In this section we apply method which is described in Section 2.2, on Eq. (1). Consider the traveling waves transformation

$$\begin{array}{}v(x,y,t)=U(\xi ),\phantom{\rule{1em}{0ex}}\xi =x+y+\omega t,\end{array}$$(33)

where *ω* is arbitrary constant which can be determined later. By using the above transformation to the Eq. (1) into the following ordinary differential equation and integrated we have

$$\begin{array}{}(4\omega -\beta ){U}^{\prime}+6{\left({U}^{\prime}\right)}^{2}+\omega {U}^{\u2034}=0.\end{array}$$(34)

Now applying the homogeneous balance principle between (*U*′)^{2} and *U*′″ in Eq. (34), we get *N* = 1. We suppose the solution of Eq. (34) has the form as:

$$\begin{array}{}U={B}_{0}+{B}_{1}\frac{{\mathit{\Psi}}^{\prime}(\xi )}{\mathit{\Psi}(\xi )}.\end{array}$$(35)

Where *B*_{0}, *B*_{1} are constant such that *B*_{1} ≠ 0, substituting Eq. (35) into Eq. (34) and equating the coefficients of *Ψ*^{–4}, *Ψ*^{–3}, *Ψ*^{–2}, *Ψ*^{–1} we get the following equations:

$$\begin{array}{}(-\omega +{B}_{1}){B}_{1}({\mathit{\Psi}}^{\prime}{)}^{4}=0,\end{array}$$(36)

$$\begin{array}{}(\omega -{B}_{1}){B}_{1}({\mathit{\Psi}}^{\prime}{)}^{2}{\mathit{\Psi}}^{\u2033}=0,\end{array}$$(37)

$$\begin{array}{}& (\beta {B}_{1}-4\omega {B}_{1})({\mathit{\Psi}}^{\prime}{)}^{2}+(6{B}_{1}^{2}-3\omega {B}_{1})({\mathit{\Psi}}^{\u2033}{)}^{2}\\ & -4\omega {B}_{1}{\mathit{\Psi}}^{\prime}{\mathit{\Psi}}^{\u2034}=0,\end{array}$$(38)

$$\begin{array}{}(-\beta {B}_{1}+4\omega {B}_{1}){\mathit{\Psi}}^{\u2033}+\omega {B}_{1}{\mathit{\Psi}}^{(4)}=0.\end{array}$$(39)

From Eq. (36) and Eq. (37) we obtain, *B*_{1} = *ω*, Now integrating Eq. (39) and substituting into Eq. (38), we get:

$$\begin{array}{}\frac{{\mathit{\Psi}}^{\u2033}}{{\mathit{\Psi}}^{\prime}}=\mu ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{where}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mu =\pm \sqrt{\frac{3\beta -16\omega}{3\omega}},\end{array}$$(40)

consequently, we obtain

$$\begin{array}{}& {\mathit{\Psi}}^{\prime}(\xi )={c}_{1}{e}^{\mu \xi},\end{array}$$(41)

$$\begin{array}{}& \mathit{\Psi}={c}_{2}+\frac{{c}_{1}}{\mu}{c}_{1}{e}^{\mu \xi},\end{array}$$(42)

where *c*_{1} and *c*_{2} are constant of integration. Now substituting the values of *B*_{1}, *Ψ* and *Ψ*′ into Eq. (35), we get the exact solution of Eq. (1), as follows

$$\begin{array}{}v(x,y,t)={B}_{0}+\omega \left(\frac{\mu {c}_{1}{e}^{\mu \xi}}{\mu {c}_{2}+{c}_{1}{e}^{\mu \xi}}\right),\end{array}$$(43)

where *ξ* = *x* + *y* + *ωt*.

Figure 7 Exact solitary wave solution of Eq. (43) is plotted in different shapes at :*B*_{0} = 1, *β* = 2, *c*_{1} = 0.5, *c*_{2} = –0.5, *ω* = 0.25:(a) solitary wave and (b) one dimensional solitary wave

Simplifying Eq. (43), we obtain

$$\begin{array}{}& v(x,y,t)=\\ & {B}_{0}+\omega \left(\frac{\mu {c}_{1}\left(\mathrm{cosh}\left(\frac{\mu \xi}{2}\right)+\mathrm{sinh}\left(\frac{\mu \xi}{2}\right)\right)}{({c}_{1}+\mu {c}_{2})\mathrm{cosh}\left(\frac{\mu \xi}{2}\right)+({c}_{1}-{c}_{2}\mu )\mathrm{sinh}\left(\frac{\mu \xi}{2}\right)}\right),\end{array}$$(44)

where *ξ* = *x* + *y* + *ωt*.

Similarly we can randomly choose parameter *c*_{1} and *c*_{2}, by setting *c*_{1} = *μc*_{2}, we obtain the following solitary solution

$$\begin{array}{}& v(x,y,t)=\\ & {B}_{0}+\sqrt{\frac{3\beta \omega -16{\omega}^{2}}{12}}\left(\mathrm{tanh}\left(\sqrt{\frac{3\beta -16\omega}{12\omega}}\xi \right)\pm 1\right),\end{array}$$(45)

where *ξ* = *x* + *y* + *ωt*.

Now again we choose *c*_{1} = –*c*_{2}*μ*, we obtain following solitary waves solution.

$$\begin{array}{}& v(x,y,t)=\\ & {B}_{0}+\sqrt{\frac{3\beta \omega -16{\omega}^{2}}{12}}\left(\mathrm{coth}\left(\sqrt{\frac{3\beta -16\omega}{12\omega}}\xi \right)\pm 1\right),\end{array}$$(46)

where *ξ* = *x* + *y* + *ωt* and *B*_{0} is left as a free parameter.

Figure 8 Exact solitary wave solution of Eq. (46) is plotted in different shapes at :*B*_{0} = 1, *β* = 1, *ω* = 0.5:(a) periodic solitary wave and (b) one-dimensional solitary wave

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.