In this section, we will use the exp(–Ф(*η*))-expansion method to construct the exact solutions and then the solitary wave solutions to the fourth order Boussinesq equation. Let us consider the equation

{\mathit{u}}_{\mathit{t}\phantom{\rule{0.12em}{0ex}}\mathit{t}}-{\mathit{u}}_{\mathit{x}\phantom{\rule{0.12em}{0ex}}\mathit{x}}-{\mathit{u}}_{\mathit{x}\phantom{\rule{0.12em}{0ex}}\mathit{x}\phantom{\rule{0.12em}{0ex}}\mathit{x}\phantom{\rule{0.12em}{0ex}}\mathit{x}}-3\phantom{\rule{0.12em}{0ex}}{\left({\mathit{u}}^{2}\right)}_{\mathit{x}\phantom{\rule{0.12em}{0ex}}\mathit{x}}=0.

(11)

The above model (11) was introduced by Boussinesq to illustrate the propagation of long waves in shallow water (Lai et al. 2008), where *u*(*x*, *t*) is the elevation of the free surface of the fluid, where the subscripts denoting partial derivatives. The equation also arises in many other physical applications, such as, nonlinear lattice waves, iron sound waves in plasma, and vibrations in a nonlinear string. It was also applied to the study of the percolation of water in porous subsurface strata.

Equation (11) possesses solitary waves, extract from traveling wave solutions and Boussinesq was the first who gave a scientific explanation of their existence. We utilize the traveling wave variable *u*(*η*) = *u*(*x*, *t*), *η* = *x* - *Vt* and this operation changes (11) to the following ODE:

{\mathit{V}}^{2}{\mathit{u}}^{\u2033}-{\mathit{u}}^{\u2033}-{\mathit{u}}^{\left(4\right)}-3{\left({\mathit{u}}^{2}\right)}^{\u2033}=0.

(12)

Integrating Eq. (12) twice with respect to *η* yields:

\left({\mathit{V}}^{2}-1\right)\phantom{\rule{0.12em}{0ex}}\mathit{u}-{\mathit{u}}^{\u2033}-3\phantom{\rule{0.12em}{0ex}}{\mathit{u}}^{2}+\mathit{C}=0,

(13)

where *C* is an integration constant to be determined.

Balancing the highest order nonlinear term *u*^{2} and linear term of the highest order *u*^{′′} appearing in (13), yields *N* = 2. Therefore, the solution of Eq. (13) takes the form

\mathit{u}\left(\mathit{\eta}\right)={\mathit{A}}_{0}+{\mathit{A}}_{1}exp\left(-\mathrm{\Phi}\left(\mathit{\eta}\right)\right)+{\mathit{A}}_{2}exp{\left(-\mathrm{\Phi}\left(\mathit{\eta}\right)\right)}^{2},

(14)

where *A*_{0}, *A*_{1}, *A*_{2} are arbitrary constants such that *A*_{2} ≠ 0.

We substitute Eq. (14) into Eq. (13) and taking consideration Eq. (5), it generates a polynomial and then setting the coefficients of exp(–Ф(*η*)) to zero, yields

-6{\mathit{A}}_{2}-3{{\mathit{A}}_{2}}^{2}=0,

(17)

-2{\mathit{A}}_{1}-10{\mathit{A}}_{2}\mathit{\lambda}-6{\mathit{A}}_{1}{\mathit{A}}_{2}=0,

(18)

-{\mathit{A}}_{2}-4{\mathit{A}}_{2}{\mathit{\lambda}}^{2}-3{\mathit{A}}_{1}\mathit{\lambda}-3{{\mathit{A}}_{1}}^{2}-6{\mathit{A}}_{0}{\mathit{A}}_{2}+{\mathit{V}}^{2}{\mathit{A}}_{2}-8{\mathit{A}}_{2}\mathit{\mu}=0,

(19)

