\[-4pt]G_{23} &=& \frac{{2\,h\sinh \left( {\sqrt {f^2 - 4gh} \,\phi } \right)}}{{ - f\sinh \,\left( {\sqrt {f^2 - 4gh} \,\phi } \right) + \sqrt {f^2 - 4gh} \cosh \,\left( {\sqrt {f^2 - 4gh} \,\phi } \right)\,\, \pm \,\,\sqrt {f^2 - 4gh} }},\\[-4pt]G_{24}\! &=&\! \frac{{4h\sinh \left(\! {\frac{{\sqrt {f^2 - 4gh} }}{4}\phi } \!\right)\cosh \left(\! {\frac{{\sqrt {f^2 - 4gh} }}{4}\phi }\! \right)}}{{ - 2f\sinh\left(\! {\frac{{\sqrt {f^2 - 4gh} }}{4}\phi }\! \right)\cosh\left(\! {\frac{{\sqrt {f^2 - 4gh} }}{4}\phi }\! \right)\! +\! 2\sqrt {f^2 - 4gh} \cosh ^2 \left(\! {\frac{{\sqrt {f^2 - 4gh} }}{4}\phi } \!\right)\! -\! \sqrt {f^2 - 4gh} }},\end{eqnarray*}

\begin{matrix} G_{20} & = & \frac{2 h \cosh (\frac{\sqrt{f^{2} - 4 g h}}{2} ϕ)}{\sqrt{f^{2} - 4 g h} \sinh (\frac{\sqrt{f^{2} - 4 g h}}{2} ϕ) - f \cosh (\frac{\sqrt{f^{2} - 4 g h}}{2} ϕ)}, \\ G_{21} & = & \frac{- 2 h \sinh (\frac{\sqrt{f^{2} - 4 g h}}{2} ϕ)}{f \sinh (\frac{\sqrt{f^{2} - 4 g h}}{2} ϕ) - \sqrt{f^{2} - 4 g h} \cosh (\frac{\sqrt{f^{2} - 4 g h}}{2} ϕ)}, \\ G_{22} & = & \frac{2 h \cosh (\sqrt{f^{2} - 4 g h} ϕ)}{\sqrt{f^{2} - 4 g h} \sinh (\sqrt{f^{2} - 4 g h} ϕ) - r \cosh (\sqrt{f^{2} - 4 g h} ϕ) \pm i \sqrt{f^{2} - 4 g h}}, \\ G_{23} & = & \frac{2 h \sinh (\sqrt{f^{2} - 4 g h} ϕ)}{- f \sinh (\sqrt{f^{2} - 4 g h} ϕ) + \sqrt{f^{2} - 4 g h} \cosh (\sqrt{f^{2} - 4 g h} ϕ) \pm \sqrt{f^{2} - 4 g h}}, \\ G_{24} & = & \frac{4 h \sinh (\frac{\sqrt{f^{2} - 4 g h}}{4} ϕ) \cosh (\frac{\sqrt{f^{2} - 4 g h}}{4} ϕ)}{- 2 f \sinh (\frac{\sqrt{f^{2} - 4 g h}}{4} ϕ) \cosh (\frac{\sqrt{f^{2} - 4 g h}}{4} ϕ) + 2 \sqrt{f^{2} - 4 g h} \cosh^{2} (\frac{\sqrt{f^{2} - 4 g h}}{4} ϕ) - \sqrt{f^{2} - 4 g h}}, \end{matrix}

Family 2.3: when h = 0 and gh ≠ 0, the solution Eq. (5) becomes:

\begin{equation*}G_{25} = \frac{{ - \,f\,c_1 }}{{g\left( {c_1 + \cosh \left( {f\phi } \right) - \sinh \left( {f\phi } \right)} \right)\,\,}},\end{equation*}

G_{25} = \frac{- f c_{1}}{g (c_{1} + \cosh (f ϕ) - \sinh (f ϕ))},

\begin{equation*}G_{26} = \frac{{ - \,f\left( {\cosh \left( {f\phi } \right) + \sinh \left( {f\phi } \right)} \right)}}{{g\left( {c_1 + \cosh \left( {f\phi } \right) + \sinh \left( {f\phi } \right)} \right)\,\,}},\end{equation*}

G_{26} = \frac{- f (\cosh (f ϕ) + \sinh (f ϕ))}{g (c_{1} + \cosh (f ϕ) + \sinh (f ϕ))},

where c₁ is an arbitrary constant.

Family 2.4: when g ≠ 0 and h = f = 0, the solution of Eq. (5) becomes:

\begin{equation*}G_{27} = \frac{{ - 1}}{{g\phi + l_1 }},\end{equation*}

G_{27} = \frac{- 1}{g ϕ + l_{1}},

where l₁ is an arbitrary constant.

III. APPLICATIONS OF THE METHOD

In this section, we have constructed new exact traveling wave solutions for the Sawada-Kotera equation by using the method.

A. The Sawada-Kotera equation

We consider the Sawada-Kotera equation followed by Liu and Dai:⁵⁸

\begin{equation}u_t + 45u^2 u_x + 15u_x u_{xx} + 15uu_{xxx} + u_{xxxxx} = 0.\end{equation}

u_{t} + 45 u^{2} u_{x} + 15 u_{x} u_{x x} + 15 u u_{x x x} + u_{x x x x x} = 0 .

(6)

Now, we use the transformation Eq. (2) into the Eq. (6), which yields:

\begin{equation}- Q\,q' + 45\,q^2 \,q' + 15\,q'\,q'' + 15\,q\,q''' + q^{\left( 5 \right)} = 0,\end{equation}

- Q q^{'} + 45 q^{2} q^{'} + 15 q^{'} q^{''} + 15 q q^{'''} + q^{(5)} = 0,

(7)

Eq. (7) is integrable, therefore, integrating with respect ϕ once yields:

\begin{equation}C - Qq + 15q^3 + 15qq'' + q^{\left( 4 \right)} = 0,\end{equation}

C - Q q + 15 q^{3} + 15 q q^{''} + q^{(4)} = 0,

(8)

where C is an integral constant which is to be determined later.

Taking the homogeneous balance between q³and q⁽⁴⁾ in Eq. (8), we obtain m = 2.

Therefore, the solution of Eq. (8) is of the form:

\begin{equation}q( \phi ) = b_2 ( {G'/G} )^2 + b_1 ( {G'/G} ) + b_0,\quad b_2 \ne 0.\end{equation}

q (ϕ) = b_{2} {(G^{'} / G)}^{2} + b_{1} (G^{'} / G) + b_{0}, b_{2} \neq 0 .

(9)

Using Eq. (5), Eq. (9) can be re-written as:

\begin{equation}q( \phi ) = b_2 ( {f + h\,G^{ - 1} + g\,G} )^2 + b_1 ( {f + h\,G^{ - 1} + g\,G} ) + b_0,\end{equation}

q (ϕ) = b_{2} {(f + h G^{- 1} + g G)}^{2} + b_{1} (f + h G^{- 1} + g G) + b_{0},

(10)

where f, g and h are free parameters.

By substituting Eq. (10) into Eq. (8), collecting all coefficients of G^k and G^−k (k = 0, 1, 2, ...) and setting them equal to zero, we obtain a set of algebraic equations for b₀, b₁, b₂, f, g, h, C and Q (algebraic equations are not shown, for simplicity). Solving the system of algebraic equations with the help of algebraic software Maple, we obtain

\begin{eqnarray*}b_0 &=& \frac{{ - f^2 }}{3} + \frac{{4gh}}{3},\quad b_1 = 2p,\quad b_2 = - 2,\,Q = f^4 - 8f^2 gh + 16g^2 h^2,\nonumber\\[6pt]C &=& \frac{{32f^2 g^2 h^2 }}{3} - \frac{{128g^3 h^3 }}{9} + \frac{{2f^6 }}{9} - \frac{{8f^4 gh}}{3},\end{eqnarray*}

\begin{matrix} b_{0} & = & \frac{- f^{2}}{3} + \frac{4 g h}{3}, b_{1} = 2 p, b_{2} = - 2, Q = f^{4} - 8 f^{2} g h + 16 g^{2} h^{2}, \\ C & = & \frac{32 f^{2} g^{2} h^{2}}{3} - \frac{128 g^{3} h^{3}}{9} + \frac{2 f^{6}}{9} - \frac{8 f^{4} g h}{3}, \end{matrix}

Family 3.1: The periodic form solutions of Eq. (6) (when f² − 4gh < 0 and f g ≠ 0 or g h ≠ 0) are:

\begin{equation*}q_1 = - 2\left( {\frac{{2\Psi ^2 \sec ^2 \left( {\Psi \,\phi } \right)}}{{ - f + 2\Psi \tan \left( {\Psi \,\phi } \right)}}} \right)^2 + 2f\left( {\frac{{2\Psi ^2 \sec ^2 \left( {\Psi \,\phi } \right)}}{{ - f + 2\Psi \tan \left( {\Psi \,\phi } \right)}}} \right) + \frac{1}{3}( {4gh - f^2 } ),\end{equation*}

q_{1} = - 2 {(\frac{2 Ψ^{2} \sec^{2} (Ψ ϕ)}{- f + 2 Ψ \tan (Ψ ϕ)})}^{2} + 2 f (\frac{2 Ψ^{2} \sec^{2} (Ψ ϕ)}{- f + 2 Ψ \tan (Ψ ϕ)}) + \frac{1}{3} (4 g h - f^{2}),

where |$\Psi = \frac{1}{2}\sqrt {4gh - f^2 }$| $Ψ = \frac{1}{2} \sqrt{4 g h - f^{2}}$ ⁠, ϕ = x − (f⁴ − 8f²gh + 16g²h²)t and f, g, h are arbitrary constants.

