A Highly Nonlinear Shallow-Water Model Arising from the Full Water Waves with the Coriolis Effect

Abstract

In the present paper we apply the method of double asymptotic expansion to formally derive a highly nonlinear shallow-water model propagating in the equatorial ocean regions with the Coriolis effect due to the Earth’s rotation. The asymptotic procedures of the derivation of this equation needs to be implemented under a suitable scaling, which is proved to be optimal in the sense that there are no non-local terms in the equation of the surface elevation. Moreover, we give the local well-posedness of the corresponding Cauchy problem and wave breaking criterion.

Appendix

In this section, we provide a detailed computation of the orders $O(\varepsilon ^{5}\mu ^{0}),O(\varepsilon ^{6}\mu ^{0})$ $\text{ and }\ O(\varepsilon ^{3}\mu ^{1})$-approximation. From the fifth equality of (2.5), we obtain

$$\begin{aligned} \eta _{30,\tau }= & {} 2c_{1}(\eta _{00}\eta _{30}+\eta _{10}\eta _{20})_{\xi }+\frac{(3c+2c_{1})(c+c_{1})}{c+\Omega }(\eta ^{2}_{00}\eta _{20}+\eta _{00}\eta ^{2}_{10})_{\xi }\nonumber \\{} & {} +\frac{64 cc_{1}+24c_{1}^{2}+45c^{2}-15}{6(\Omega +c)}(c+c_{1})(\eta ^{3}_{00}\eta _{10})_{\xi }+\frac{B_{2}}{2(\Omega +c)}(\eta ^{5}_{00})_{\xi }. \end{aligned}$$

(4.1)

It follows from $u_{40,\xi }=c\eta _{40,\xi }-\eta _{30,\tau }-(u_{00}\eta _{30}+u_{10}\eta _{20}+u_{20}\eta _{10}+u_{30}\eta _{00})_{\xi }$ in [20] and (4.1) that

$$\begin{aligned} u_{40,\xi }= & {} c\eta _{40,\xi }-2(c+c_{1})(\eta _{00}\eta _{30}+\eta _{10}\eta _{20})_{\xi }-\frac{2c_{1}-3\Omega }{\Omega +c}(c+c_{1})(\eta ^{2}_{00}\eta _{20}+\eta _{00}\eta ^{2}_{10})_{\xi }\\{} & {} -\frac{64 cc_{1}+24c_{1}^{2}+45c^{2}+24\Omega ^{2}-3}{6(\Omega +c)^{2}}(c+c_{1})(\eta ^{3}_{00}\eta _{10})_{\xi }-B_{3}(\eta ^{5}_{00})_{\xi }, \end{aligned}$$

with

$$\begin{aligned} B_{3}=\frac{B_{2}}{2(\Omega +c)}-\frac{64 cc_{1}+24c_{1}^{2}+45c^{2}+24\Omega ^{2}-3}{24(\Omega +c)^{2}}(c+c_{1}), \end{aligned}$$

which implies

$$\begin{aligned} u_{40}= & {} c\eta _{40}-2(c+c_{1})(\eta _{00}\eta _{30}+\eta _{10}\eta _{20})-\frac{2c_{1}-3\Omega }{\Omega +c}(c+c_{1})(\eta ^{2}_{00}\eta _{20}+\eta _{00}\eta ^{2}_{10})\nonumber \\{} & {} -\frac{64 cc_{1}+24c_{1}^{2}+45c^{2}+24\Omega ^{2}-3}{6(\Omega +c)^{2}}(c+c_{1})\eta ^{3}_{00}\eta _{10}-B_{3}\eta ^{5}_{00}. \end{aligned}$$

(4.2)

It thus deduces from (4.1) that

$$\begin{aligned} u_{40,\tau }= & {} c\eta _{40,\tau }-4c_{1}(c+c_{1})(\eta ^{2}_{00}\eta _{30}+2\eta _{00}\eta _{10}\eta _{20}+\frac{1}{3}\eta ^{3}_{10})_{\xi }\nonumber \\{} & {} -\frac{2(3c^{2}+5c c_{1}+4c_{1}^{2}-3\Omega c_{1})}{\Omega +c}(c+c_{1})(\eta ^{3}_{00}\eta _{20}+\frac{3}{2}\eta ^{2}_{00}\eta ^{2}_{10})_{\xi }\nonumber \\{} & {} -B_{1}(\eta ^{4}_{00}\eta _{10})_{\xi }-B_{4}\eta ^{5}_{00}\eta _{00,\xi }, \end{aligned}$$

(4.3)

with

$$\begin{aligned} B_{4}= & {} \frac{5B_{2}(c+c_{1})}{\Omega +c}+\frac{(2c_{1}-3\Omega )(64 cc_{1}+24c_{1}^{2}+45c^{2}-15)}{6(\Omega +c)^{3}}(c+c_{1})^{2}\\{} & {} +\frac{(2c_{1}+3c)(64 cc_{1}+24c_{1}^{2}+45c^{2}+24\Omega ^{2}-3)}{6(\Omega +c)^{3}}(c+c_{1})^{2}\\= & {} -\frac{3c^{4}(c^{2}-2)(37c^{12}-145c^{10}+366c^{8}-378c^{6}+399c^{4}-121c^{2}+66)}{4(c^{2}+1)^{8}}. \end{aligned}$$