\mathit{C}+{\mathit{V}}^{2}{\mathit{A}}_{0}-{\mathit{A}}_{0}-2{\mathit{A}}_{2}{\mathit{\mu}}^{2}-3{{\mathit{A}}_{0}}^{2}-3{{\mathit{A}}_{1}}^{2}-{\mathit{A}}_{1}\mathit{\lambda \mu}=0,

(20)

Solutions of Eqs. (17)-(20), yield

\begin{array}{l}\mathit{C}=-4{\mathit{\mu}}^{2}-3{{\mathit{A}}_{0}}^{2}-2{\mathit{\lambda}}^{2}\mathit{\mu}-{\mathit{\lambda}}^{2}{\mathit{A}}_{\phantom{\rule{0.12em}{0ex}}0}-8{\mathit{A}}_{\phantom{\rule{0.12em}{0ex}}0}\mathit{\mu},\\ \mathit{V}=\pm \sqrt{{\mathit{\lambda}}^{2}+1+8\mathit{\mu}+6{\mathit{A}}_{\phantom{\rule{0.12em}{0ex}}0}},{\mathit{A}}_{1}=-2\mathit{\lambda},{\mathit{A}}_{1}=-2\end{array}.

where *λ*, *μ* and *A*_{0} are arbitrary constants.

Substituting the values of *V*, *A*_{0}, *A*_{1}, *A*_{2} into Eq. (14), yields

\mathit{u}\left(\mathit{\eta}\right)={\mathit{A}}_{0}-2\mathit{\lambda}exp\left(-\mathrm{\Phi}\left(\mathit{\eta}\right)\right)-2exp{\left(-\mathrm{\Phi}\left(\mathit{\eta}\right)\right)}^{2},

(21)

where \mathit{\eta}=\mathit{x}\mp \sqrt{{\mathit{\lambda}}^{2}+1+8\mathit{\mu}+6{\mathit{A}}_{0}}\phantom{\rule{0.24em}{0ex}}\mathit{t}.

By using the solutions of Eq. (5) into Eq. (21), we obtain the succeeding traveling wave solutions of the fourth order Boussinesq equation:

Type 1: When *μ* ≠ 0, *λ*^{2} - 4*μ* > 0,

\begin{array}{r}\hfill {\mathit{u}}_{1}\left(\mathit{\eta}\right)={\mathit{A}}_{0}+\frac{4\mathit{\lambda \mu}}{\sqrt{{\mathit{\lambda}}^{2}-4\mathit{\mu}}tanh\left(\frac{\sqrt{{\mathit{\lambda}}^{2}-4\mathit{\mu}}}{2}\left(\mathit{\eta}+\mathit{E}\right)\right)+\mathit{\lambda}}\\ \hfill -\frac{8\phantom{\rule{0.12em}{0ex}}{\mathit{\mu}}^{2}}{{\left\{\sqrt{{\mathit{\lambda}}^{2}-4\mathit{\mu}}tanh\left(\frac{\sqrt{{\mathit{\lambda}}^{2}-4\mathit{\mu}}}{2}\left(\mathit{\eta}+\mathit{E}\right)\right)+\mathit{\lambda}\right\}}^{2}}\end{array},

where \mathit{\eta}=\mathit{x}\mp \sqrt{{\mathit{\lambda}}^{2}+8\mathit{\mu}+6{\mathit{A}}_{0}+1}\phantom{\rule{0.24em}{0ex}}\mathit{t} and *E* is an arbitrary constant.