{\fontsize{7.9}{12}\selectfont{\begin{eqnarray*}q_2 &=& - 2\left( {\frac{{2\Psi ^2 \csc ^2 \left( {\Psi \,\phi } \right)}}{{f + 2\Psi \cot \left( {\Psi \,\phi } \right)}}} \right)^2 + 2f\left( {\frac{{2\Psi ^2 \csc ^2 \left( {\Psi \,\phi } \right)}}{{f + 2\Psi \cot \left( {\Psi \,\phi } \right)}}} \right) + \frac{1}{3}\left( {4gh - f^2 } \right),\\[6pt]q_3 &=& - 2\left( {\frac{{4\Psi ^2 \sec \left( {2\Psi \,\phi } \right)\left( {1 \pm \sin \left( {2\Psi \,\phi } \right)} \right)}}{{ - f\cos \left( {2\Psi \,\phi } \right) + 2\Psi \sin \left( {2\Psi \,\phi } \right) \pm 2\Psi }}} \right)^2 + 2f\left( {\frac{{4\Psi ^2 \sec \left( {2\Psi \,\phi } \right)\left( {1 \pm \sin \left( {2\Psi \,\phi } \right)} \right)}}{{ - f\cos \left( {2\Psi \,\phi } \right) + 2\Psi \sin \left( {2\Psi \,\phi } \right) \pm 2\Psi }}} \right) + \frac{1}{3}\left( {4gh - f^2 } \right),\\[6pt]q_4 &=& - 2\left( {\frac{{4\Psi ^2 \csc \left( {2\Psi \,\phi } \right)\left( {1 \pm \cos \left( {2\Psi \,\phi } \right)} \right)}}{{f\sin \left( {2\Psi \,\phi } \right) + 2\Psi \cos \left( {2\Psi \,\phi } \right) \pm 2\Psi }}} \right)^2 + 2f\left( {\frac{{4\Psi ^2 \csc \left( {2\Psi \,\phi } \right)\left( {1 \pm \cos \left( {2\Psi \,\phi } \right)} \right)}}{{f\sin \left( {2\Psi \,\phi } \right) + 2\Psi \cos \left( {2\Psi \,\phi } \right) \pm 2\Psi }}} \right) + \frac{1}{3}\left( {4gh - f^2 } \right),\\[6pt]q_5 &=& - 2\left( {\frac{{2\Psi ^2 \csc \left( {\Psi \,\phi } \right)}}{{f\sin \left( {\Psi \,\phi } \right) + 2\Psi \cos \left( {\Psi \,\phi } \right)}}} \right)^2 + 2f\left( {\frac{{2\Psi ^2 \csc \left( {\Psi \,\phi } \right)}}{{f\sin \left( {\Psi \,\phi } \right) + 2\Psi \cos \left( {\Psi \,\phi } \right)}}} \right) + \frac{1}{3}( {4gh - f^2 } ),\\[6pt]q_6 &=& - 2\left( {\frac{{4A\Psi ^2 \left( {\sqrt {A^2 - B^2 } \cos \left( {2\Psi \,\phi } \right) - B\sin \left( {2\Psi \,\phi } \right) - A} \right)\left( {A\sin \left( {2\Psi \,\phi } \right) + B} \right)}}{{\left( {A^2 \cos ^2 \left( {2\Psi \phi } \right) - A^2 - B^2 - 2AB\sin \left( {2\Psi \,\phi } \right)} \right)\left( {fA\sin \left( {2\Psi \,\phi } \right) + 2A\Psi \cos \left( {2\Psi \,\phi } \right) + fB - 2\Psi \sqrt {A^2 - B^2 } } \right)}}} \right)^2 \\&& +\, 2f\left( {\frac{{4A\Psi ^2 \left( {\sqrt {A^2 - B^2 } \cos \left( {2\Psi \,\phi } \right) - B\sin \left( {2\Psi \,\phi } \right) - A} \right)\left( {A\sin \left( {2\Psi \,\phi } \right) + B} \right)}}{{\left( {A^2 \cos ^2 \left( {2\Psi \,\phi } \right) - A^2 - B^2 - 2AB\sin \left( {2\Psi \,\phi } \right)} \right)\left( {fA\sin \left( {2\Psi \,\phi } \right) + 2A\Psi \cos \left( {2\Psi \,\phi } \right) + fB - 2\Psi \sqrt {A^2 - B^2 } } \right)}}} \right)\\&& +\, \frac{1}{3}\left( {4gh - f^2 } \right), \\q_7 &=& - 2\left( {\frac{{4A\Psi ^2 \left( {\sqrt {A^2 - B^2 } \cos \left( {2\Psi \,\phi } \right) + B\sin \left( {2\Psi \,\phi } \right) + A} \right)\left( {A\sin \left( {2\Psi \,\phi } \right) + B} \right)}}{{\left( {A^2 \cos ^2 \left( {2\Psi \,\phi } \right) - A^2 - B^2 - 2AB\sin \left( {2\Psi \,\phi } \right)} \right)\left( {fA\sin \left( {2\Psi \,\phi } \right) - 2A\Psi \cos \left( {2\Psi \,\phi } \right) + fB - 2\Psi \sqrt {A^2 - B^2 } } \right)}}} \right)^2 \\&& +\, 2f\,\left( {\frac{{4A\Psi ^2 \left( {\sqrt {A^2 - B^2 } \cos \left( {2\Psi \,\phi } \right) + B\sin \left( {2\Psi \,\phi } \right) + A} \right)\left( {A\sin \left( {2\Psi \,\phi } \right) + B} \right)}}{{\left( {A^2 \cos ^2 \left( {2\Psi \,\phi } \right) - A^2 - B^2 - 2AB\sin \left( {2\Psi \,\phi } \right)} \right)\left( {fA\sin \left( {2\Psi \,\phi } \right) - 2A\Psi \cos \left( {2\Psi \,\phi } \right) + fB - 2\Psi \sqrt {A^2 - B^2 } } \right)}}} \right)\\&& +\, \frac{1}{3}\left( {4gh - f^2 } \right),\end{eqnarray*}}}