Type 2: When *μ* ≠ 0, *λ*^{2} - 4*μ* < 0,

\begin{array}{ll}{\mathit{u}}_{2}\left(\mathit{\eta}\right)& ={\mathit{A}}_{0}-\frac{4\mathit{\lambda \mu}}{\sqrt{4\mathit{\mu}-{\mathit{\lambda}}^{2}}tan\left(\frac{\sqrt{4\mathit{\mu}-{\mathit{\lambda}}^{2}}}{2}\left(\mathit{\eta}+\mathit{E}\right)\right)-\mathit{\lambda}}\\ -\frac{8\phantom{\rule{0.12em}{0ex}}{\mathit{\mu}}^{2}}{{\left\{\sqrt{4\mathit{\mu}-{\mathit{\lambda}}^{2}}tan\left(\frac{\sqrt{4\mathit{\mu}-{\mathit{\lambda}}^{2}}}{2}\left(\mathit{\eta}+\mathit{E}\right)\right)-\mathit{\lambda}\right\}}^{2}}\end{array},

where \mathit{\eta}=\mathit{x}\mp \sqrt{{\mathit{\lambda}}^{2}+8\mathit{\mu}+6{\mathit{A}}_{0}+1}\phantom{\rule{0.24em}{0ex}}\mathit{t} and *E* is an arbitrary constant.

Type 3: When *μ* = 0, *λ* ≠ 0, and *λ*^{2} - 4*μ* > 0,

{\mathit{u}}_{3}\left(\mathit{\eta}\right)={\mathit{A}}_{0}-\frac{2{\mathit{\lambda}}^{2}}{exp\left(\mathit{\lambda}\left(\mathit{\eta}+\mathit{E}\right)\right)-1}-\frac{2\phantom{\rule{0.12em}{0ex}}{\mathit{\lambda}}^{2}}{{\left\{exp\left(\mathit{\lambda}\left(\mathit{\eta}+\mathit{E}\right)\right)-1\right\}}^{2}},

where \mathit{\eta}=\mathit{x}\mp \sqrt{{\mathit{\lambda}}^{2}+8\mathit{\mu}+6{\mathit{A}}_{0}+1}\phantom{\rule{0.24em}{0ex}}\mathit{t} and *E* is an arbitrary constant.

Type 4: When *μ* ≠ 0, *λ* ≠ 0, and *λ*^{2} - 4*μ* = 0,

{\mathit{u}}_{4}\left(\mathit{\eta}\right)={\mathit{A}}_{0}+\frac{{\mathit{\lambda}}^{3}\left(\mathit{\eta}+\mathit{E}\right)}{\mathit{\lambda}\left(\mathit{\eta}+\mathit{E}\right)+2}-\frac{1}{2}{\left(\frac{{\mathit{\lambda}}^{2}\left(\mathit{\eta}+\mathit{E}\right)}{\mathit{\lambda}\left(\mathit{\eta}+\mathit{E}\right)+2}\right)}^{2},

where \mathit{\eta}=\mathit{x}\mp \sqrt{{\mathit{\lambda}}^{2}+8\mathit{\mu}+6{\mathit{A}}_{0}+1}\phantom{\rule{0.24em}{0ex}}\mathit{t} and *E* is an arbitrary constant.

Type 5: When *μ* = 0, *λ* = 0, and *λ*^{2} - 4*μ* = 0,

{\mathit{u}}_{5}\left(\mathit{\eta}\right)={\mathit{A}}_{0}-\frac{2}{{\left(\mathit{\eta}+\mathit{E}\right)}^{2}},

where \mathit{\eta}=\mathit{x}\mp \sqrt{{\mathit{\lambda}}^{2}+8\mathit{\mu}+6{\mathit{A}}_{0}+1}\phantom{\rule{0.24em}{0ex}}\mathit{t} and *E* is an arbitrary constant.

Solitary wave solutions represent an important type of solutions for nonlinear partial differential equations (PDEs) as many nonlinear partial differential equations have been found to have a variety of solitary wave solutions. It is familiar that searching of exact solutions of nonlinear partial differential equations plays a significant role in the study of nonlinear physical phenomena. Exact traveling wave solutions are useful for verifying the accuracy and stability of popular numerical schemes such as the finite difference and finite element methods. The solitary wave solutions obtained in this article are encouraging, applicable, and could be helpful in analyzing long wave propagation on the surface of a fluid layer under the action of gravity, iron sound waves in plasma, and vibrations in a nonlinear string.