\begin{matrix} q_{2} & = & - 2 {(\frac{2 Ψ^{2} \csc^{2} (Ψ ϕ)}{f + 2 Ψ \cot (Ψ ϕ)})}^{2} + 2 f (\frac{2 Ψ^{2} \csc^{2} (Ψ ϕ)}{f + 2 Ψ \cot (Ψ ϕ)}) + \frac{1}{3} (4 g h - f^{2}), \\ q_{3} & = & - 2 {(\frac{4 Ψ^{2} \sec (2 Ψ ϕ) (1 \pm \sin (2 Ψ ϕ))}{- f \cos (2 Ψ ϕ) + 2 Ψ \sin (2 Ψ ϕ) \pm 2 Ψ})}^{2} + 2 f (\frac{4 Ψ^{2} \sec (2 Ψ ϕ) (1 \pm \sin (2 Ψ ϕ))}{- f \cos (2 Ψ ϕ) + 2 Ψ \sin (2 Ψ ϕ) \pm 2 Ψ}) + \frac{1}{3} (4 g h - f^{2}), \\ q_{4} & = & - 2 {(\frac{4 Ψ^{2} \csc (2 Ψ ϕ) (1 \pm \cos (2 Ψ ϕ))}{f \sin (2 Ψ ϕ) + 2 Ψ \cos (2 Ψ ϕ) \pm 2 Ψ})}^{2} + 2 f (\frac{4 Ψ^{2} \csc (2 Ψ ϕ) (1 \pm \cos (2 Ψ ϕ))}{f \sin (2 Ψ ϕ) + 2 Ψ \cos (2 Ψ ϕ) \pm 2 Ψ}) + \frac{1}{3} (4 g h - f^{2}), \\ q_{5} & = & - 2 {(\frac{2 Ψ^{2} \csc (Ψ ϕ)}{f \sin (Ψ ϕ) + 2 Ψ \cos (Ψ ϕ)})}^{2} + 2 f (\frac{2 Ψ^{2} \csc (Ψ ϕ)}{f \sin (Ψ ϕ) + 2 Ψ \cos (Ψ ϕ)}) + \frac{1}{3} (4 g h - f^{2}), \\ q_{6} & = & - 2 {(\frac{4 A Ψ^{2} (\sqrt{A^{2} - B^{2}} \cos (2 Ψ ϕ) - B \sin (2 Ψ ϕ) - A) (A \sin (2 Ψ ϕ) + B)}{(A^{2} \cos^{2} (2 Ψ ϕ) - A^{2} - B^{2} - 2 A B \sin (2 Ψ ϕ)) (f A \sin (2 Ψ ϕ) + 2 A Ψ \cos (2 Ψ ϕ) + f B - 2 Ψ \sqrt{A^{2} - B^{2}})})}^{2} \\ + 2 f (\frac{4 A Ψ^{2} (\sqrt{A^{2} - B^{2}} \cos (2 Ψ ϕ) - B \sin (2 Ψ ϕ) - A) (A \sin (2 Ψ ϕ) + B)}{(A^{2} \cos^{2} (2 Ψ ϕ) - A^{2} - B^{2} - 2 A B \sin (2 Ψ ϕ)) (f A \sin (2 Ψ ϕ) + 2 A Ψ \cos (2 Ψ ϕ) + f B - 2 Ψ \sqrt{A^{2} - B^{2}})}) \\ + \frac{1}{3} (4 g h - f^{2}), \\ q_{7} & = & - 2 {(\frac{4 A Ψ^{2} (\sqrt{A^{2} - B^{2}} \cos (2 Ψ ϕ) + B \sin (2 Ψ ϕ) + A) (A \sin (2 Ψ ϕ) + B)}{(A^{2} \cos^{2} (2 Ψ ϕ) - A^{2} - B^{2} - 2 A B \sin (2 Ψ ϕ)) (f A \sin (2 Ψ ϕ) - 2 A Ψ \cos (2 Ψ ϕ) + f B - 2 Ψ \sqrt{A^{2} - B^{2}})})}^{2} \\ + 2 f (\frac{4 A Ψ^{2} (\sqrt{A^{2} - B^{2}} \cos (2 Ψ ϕ) + B \sin (2 Ψ ϕ) + A) (A \sin (2 Ψ ϕ) + B)}{(A^{2} \cos^{2} (2 Ψ ϕ) - A^{2} - B^{2} - 2 A B \sin (2 Ψ ϕ)) (f A \sin (2 Ψ ϕ) - 2 A Ψ \cos (2 Ψ ϕ) + f B - 2 Ψ \sqrt{A^{2} - B^{2}})}) \\ + \frac{1}{3} (4 g h - f^{2}), \end{matrix}

where A and B are two non-zero real constants and satisfies A² − B² > 0.

\begin{eqnarray*}q_8 &=& - 2\left( {\frac{{2\Psi ^2 \sec \left( {\Psi \,\phi } \right)\left( {f\cos \left( {\Psi \,\phi } \right) + 2\Psi \sin \left( {\Psi \,\phi } \right)} \right)}}{{2\left( {f^2 - 2gh} \right)\cos ^2 \left( {\Psi \,\phi } \right) + 4\Psi f\sin \left( {\Psi \,\phi } \right)\cos \left( {\Psi \,\phi } \right) + 4\Psi ^2 }}} \right)^2 \\[3pt]&&+\, 2f\left( {\frac{{2\Psi ^2 \sec \left( {\Psi \,\phi } \right)\left( {f\cos \left( {\Psi \,\phi } \right) + 2\Psi \sin \left( {\Psi \,\phi } \right)} \right)}}{{2\left( {f^2 - 2gh} \right)\cos ^2 \left( {\Psi \,\phi } \right) + 4\Psi f\sin \left( {\Psi \,\phi } \right)\cos \left( {\Psi \,\phi } \right) + 4\Psi ^2 }}} \right) + \frac{1}{3}\left( {4gh - f^2 } \right), \\[3pt]q_9 &=& - 2\left( {\frac{{2\Psi ^2 \csc \left( {\Psi \,\phi } \right)\left( {f\sin \left( {\Psi \,\phi } \right) - 2\Psi \cos \left( {\Psi \,\phi } \right)} \right)}}{{2\left( {f^2 - 2gh} \right)\cos ^2 \left( {\Psi \,\phi } \right) + 4\Psi f\sin \left( {\Psi \,\phi } \right)\cos \left( {\Psi \,\phi } \right) - f^2 }}} \right)^2 \\[3pt]&&+\, 2f\left( {\frac{{2\Psi ^2 \csc \left( {\Psi \,\phi } \right)\left( {f\sin \left( {\Psi \,\phi } \right) - 2\Psi \cos \left( {\Psi \,\phi } \right)} \right)}}{{2\left( {f^2 - 2gh} \right)\cos ^2 \left( {\Psi \,\phi } \right) + 4\Psi f\sin \left( {\Psi \,\phi } \right)\cos \left( {\Psi \,\phi } \right) - f^2 }}} \right) + \frac{1}{3}\left( {4gh - f^2 } \right), \\[3pt]q_{10} &=& - 2\left( {\frac{{2\Psi ^2 \sec \left( {2\Psi \,\phi } \right)\left( {1 \pm \sin \left( {2\Psi \,\phi } \right)} \right)\left( {f\cos \left( {2\Psi \,\phi } \right) + 2\Psi \sin \left( {2\Psi \,\phi } \right) \pm 2\Psi } \right)}}{{\left( {f^2 - 2gh} \right)\cos ^2 \left( {2\Psi \,\phi } \right) + 2\Psi \left( {1 \pm \sin \left( {2\Psi \,\phi } \right)} \right)\left( {2\Psi \pm f\cos \left( {2\Psi \,\phi } \right)} \right)}}} \right)^2 \\[2pt]&&+\, 2f\left( {\frac{{2\Psi ^2 \sec \left( {2\Psi \,\phi } \right)\left( {1 \pm \sin \left( {2\Psi \,\phi } \right)} \right)\left( {f\cos \left( {2\Psi \,\phi } \right) + 2\Psi \sin \left( {2\Psi \,\phi } \right) \pm 2\Psi } \right)}}{{\left( {f^2 - 2gh} \right)\cos ^2 \left( {2\Psi \,\phi } \right) + 2\Psi \left( {1 \pm \sin \left( {2\Psi \,\phi } \right)} \right)\left( {2\Psi \pm f\cos \left( {2\Psi \,\phi } \right)} \right)}}} \right)\\&& +\, \frac{1}{3}\left( {4gh - f^2 } \right), \\q_{11} &=& - 2\left( {\frac{{2\Psi ^2 \csc \left( {2\Psi \,\phi } \right)\left( { - f\sin \left( {2\Psi \,\phi } \right) + 2\Psi \cos \left( {2\Psi \,\phi } \right) \pm 2\Psi } \right)}}{{\left( {2gh - f^2 } \right)\cos \left( {2\Psi \,\phi } \right) - 2f\Psi \sin \left( {2\Psi \,\phi } \right) \pm 2gh}}} \right)^2 \\&& +\, 2f\left( {\frac{{2\Psi ^2 \csc \left( {2\Psi \,\phi } \right)\left( { - f\sin \left( {2\Psi \,\phi } \right) + 2\Psi \cos \left( {2\Psi \,\phi } \right) \pm 2\Psi } \right)}}{{\left( {2gh - f^2 } \right)\cos \left( {2\Psi \,\phi } \right) - 2f\Psi \sin \left( {2\Psi \,\phi } \right) \pm 2gh}}} \right) + \frac{1}{3}\left( {4gh - f^2 } \right), \\q_{12} &=& - 2\left( {\frac{{2\Psi ^2 \csc \left( {\Psi \,\phi } \right)\left( {f\sin \left( {\Psi \,\phi } \right) - 2\Psi \cos \left( {\Psi \,\phi } \right)} \right)}}{{2\left( {f^2 - 2gh} \right)\cos ^2 \left( {\Psi \,\phi } \right) + 4f\Psi \sin \left( {\Psi \,\phi } \right)\cos \left( {\Psi \,\phi } \right) - f^2 }}} \right)^2 \\&& +\, 2f\left( {\frac{{2\Psi ^2 \csc \left( {\Psi \,\phi } \right)\left( {f\sin \left( {\Psi \,\phi } \right) - 2\Psi \cos \left( {\Psi \,\phi } \right)} \right)}}{{2\left( {f^2 - 2gh} \right)\cos ^2 \left( {\Psi \,\phi } \right) + 4f\Psi \sin \left( {\Psi \,\phi } \right)\cos \left( {\Psi \,\phi } \right) - f^2 }}} \right) + \frac{1}{3}\left( {4gh - f^2 } \right), \\\end{eqnarray*}

\begin{matrix} q_{8} & = & - 2 {(\frac{2 Ψ^{2} \sec (Ψ ϕ) (f \cos (Ψ ϕ) + 2 Ψ \sin (Ψ ϕ))}{2 (f^{2} - 2 g h) \cos^{2} (Ψ ϕ) + 4 Ψ f \sin (Ψ ϕ) \cos (Ψ ϕ) + 4 Ψ^{2}})}^{2} \\ + 2 f (\frac{2 Ψ^{2} \sec (Ψ ϕ) (f \cos (Ψ ϕ) + 2 Ψ \sin (Ψ ϕ))}{2 (f^{2} - 2 g h) \cos^{2} (Ψ ϕ) + 4 Ψ f \sin (Ψ ϕ) \cos (Ψ ϕ) + 4 Ψ^{2}}) + \frac{1}{3} (4 g h - f^{2}), \\ q_{9} & = & - 2 {(\frac{2 Ψ^{2} \csc (Ψ ϕ) (f \sin (Ψ ϕ) - 2 Ψ \cos (Ψ ϕ))}{2 (f^{2} - 2 g h) \cos^{2} (Ψ ϕ) + 4 Ψ f \sin (Ψ ϕ) \cos (Ψ ϕ) - f^{2}})}^{2} \\ + 2 f (\frac{2 Ψ^{2} \csc (Ψ ϕ) (f \sin (Ψ ϕ) - 2 Ψ \cos (Ψ ϕ))}{2 (f^{2} - 2 g h) \cos^{2} (Ψ ϕ) + 4 Ψ f \sin (Ψ ϕ) \cos (Ψ ϕ) - f^{2}}) + \frac{1}{3} (4 g h - f^{2}), \\ q_{10} & = & - 2 {(\frac{2 Ψ^{2} \sec (2 Ψ ϕ) (1 \pm \sin (2 Ψ ϕ)) (f \cos (2 Ψ ϕ) + 2 Ψ \sin (2 Ψ ϕ) \pm 2 Ψ)}{(f^{2} - 2 g h) \cos^{2} (2 Ψ ϕ) + 2 Ψ (1 \pm \sin (2 Ψ ϕ)) (2 Ψ \pm f \cos (2 Ψ ϕ))})}^{2} \\ + 2 f (\frac{2 Ψ^{2} \sec (2 Ψ ϕ) (1 \pm \sin (2 Ψ ϕ)) (f \cos (2 Ψ ϕ) + 2 Ψ \sin (2 Ψ ϕ) \pm 2 Ψ)}{(f^{2} - 2 g h) \cos^{2} (2 Ψ ϕ) + 2 Ψ (1 \pm \sin (2 Ψ ϕ)) (2 Ψ \pm f \cos (2 Ψ ϕ))}) \\ + \frac{1}{3} (4 g h - f^{2}), \\ q_{11} & = & - 2 {(\frac{2 Ψ^{2} \csc (2 Ψ ϕ) (- f \sin (2 Ψ ϕ) + 2 Ψ \cos (2 Ψ ϕ) \pm 2 Ψ)}{(2 g h - f^{2}) \cos (2 Ψ ϕ) - 2 f Ψ \sin (2 Ψ ϕ) \pm 2 g h})}^{2} \\ + 2 f (\frac{2 Ψ^{2} \csc (2 Ψ ϕ) (- f \sin (2 Ψ ϕ) + 2 Ψ \cos (2 Ψ ϕ) \pm 2 Ψ)}{(2 g h - f^{2}) \cos (2 Ψ ϕ) - 2 f Ψ \sin (2 Ψ ϕ) \pm 2 g h}) + \frac{1}{3} (4 g h - f^{2}), \\ q_{12} & = & - 2 {(\frac{2 Ψ^{2} \csc (Ψ ϕ) (f \sin (Ψ ϕ) - 2 Ψ \cos (Ψ ϕ))}{2 (f^{2} - 2 g h) \cos^{2} (Ψ ϕ) + 4 f Ψ \sin (Ψ ϕ) \cos (Ψ ϕ) - f^{2}})}^{2} \\ + 2 f (\frac{2 Ψ^{2} \csc (Ψ ϕ) (f \sin (Ψ ϕ) - 2 Ψ \cos (Ψ ϕ))}{2 (f^{2} - 2 g h) \cos^{2} (Ψ ϕ) + 4 f Ψ \sin (Ψ ϕ) \cos (Ψ ϕ) - f^{2}}) + \frac{1}{3} (4 g h - f^{2}), \end{matrix}

Family 3.2: The soliton and soliton-like solutions of Eq. (6) (when f² − 4gh > 0 and f g ≠ 0 or g h ≠ 0) are:

\begin{equation*}q_{13} = - 2\left( {\frac{{2\Phi ^2 \sec h^2 \left( {\Phi \,\phi } \right)}}{{f + 2\Phi \tanh \left( {\Phi \,\phi } \right)}}} \right)^2 + 2f\left( {\frac{{2\Phi ^2 \sec h^2 \left( {\Phi \,\phi } \right)}}{{f + 2\Phi \tanh \left( {\Phi \,\phi } \right)}}} \right) - \frac{1}{3}\left( {f^2 - 4gh} \right),\end{equation*}

q_{13} = - 2 {(\frac{2 Φ^{2} \sec h^{2} (Φ ϕ)}{f + 2 Φ \tanh (Φ ϕ)})}^{2} + 2 f (\frac{2 Φ^{2} \sec h^{2} (Φ ϕ)}{f + 2 Φ \tanh (Φ ϕ)}) - \frac{1}{3} (f^{2} - 4 g h),

where |$\Phi = \frac{1}{2}\sqrt {f^2 - 4gh},$| $Φ = \frac{1}{2} \sqrt{f^{2} - 4 g h},$ ϕ = x − (f⁴ − 8f²gh + 16g²h²)t and f, g, h are arbitrary constants.

{\fontsize{9.4}{12}\selectfont{\begin{eqnarray*}q_{14} &=& - 2\left( {\frac{{2\Phi ^2 \csc h^2 \left( {\Phi \,\phi } \right)}}{{f + 2\Phi \coth \left( {\Phi \,\phi } \right)}}} \right)^2 + 2f\left( {\frac{{2\Phi ^2 \csc h^2 \left( {\Phi \,\phi } \right)}}{{f + 2\Phi \coth \left( {\Phi \,\phi } \right)}}} \right) - \frac{1}{3}\left( {f^2 - 4gh} \right),\\[3pt]q_{15}\! &=&\! - 2\left( {\frac{{4\Phi ^2 \sec h\left( {2\Phi \,\phi } \right)\left( {1 \mp i\sinh \left( {2\Phi \,\phi } \right)} \right)}}{{f\cosh \left( {2\Phi \,\phi } \right) + 2\Phi \sinh \left( {2\Phi \,\phi } \right) \pm i2\Phi }}} \right)^2\! +\! 2f\left( {\frac{{4\Phi ^2 \sec h\left( {2\Phi \,\phi } \right)\left( {1 \mp i\sinh \left( {2\Phi \,\phi } \right)} \right)}}{{f\cosh \left( {2\Phi \,\phi } \right) + 2\Phi \sinh \left( {2\Phi \,\phi } \right) \pm i2\Phi }}} \right)\\[3pt]&& - \frac{1}{3}\left( {f^2 - 4gh} \right),\\[3pt]q_{16} &=& - 2\left( {\frac{{4\Phi ^2 \csc h\left( {2\Phi \,\phi } \right)\left( {1 \pm \cosh \left( {2\Phi \,\phi } \right)} \right)}}{{f\sinh \left( {2\Phi \,\phi } \right) + 2\Phi \cosh \left( {2\Phi \,\phi } \right) \pm 2\Phi }}} \right)^2 + 2f\left( {\frac{{4\Phi ^2 \csc h\left( {2\Phi \,\phi } \right)\left( {1 \pm \cosh \left( {2\Phi \,\phi } \right)} \right)}}{{f\sinh \left( {2\Phi \,\phi } \right) + 2\Phi \cosh \left( {2\Phi \,\phi } \right) \pm 2\Phi }}} \right)\\[3pt]&& - \frac{1}{3}\left( {f^2 - 4gh} \right),\\[3pt]q_{17} &=& - 2\left( {\frac{{\Phi ^2 \sec h^2 \left( {\Phi \,\phi /2} \right)}}{{2\left( {\cosh ^2 \left( {\Phi \,\,\phi /2} \right) - 1} \right)\left( {f + \Phi \tanh \left( {\Phi \,\,\phi /2} \right) + \coth \left( {\Phi \,\phi /2} \right)} \right)}}} \right)^2 \\&& +\, 2f\left( {\frac{{\Phi \,^2 \sec h^2 \left( {\Phi \,\phi /2} \right)}}{{2\left( {\cosh ^2 \left( {\Phi \,\,\phi /2} \right) - 1} \right)\left( {f + \Phi \tanh \left( {\Phi \,\,\phi /2} \right) + \coth \left( {\Phi \,\phi /2} \right)} \right)}}} \right) - \frac{1}{3}\left( {f^2 - 4gh} \right), \\q_{18} &=& - 2\left( {\frac{{4A\Phi \,^2 \left( {A - B\sinh \left( {2\Phi \,\,\phi } \right) - \sqrt {A^2 + B^2 } \cosh \left( {2\Phi \,\,\phi } \right)} \right)}}{{\left( {A\sinh \left( {2\Phi \,\,\phi } \right) + B} \right)\left( {f\,A\sinh \left( {2\Phi \,\,\phi } \right) + f\,B - 2\Phi \,\sqrt {A^2 + B^2 } + 2A\Phi \cosh \left( {2\Phi \,\,\phi } \right)} \right)}}} \right)^2 \\&& +\, 2f\left( {\frac{{4A\Phi \,^2 \left( {A - B\sinh \left( {2\Phi \,\,\phi } \right) - \sqrt {A^2 + B^2 } \cosh \left( {2\Phi \,\phi } \right)} \right)}}{{\left( {A\sinh \left( {2\Phi \,\,\phi } \right) + B} \right)\left( {fA\sinh \left( {2\Phi \,\,\phi } \right) + f\,B - 2\Phi \,\sqrt {A^2 + B^2 } + 2A\Phi \cosh \left( {2\Phi \,\,\phi } \right)} \right)}}} \right)\\&& -\, \frac{1}{3}\left( {f^2 - 4gh} \right), \\q_{19} &=& - 2\left( {\frac{{4A\Phi \,^2 \left( {A - B\sinh \left( {2\Phi \,\,\phi } \right) + \sqrt {A^2 + B^2 } \cosh \left( {2\Phi \,\,\phi } \right)} \right)}}{{\left( {A\sinh \left( {2\Phi \,\,\phi } \right) + B} \right)\left( {f\,A\sinh \left( {2\Phi \,\,\phi } \right) + f\,B + 2\Phi \,\sqrt {A^2 + B^2 } + 2A\Phi \cosh \left( {2\Phi \,\,\phi } \right)} \right)}}} \right)^2 \\&& +\, 2f\left( {\frac{{4A\Phi \,^2 \left( {A - B\sinh \left( {2\Phi \,\,\phi } \right) + \sqrt {A^2 + B^2 } \cosh \left( {2\Phi \,\phi } \right)} \right)}}{{\left( {A\sinh \left( {2\Phi \,\,\phi } \right) + B} \right)\left( {f\,A\sinh \left( {2\Phi \,\,\phi } \right) + f\,B + 2\Phi \,\sqrt {A^2 + B^2 } + 2A\Phi \cosh \left( {2\Phi \,\,\phi } \right)} \right)}}} \right)\\&& -\, \frac{1}{3}\left( {f^2 - 4gh} \right),\end{eqnarray*}}}

\begin{matrix} q_{14} & = & - 2 {(\frac{2 Φ^{2} \csc h^{2} (Φ ϕ)}{f + 2 Φ \coth (Φ ϕ)})}^{2} + 2 f (\frac{2 Φ^{2} \csc h^{2} (Φ ϕ)}{f + 2 Φ \coth (Φ ϕ)}) - \frac{1}{3} (f^{2} - 4 g h), \\ q_{15} & = & - 2 {(\frac{4 Φ^{2} \sec h (2 Φ ϕ) (1 \mp i \sinh (2 Φ ϕ))}{f \cosh (2 Φ ϕ) + 2 Φ \sinh (2 Φ ϕ) \pm i 2 Φ})}^{2} + 2 f (\frac{4 Φ^{2} \sec h (2 Φ ϕ) (1 \mp i \sinh (2 Φ ϕ))}{f \cosh (2 Φ ϕ) + 2 Φ \sinh (2 Φ ϕ) \pm i 2 Φ}) \\ - \frac{1}{3} (f^{2} - 4 g h), \\ q_{16} & = & - 2 {(\frac{4 Φ^{2} \csc h (2 Φ ϕ) (1 \pm \cosh (2 Φ ϕ))}{f \sinh (2 Φ ϕ) + 2 Φ \cosh (2 Φ ϕ) \pm 2 Φ})}^{2} + 2 f (\frac{4 Φ^{2} \csc h (2 Φ ϕ) (1 \pm \cosh (2 Φ ϕ))}{f \sinh (2 Φ ϕ) + 2 Φ \cosh (2 Φ ϕ) \pm 2 Φ}) \\ - \frac{1}{3} (f^{2} - 4 g h), \\ q_{17} & = & - 2 {(\frac{Φ^{2} \sec h^{2} (Φ ϕ / 2)}{2 (\cosh^{2} (Φ ϕ / 2) - 1) (f + Φ \tanh (Φ ϕ / 2) + \coth (Φ ϕ / 2))})}^{2} \\ + 2 f (\frac{Φ^{2} \sec h^{2} (Φ ϕ / 2)}{2 (\cosh^{2} (Φ ϕ / 2) - 1) (f + Φ \tanh (Φ ϕ / 2) + \coth (Φ ϕ / 2))}) - \frac{1}{3} (f^{2} - 4 g h), \\ q_{18} & = & - 2 {(\frac{{4 A Φ}^{2} (A - B \sinh (2 Φ ϕ) - \sqrt{A^{2} + B^{2}} \cosh (2 Φ ϕ))}{(A \sinh (2 Φ ϕ) + B) (f A \sinh (2 Φ ϕ) + f B - 2 Φ \sqrt{A^{2} + B^{2}} + 2 A Φ \cosh (2 Φ ϕ))})}^{2} \\ + 2 f (\frac{{4 A Φ}^{2} (A - B \sinh (2 Φ ϕ) - \sqrt{A^{2} + B^{2}} \cosh (2 Φ ϕ))}{(A \sinh (2 Φ ϕ) + B) (f A \sinh (2 Φ ϕ) + f B - 2 Φ \sqrt{A^{2} + B^{2}} + 2 A Φ \cosh (2 Φ ϕ))}) \\ - \frac{1}{3} (f^{2} - 4 g h), \\ q_{19} & = & - 2 {(\frac{4 A Φ^{2} (A - B \sinh (2 Φ ϕ) + \sqrt{A^{2} + B^{2}} \cosh (2 Φ ϕ))}{(A \sinh (2 Φ ϕ) + B) (f A \sinh (2 Φ ϕ) + f B + 2 Φ \sqrt{A^{2} + B^{2}} + 2 A Φ \cosh (2 Φ ϕ))})}^{2} \\ + 2 f (\frac{{4 A Φ}^{2} (A - B \sinh (2 Φ ϕ) + \sqrt{A^{2} + B^{2}} \cosh (2 Φ ϕ))}{(A \sinh (2 Φ ϕ) + B) (f A \sinh (2 Φ ϕ) + f B + 2 Φ \sqrt{A^{2} + B^{2}} + 2 A Φ \cosh (2 Φ ϕ))}) \\ - \frac{1}{3} (f^{2} - 4 g h), \end{matrix}

where A and B are two non-zero real constants and satisfies A² − B² < 0.

{\fontsize{9.2}{12}\selectfont{\begin{eqnarray*}q_{20} &=& - 2\left( {\frac{{2\Phi \,^2 \sec h\left( {\Phi \,\,\phi } \right)}}{{2\Phi \sinh \left( {\Phi \,\,\phi } \right) - f\cosh \left( {\Phi \,\phi } \right)}}} \right)^2 + 2f\left( {\frac{{2\Phi \,^2 \sec h\left( {\Phi \,\,\phi } \right)}}{{2\Phi \sinh \left( {\Phi \,\,\phi } \right) - f\cosh \left( {\Phi \,\phi } \right)}}} \right) -\, \frac{1}{3}\left( {f^2 - 4gh} \right),\\q_{21} &=& - 2\left( {\frac{{2\Phi \,^2 \csc h\left( {\Phi \,\,\phi } \right)}}{{2\Phi \cosh \left( {\Phi \,\,\phi } \right) - f\sinh \left( {\Phi \,\phi } \right)}}} \right)^2 + 2f\left( {\frac{{2\Phi \,^2 \csc h\left( {\Phi \,\,\phi } \right)}}{{2\Phi \cosh \left( {\Phi \,\,\phi } \right) - f\sinh \left( {\Phi \,\phi } \right)}}} \right) - \frac{1}{3}\left( {f^2 - 4gh} \right),\\[-2pt]q_{22} &=& - 2\left( {\frac{{4\Phi \,^2 \sec h\left( {2\Phi \,\,\phi } \right)\left( {1 \mp i\sinh \left( {2\Phi \,\,\phi } \right)} \right)}}{{f\cosh \left( {2\Phi \,\phi } \right) - 2\Phi \sinh \left( {2\Phi \,\,\phi } \right) \mp i2\Phi \,}}} \right)^2 + 2f\left( {\frac{{4\Phi \,^2 \sec h\left( {2\Phi \,\,\phi } \right)\left( {1 \mp i\sinh \left( {2\Phi \,\,\phi } \right)} \right)}}{{f\cosh \left( {2\Phi \,\phi } \right) - 2\Phi \sinh \left( {2\Phi \,\,\phi } \right) \mp i2\Phi \,}}} \right)\\[-2pt]&& -\, \frac{1}{3}\left( {f^2 - 4gh} \right),\\[-2pt]q_{23} &=& - 2\left( {\frac{{4\Phi \,^2 \csc h\left( {2\Phi \,\,\phi } \right)\left( {1 \pm \cosh \left( {2\Phi \,\,\phi } \right)} \right)}}{{2\Phi \cosh \left( {2\Phi \,\phi } \right) - f\sinh \left( {2\Phi \,\,\phi } \right) \pm 2\Phi \,}}} \right)^2 + 2f\left( {\frac{{4\Phi \,^2 \csc h\left( {2\Phi \,\,\phi } \right)\left( {1 \pm \cosh \left( {2\Phi \,\,\phi } \right)} \right)}}{{2\Phi \cosh \left( {2\Phi \,\phi } \right) - f\sinh \left( {2\Phi \,\,\phi } \right) \pm 2\Phi \,}}} \right)\\[-2pt]&& -\, \frac{1}{3}\left( {f^2 - 4gh} \right),\\[-2pt]q_{24} &=& - 2\left( {\frac{{2\Phi \,^2 \csc h\left( {\Phi \,\,\phi } \right)}}{{2\Phi \cosh \left( {\Phi \,\,\phi } \right) - f\sinh \left( {\Phi \,\phi } \right)}}} \right)^2 + 2f\left( {\frac{{2\Phi \,^2 \csc h\left( {\Phi \,\,\phi } \right)}}{{2\Phi \cosh \left( {\Phi \,\,\phi } \right) - f\sinh \left( {\Phi \,\phi } \right)}}} \right) -\, \frac{1}{3}\left( {f^2 - 4gh} \right),\end{eqnarray*}}}

\begin{matrix} q_{20} & = & - 2 {(\frac{{2 Φ}^{2} \sec h (Φ ϕ)}{2 Φ \sinh (Φ ϕ) - f \cosh (Φ ϕ)})}^{2} + 2 f (\frac{{2 Φ}^{2} \sec h (Φ ϕ)}{2 Φ \sinh (Φ ϕ) - f \cosh (Φ ϕ)}) - \frac{1}{3} (f^{2} - 4 g h), \\ q_{21} & = & - 2 {(\frac{{2 Φ}^{2} \csc h (Φ ϕ)}{2 Φ \cosh (Φ ϕ) - f \sinh (Φ ϕ)})}^{2} + 2 f (\frac{{2 Φ}^{2} \csc h (Φ ϕ)}{2 Φ \cosh (Φ ϕ) - f \sinh (Φ ϕ)}) - \frac{1}{3} (f^{2} - 4 g h), \\ q_{22} & = & - 2 {(\frac{{4 Φ}^{2} \sec h (2 Φ ϕ) (1 \mp i \sinh (2 Φ ϕ))}{f \cosh (2 Φ ϕ) - 2 Φ \sinh (2 Φ ϕ) \mp i 2 Φ})}^{2} + 2 f (\frac{{4 Φ}^{2} \sec h (2 Φ ϕ) (1 \mp i \sinh (2 Φ ϕ))}{f \cosh (2 Φ ϕ) - 2 Φ \sinh (2 Φ ϕ) \mp i 2 Φ}) \\ - \frac{1}{3} (f^{2} - 4 g h), \\ q_{23} & = & - 2 {(\frac{{4 Φ}^{2} \csc h (2 Φ ϕ) (1 \pm \cosh (2 Φ ϕ))}{2 Φ \cosh (2 Φ ϕ) - f \sinh (2 Φ ϕ) \pm 2 Φ})}^{2} + 2 f (\frac{{4 Φ}^{2} \csc h (2 Φ ϕ) (1 \pm \cosh (2 Φ ϕ))}{2 Φ \cosh (2 Φ ϕ) - f \sinh (2 Φ ϕ) \pm 2 Φ}) \\ - \frac{1}{3} (f^{2} - 4 g h), \\ q_{24} & = & - 2 {(\frac{{2 Φ}^{2} \csc h (Φ ϕ)}{2 Φ \cosh (Φ ϕ) - f \sinh (Φ ϕ)})}^{2} + 2 f (\frac{{2 Φ}^{2} \csc h (Φ ϕ)}{2 Φ \cosh (Φ ϕ) - f \sinh (Φ ϕ)}) - \frac{1}{3} (f^{2} - 4 g h), \end{matrix}

Family 3.3: The soliton and soliton-like solutions of Eq. (6) (when h = 0 and f g ≠ 0) are:

\begin{eqnarray*}q_{25} &=& - 2\left( {\frac{{f\left( {\cosh \left( {f\phi } \right) - \sinh \left( {f\phi } \right)} \right)}}{{c_1 + \cosh \left( {f\phi } \right) - \sinh \left( {f\phi } \right)\,\,}}} \right)^2 + 2f\left( {\frac{{f\left( {\cosh \left( {f\phi } \right) - \sinh \left( {f\phi } \right)} \right)}}{{c_1 + \cosh \left( {f\phi } \right) - \sinh \left( {f\phi } \right)\,\,}}} \right)\\&& -\, \frac{1}{3}\left( {f^2 - 4gh} \right),\\q_{26} &=& - 2\,\left( {\frac{{f\,c_1 }}{{c_1 + \cosh \left( {f\phi } \right) + \sinh \left( {f\phi } \right)\,\,}}} \right)^2 + 2f\left( {\frac{{f\,c_1 }}{{c_1 + \cosh \left( {f\phi } \right) + \sinh \left( {f\phi } \right)\,\,}}} \right)\\&& -\, \frac{1}{3}\left( {f^2 - 4gh} \right),\end{eqnarray*}

\begin{matrix} q_{25} & = & - 2 {(\frac{f (\cosh (f ϕ) - \sinh (f ϕ))}{c_{1} + \cosh (f ϕ) - \sinh (f ϕ)})}^{2} + 2 f (\frac{f (\cosh (f ϕ) - \sinh (f ϕ))}{c_{1} + \cosh (f ϕ) - \sinh (f ϕ)}) \\ - \frac{1}{3} (f^{2} - 4 g h), \\ q_{26} & = & - 2 {(\frac{f c_{1}}{c_{1} + \cosh (f ϕ) + \sinh (f ϕ)})}^{2} + 2 f (\frac{f c_{1}}{c_{1} + \cosh (f ϕ) + \sinh (f ϕ)}) \\ - \frac{1}{3} (f^{2} - 4 g h), \end{matrix}

where ϕ = x − (f⁴ − 8f²gh + 16g²h²)t and c₁, f, g, h are arbitrary constants.

Family 3.4: The rational function solution (when g ≠ 0 and h = f = 0) is:

\begin{equation*}q_{27} = - 2\left( {\frac{g}{{g\phi + l_1 }}} \right)^2 - 2f\left( {\frac{g}{{g\phi + l_1 }}} \right) - \frac{1}{3}\left( {f^2 - 4gh} \right),\end{equation*}

q_{27} = - 2 {(\frac{g}{g ϕ + l_{1}})}^{2} - 2 f (\frac{g}{g ϕ + l_{1}}) - \frac{1}{3} (f^{2} - 4 g h),

where l₁ is an arbitrary constant and ϕ = x − (f⁴ − 8f²gh + 16g²h²)t.

IV. GRAPHICAL PRESENTATIONS OF SOME TRAVELING WAVE SOLUTIONS

The graphical descriptions of some solutions are represented in figures 1–12 with the aid of commercial software Maple.

FIG. 1.

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Solitons solution for f = 1, g = 4, h = 0.25.

FIG. 2.

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Solitons solution for f = 6, g = 19, h = 0.5.

FIG. 3.

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Solitons solution for f = 1, g = 2, h = 0.25.

FIG. 4.

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Solitons solution for f = 3, g = 10, h = 0.25.

FIG. 5.

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Solitons solution for f = 8, g = 33, h = 0.5.

FIG. 6.

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Solitons solution for f = 7, g = 24, h = 0.5.

FIG. 7.

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Solitons solution for f = 0.75, g = 0.25, h = 0.

FIG. 8.

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Solitons solution for f = 2, g = 5, h = 0.25.

FIG. 9.

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Solitons solution for f = 3, g = 4, h = 0.5.

FIG. 10.

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Solitons solution for f = 3, g = 6.5, h = 0.5.

FIG. 11.

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Solitons solution for f = 5, g = 13, h = 0.5.

FIG. 12.

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Periodic solution for f = 0, g = 0.25, h = 0.

ACKNOWLEDGMENTS

This article is supported by the USM short term grant (Ref. No. 304/PMATHS/6310072) and authors would like to express their thanks to the School of Mathematical Sciences, USM for providing related research facilities.

REFERENCES

1.

J.

Weiss

,

M.

Tabor

and

G.

Carnevale

, “

The Painleve property for partial differential equations

,”

J. Math. Phys.

24

(

3

),

522

–

526

(

1982

).

https://doi.org/10.1063/1.525721

Google Scholar

Crossref

2.

S. L.

Zhang

,

B.

Wu

and

S. Y.

Lou

, “

Painleve analysis and special solutions of generalized Broer-Kaup equations

,”

Phys. Lett.

300

(

1

),

40

–

48

(

2002

).

https://doi.org/10.1016/S0375-9601(02)00688-6

Google Scholar

Crossref

3.

R.

Hirota

, “

Exact solution of the KdV equation for multiple collisions of solutions

,”

Physics Review Letters

27

,

1192

–

1194

(

1971

).

https://doi.org/10.1103/PhysRevLett.27.1192

Google Scholar

Crossref

4.

N. A.

Kudryashov

, “

Exact solutions of the generalized Kuramoto-Sivashinsky equation

,”

Phys. Lett.

147

(

5–6

),

287

–

291

(

1990

).

https://doi.org/10.1016/0375-9601(90)90449-X

Google Scholar

Crossref

5.

M. J.

Ablowitz

and

P. A.

Clarkson

,

Solitons, nonlinear evolution equations and inverse scattering transform

(

Cambridge Univ. Press

,

Cambridge

,

1991

).

Google Scholar

Crossref

6.

C.

Rogers

and

W. F.

Shadwick

,

Backlund transformations

(

Aca. Press

,

New York

,

1982

).

Google Scholar

7.

L.

Jianming

,

D.

Jie

, and

Y.

Wenjun

, “

Backlund transformation and new exact solutions of the Sharma-Tasso-Olver equation

,”

Abstract and Applied Analysis

2011

,

935710

(

2011

).

https://doi.org/10.1155/2011/935710

Google Scholar

Crossref

8.

Y.

Huang

, and

Y.

Shang

, “

The backlund transformations and abundant exact explicit solutions for a general nonintegrable nonlinear convection-diffusion equation

,”

Abstract and Applied Analysis

2012

,

489043

(

2012

).

https://doi.org/10.1155/2012/489043

Google Scholar

Crossref

9.

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

289

,

69

–

74

(

2001

).

https://doi.org/10.1016/S0375-9601(01)00580-1

Google Scholar

Crossref

10.

Z. I. A.

Al-Muhiameed

, and

E. A. B.

Abdel-Salam

, “

Generalized Jacobi elliptic function solution to a class of nonlinear Schrodinger-type equations

,”

Mathematical Problems in Engineering

2011

,

575679

(

2011

).

https://doi.org/10.1155/2011/575679

Google Scholar

Crossref

11.

W.

Zhang

, “

Extended Jacobi elliptic function expansion method to the ZK-Mew equation

,”

International Journal of Differential Equations

2011

,

451420

(

2011

).

https://doi.org/10.1155/2011/451420

Google Scholar

Crossref

12.

W.

Malfliet

, “

Solitary wave solutions of nonlinear wave equations

,”

Am. J. Phys.

60

,

650

–

654

(

1992

).

https://doi.org/10.1119/1.17120

Google Scholar

Crossref

13.

A. M.

Wazwaz

, “

The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations

,”

Applied Mathematics and Computation.

188

,

1467

–

1475

(

2007

).

https://doi.org/10.1016/j.amc.2006.11.013

Google Scholar

Crossref

14.

A. A.

Soliman

and

H. A.

Abdo

, “

New exact Solutions of nonlinear variants of the RLW, the PHI-four and Boussinesq equations based on modified extended direct algebraic method

,”

International Journal of Nonlinear Science

7

(

3

),

274

–

282

(

2009

).

Google Scholar

15.

A. H.

Salas

, and

C. A.

Gomez

, “

Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation

,”

Mathematical Problems in Engineering

2010

,

194329

(

2010

).

https://doi.org/10.1155/2010/194329

Google Scholar

Crossref

16.

S. J.

Liao

, “

The Homotopy Analysis Method and its applications in mechanics

,” Ph. D. Dissertation (in English),

Shanghai Jiao Tong Univ

, (

1992

).

Google Scholar

17.

S. T.

Mohyud-Din

and

M. A

Noor

, “

Homotopy perturbation method for solving partial differential equations

,”

Zeitschrift fur Naturforschung A, A J. Phys. Sci.

,

64

(

a

),

157

–

170

(

2009

).

Google Scholar

18.

J. H.

He

, “

Homotopy perturbation method with an auxiliary term

,”

Abstract and Applied Analysis

2012

,

857612

(

2012

).

https://doi.org/10.1155/2012/857612

Google Scholar

Crossref

19.

M. S. H.

Chowdhury

,

I.

Hashim

,

S.

Momani

, and

M. M.

Rahman

, “

Application of multistage homotopy perturbation method to the chaotic genesio system

,”

Abstract and Applied Analysis

2012

,

974293

(

2012

).

https://doi.org/10.1155/2012/974293

Google Scholar

Crossref

20.

J. H.

He

and

X. H.

Wu

, “

Exp-function method for nonlinear wave equations

,”

Chaos Solitons and Fractals

30

,

700

–

708

(

2006

).

https://doi.org/10.1016/j.chaos.2006.03.020

Google Scholar

Crossref

21.

H.

Naher

,

F. A.

Abdullah

, and

M. A.

Akbar

, “

New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method

,”

Journal of Applied Mathematics

2012

,

575387

(

2012

).

https://doi.org/10.1155/2012/575387

Google Scholar

Crossref

22.

S. T.

Mohyud-din

,

M. A.

Noor

and

K. I.

Noor

, “

Exp-function method for traveling wave solutions of modified Zakharov-Kuznetsov equation

,”

Journal of King Saud University

22

,

213

–

216

(

2010

).

https://doi.org/10.1016/j.jksus.2010.04.015

Google Scholar

Crossref

23.

H.

Naher

,

F. A.

Abdullah

and

M. A.

Akbar

, “

The exp-function method for new exact solutions of the nonlinear partial differential equations

,”

International Journal of the Physical Sciences.

6

(

29

),

6706

–

6716

(

2011

).

Google Scholar

24.

A.

Yildirim

, and

Z.

Pinar

, “

Application of the exp-function method for solving nonlinear reaction-diffusion equations arising in mathematical biology

,”

Computers and Mathematics with Applications

60

,

1873

–

1880

, (

2010

).

https://doi.org/10.1016/j.camwa.2010.07.020

Google Scholar

Crossref

25.

J. M.

Kim

, and

C.

Chun

, “

New exact solutions to the KdV-Burgers-Kuramoto equation with the Exp-function method

,”

Abstract and Applied Analysis

2012

,

892420

(

2012

).

https://doi.org/10.1155/2012/892420

Google Scholar

Crossref

26.

J. H.

He

, “

Variational iteration method for delay differential equations

,”

Communication in Nonlinear Science and Numerical Simulation

2

(

4

),

235

–

236

(

1997

).

https://doi.org/10.1016/S1007-5704(97)90008-3

Google Scholar

Crossref

27.

M. A.

Abdou

and

A. A.

Soliman

, “

Variational iteration method for solving Burger's and coupled Burger's equations

,”

J. Comput. Appl. Math.

181

,

245

–

251

(

2005

).

https://doi.org/10.1016/j.cam.2004.11.032

Google Scholar

Crossref

28.

M A.

Noor

and

S. T.

Mohyud-Din

, “

Variational iteration method for solving higher-order nonlinear boundary value problems using He's polynomials

,”

Int. J. Nonlinear Sci. Numerical Simulation

9

(

2

),

141

–

157

(

2008

).

Google Scholar

Crossref

29.

S. T.

Mohyud-Din

, “

Variational Iteration Method for Hirota-Satsuma Model Using He's Polynomials

,”

Zeitschrift für Naturforschung A, A J. Physical Sci.

65

(

6–7

),

525

–

528

(

2010

).

Google Scholar

30.

S. T.

Mohyud-Din

, “

Modified variational iteration method for integro-differential equations and coupled systems

,”

Zeitschrift für Naturforschung A, A J. Physical Sci.

65

(

4

),

277

–

284

(

2010

).

Google Scholar

Crossref

31.

J.

Kuang

, “

Variational approach to quasi-periodic solutions of nonautonomous second-order Hamiltonian systems

,”

Abstract and Applied Analysis

2012

,

271616

(

2012

).

https://doi.org/10.1155/2012/271616

Google Scholar

Crossref

32.

J.

Shao

and

F.

Meng

, “

Generalized variational principles on oscillation for nonlinear nonhomogeneous differential equations

,”

Abstract and Applied Analysis

2011

,

972656

(

2011

).

https://doi.org/10.1155/2011/972656

Google Scholar

Crossref

33.

M.

Noor

,

K.

Noor

,

A.

Waheed

, and

E. A.

Al-Said

, “

An efficient method for solving system of third-order nonlinear boundary value problems

,”

Mathematical Problems in Engineering

2011

,

250184

(

2011

).

https://doi.org/10.1155/2011/250184

Google Scholar

Crossref

34.

M.

Hasanov

, “

On the travelling waves for the generalized Schrodinger equation

,”

Abstract and Applied Analysis

2011

,

181369

(

2011

).

https://doi.org/10.1155/2011/181369

Google Scholar

Crossref

35.

Z.

Liu

,

M.

Jia

,

J. S.

Ume

, and

S. M.

Kang

, “

Positive solutions for a higher-order nonlinear neutral delay differential equation

,”

Abstract and Applied analysis

2011

,

328956

(

2011

).

https://doi.org/10.1155/2011/328956

Google Scholar

Crossref

36.

M. P.

Markakis

, “

Exact solutions for certain nonlinear autonomous ordinary differential equations of the second order and families of two-dimensional autonomous system

,”

International Journal of Differential Equations

2010

,

436860

(

2010

).

https://doi.org/10.1155/2010/436860

Google Scholar

Crossref

37.

P.

Liu

and

Y.

Wang

, “

A fold bifurcation theorem of degenerate solutions in a perturbed nonlinear equation

,”

Abstract and Applied Analysis

2011

,

302942

(

2011

).

https://doi.org/10.1155/2011/302942

Google Scholar

Crossref

38.

Q.

Liu

and

W.

Zheng

, “

Bifurcation of a Cohen-Grossberg neural network with discrete delays

,”

Abstract and Applied Analysis

2012

,

909385

(

2012

).

https://doi.org/10.1155/2012/909385

Google Scholar

Crossref

39.

G.

Schneider

, “

Justification of the NLS approximation for the kdV equation using the Miura transformation

,”

Advances in Mathematical Physics

2011

,

854719

(

2011

).

https://doi.org/10.1155/2011/854719

Google Scholar

Crossref

40.

A. H.

Salas

, “

Solving nonlinear partial differential equations by the sn-ns method

,”

Abstract and Applied Analysis

2012

,

340824

(

2012

).

https://doi.org/10.1155/2012/340824

Google Scholar

Crossref

41.

Z. I. A.

Al-Muhiameed

and

E. A. B.

Abdel-Salam

, “

Generalized hyperbolic function solution to a class of nonlinear Schrodinger-type equations

,”

Journal of Applied Mathematics

2012

,

265348

(

2012

).

https://doi.org/10.1155/2012/265348

Google Scholar

Crossref

42.

M.

Wang

,

X.

Li

, and

J.

Zhang

, “

The (G′/G) -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics

,”

Physics Letters A

372

,

417

–

423

(

2008

).

https://doi.org/10.1016/j.physleta.2007.07.051

Google Scholar

Crossref

43.

J.

Feng

,

W.

Li

, and

Q.

Wan

, “

Using (G′/G) -expansion method to seek traveling wave solution of Kolmogorov-Petrovskii-Piskunov equation

,”

Applied Mathematics and Computation

217

,

5860

–

5865

(

2011

).

https://doi.org/10.1016/j.amc.2010.12.071

Google Scholar

Crossref

44.

E. M. E.

Zayed

and

S.

Al-Joudi

, “

Applications of an Extended (G′/G) -Expansion Method to Find Exact Solutions of Nonlinear PDEs in Mathematical Physics

,”

Mathematical Problems in Engineering

2010

,

768573

(

2010

).

https://doi.org/10.1155/2010/768573

Google Scholar

Crossref

45.

H.

Naher

,

F. A.

Abdullah

, and

M. A.

Akbar

, “

The (G′/G) -expansion method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon equation

,”

Mathematical Problems in Engineering

2011

,

218216

(

2011

).

https://doi.org/10.1155/2011/218216

Google Scholar

Crossref

46.

R.

Abazari

and

R.

Abazari

, “

Hyperbolic, trigonometric and rational function solutions of Hirota-Ramani equation via (G′/G) -expansion method

,”

Mathematical Problems in Engineering

2011

,

424801

(

2011

).

https://doi.org/10.1155/2011/424801

Google Scholar

Crossref

47.

T.

Ozis

and

I.

Aslan

, “

Application of the (G′/G) -expansion method to Kawahara type equations using symbolic computation

,”

Applied Mathematics and Computation

216

,

2360

–

2365

(

2010

).

https://doi.org/10.1016/j.amc.2010.03.081

Google Scholar

Crossref

48.

H.

Naher

,

F. A.

Abdullah

, and

M. A.

Akbar

, “

New traveling wave solutions of the higher dimensional nonlinear evolution equation by the improved (G′/G) -expansion method

,”

World Applied Sciences Journal

161

,

11

–

21

(

2012

).

Google Scholar

49.

H.

Naher

and

F. A.

Abdullah

, “

Some new traveling wave solutions of the nonlinear reaction diffusion equation by using the improved (G′/G) -expansion method

,”

Mathematical Problems in Engineering

2012

,

871724

(

2012

).

https://doi.org/10.1155/2012/871724

Google Scholar

Crossref

50.

H.

Naher

and

F. A.

Abdullah

, “

Some new solutions of the combined KdV-MKdV equation by using the improved (G′/G) -expansion method

,”

World Applied Sciences Journal

16

(

11

),

1559

–

1570

(

2012

).

Google Scholar

51.

H.

Naher

and

F. A.

Abdullah

, “

The improved (G′/G) -expansion method for the (2+1)-dimensional Modified Zakharov-Kuznetsov equation

,”

Journal of Applied Mathematics

,

438928

(in press).

52.

S.

Zhu

, “

The generalizing Riccati equation mapping method in non-linear evolution equation: application to (2+1)-dimensional Boiti-Leon-Pempinelle equation

,”

Chaos, Solitons and Fractals

37

,

1335

–

1342

(

2008

).

https://doi.org/10.1016/j.chaos.2006.10.015

Google Scholar

Crossref

53.

B.

Li

,

Y.

Chen

,

H.

Xuan

, and

H.

Zhang

, “

Generalized Riccati equation expansion method and its application to the (3+1)-dimensional Jumbo-Miwa equation

,”

Applied Mathematics and Computation

152

,

581

–

595

(

2004

).

https://doi.org/10.1016/S0096-3003(03)00578-2

Google Scholar

Crossref

54.

A.

Bekir

and

A. C.

Cevikel

, “

The tanh-coth method combined with the Riccati equation for solving nonlinear coupled equation in mathematical physics

,”

Journal of King Saud University – Science

23

,

127

–

132

(

2011

).

https://doi.org/10.1016/j.jksus.2010.06.020

Google Scholar

Crossref

55.

A.

Salas

, “

Some exact solutions for the Caudrey-Dodd-Gibbon equation

,” arXiv:0805.2969v2 math-ph 21 May 2008.

56.

S.

Guo

,

L.

Mei

,

Y.

Zhou

, and

C.

Li

, “

The extended Riccati equation mapping method for variable-coefficient diffusion-reaction and mKdV equation

,”

Applied Mathematics and Computation

217

,

6264

–

6272

(

2011

).

https://doi.org/10.1016/j.amc.2010.12.116

Google Scholar

Crossref

57.

H.

Naher

and

F. A.

Abdullah

, “

The modified Benjamin-Bona-Mahony equation via the extended generalized Riccati equation mapping method

,”

Applied Mathematical Sciences

6

(

111

),

5495

–

5512

(

2012

) (in press).

Google Scholar

58.

C.

Liu

and

Z.

Dai

, “

Exact soliton solutions for the fifth-order Sawada-Kotera equation

”

Appl. Math. and Computation

206

,

272

–

275

(

2008

).

https://doi.org/10.1016/j.amc.2008.08.028

Google Scholar

Crossref

59.

Q.

Feng

and

B.

Zheng

, “

Traveling Wave Solutions For The Fifth-Order Sawada-Kotera Equation And The General Gardner Equation by (G′/G)-expansion method

,”

WSEAS Transactions on Mathematics

9

(

3

),

171

–

180

(

2010

).

Google Scholar

60.

A. M.

Wazwaz

, “

The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations

,”

Appl. Math. and Computation

184

,

1002

–

1014

(

2007

).

https://doi.org/10.1016/j.amc.2006.07.002

Google Scholar

Crossref

61.

A.

Salas

, “

Some solutions for a type of generalized Sawada-Kotera equation

,”

Appl. Math. and Computation

196

,

812

–

817

(

2008

).

https://doi.org/10.1016/j.amc.2007.07.013

Google Scholar

Crossref

2013

Author(s)

Extended generalized Riccati equation mapping method for the fifth-order Sawada-Kotera equation

I. INTRODUCTION

II. THE EXTENDED GENERALIZED RICCATI EQUATION MAPPING METHOD

III. APPLICATIONS OF THE METHOD

A. The Sawada-Kotera equation

IV. GRAPHICAL PRESENTATIONS OF SOME TRAVELING WAVE SOLUTIONS

ACKNOWLEDGMENTS

REFERENCES

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Extended generalized Riccati equation mapping method for the fifth-order Sawada-Kotera equation

I. INTRODUCTION

II. THE EXTENDED GENERALIZED RICCATI EQUATION MAPPING METHOD

III. APPLICATIONS OF THE METHOD

A. The Sawada-Kotera equation

IV. GRAPHICAL PRESENTATIONS OF SOME TRAVELING WAVE SOLUTIONS

ACKNOWLEDGMENTS

REFERENCES

Citing articles via

Submit your article

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Related Content

Resources

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