For the order $O(\varepsilon ^{5}\mu ^{0})$ terms of the governing equations (2.2), we obtain from the Taylor expansion (2.3) that

$$\begin{aligned} {\left\{ \begin{array}{ll} -cu_{50,\xi }+u_{40,\tau }+u_{40}u_{00,\xi }+u_{30}u_{10,\xi }+u_{20}u_{20,\xi }+u_{10}u_{30,\xi }\\ +u_{00}u_{40,\xi }+2\Omega W_{50}=-p_{50,\xi }, &{}{ in} \ 0<z<1,\\ 2\Omega u_{50}=p_{50,z}, &{}{ in} \ 0<z<1,\\ u_{50,\xi }+W_{50,z}=0, &{}{ in} \ 0<z<1,\\ u_{50,z}=0, &{}{ in} \ 0<z<1,\\ p_{50}+p_{40,z}\eta _{00}+p_{30,z}\eta _{10}+p_{20,z}\eta _{20}+p_{10,z}\eta _{30}+p_{00,z}\eta _{40}=\eta _{50}, &{}{ on} \ z=1,\\ W_{50}+W_{40,z}\eta _{00}+W_{30,z}\eta _{10}+W_{20,z}\eta _{20}+W_{10,z}\eta _{30}+W_{00,z}\eta _{40}\\ =-c\eta _{50,\xi }+\eta _{40,\tau }+u_{40}\eta _{00,\xi }+u_{30}\eta _{10,\xi }+u_{20}\eta _{20,\xi }+u_{10}\eta _{30,\xi }+u_{00}\eta _{40,\xi }, &{}{ on} \ z=1,\\ W_{50}=0, &{}{ on} \ z=0.\\ \end{array}\right. } \end{aligned}$$

(4.4)

In view of the fourth equation in (4.4), we find that $u_{50}$ is independent of z, so $u_{50}=u_{50}(\tau ,\xi ).$ Due to the third equation in (4.4) and the boundary condition of $W_{50}$ on $z=0$ and $z=1$, we get

$$\begin{aligned} W_{50}=W_{50}|_{z=0}+\int _{0}^{z}W_{50,z^{'}}dz^{'}=-zu_{50,\xi }, \end{aligned}$$

(4.5)

and

$$\begin{aligned} W_{50}|_{z=1}=-c\eta _{50,\xi }+\eta _{40,\tau }+(u_{40}\eta _{00}+u_{30}\eta _{10}+u_{20}\eta _{20}+u_{10}\eta _{30}+u_{00}\eta _{40})_{\xi }. \end{aligned}$$

Hence, we obtain

$$\begin{aligned} u_{50,\xi }=c\eta _{50,\xi }-\eta _{40,\tau }-(u_{40}\eta _{00}+u_{30}\eta _{10}+u_{20}\eta _{20}+u_{10}\eta _{30}+u_{00}\eta _{40})_{\xi }, \end{aligned}$$

(4.6)

and then

$$\begin{aligned} W_{50}=z(-c\eta _{50,\xi }+\eta _{40,\tau }+(u_{40}\eta _{00}+u_{30}\eta _{10}+u_{20}\eta _{20}+u_{10}\eta _{30}+u_{00}\eta _{40})_{\xi }). \end{aligned}$$

On the other hand, taking account of the second and the fifth equation in (4.4), we have

$$\begin{aligned} p_{50}= & {} p_{50}|_{z=1}+\int _{1}^{z}p_{50,z^{'}}dz^{'}\\= & {} \eta _{50}-2\Omega (u_{40}\eta _{00}+u_{30}\eta _{10}+u_{20}\eta _{20}+u_{10}\eta _{30}+u_{00}\eta _{40})+2\Omega (z-1)u_{50}, \end{aligned}$$

and then

$$\begin{aligned} p_{50,\xi }=\eta _{50,\xi }-2\Omega (u_{40}\eta _{00}+u_{30}\eta _{10}+u_{20}\eta _{20}+u_{10}\eta _{30}+u_{00}\eta _{40})_{\xi }+2\Omega (z-1)u_{50,\xi }. \end{aligned}$$

(4.7)

On account of the first equation in (4.4), we deduce from (4.5) that

$$\begin{aligned} -p_{50,\xi }=-cu_{50,\xi }+u_{40,\tau }+(u_{00}u_{40}+u_{10}u_{30}+\frac{1}{2}u^{2}_{20})_{\xi }-2\Omega zu_{50,\xi }. \end{aligned}$$

(4.8)

Combining (4.7) with (4.8), we have

$$\begin{aligned} \eta _{50,\xi }- & {} 2\Omega (u_{40}\eta _{00}+u_{30}\eta _{10}+u_{20}\eta _{20}+u_{10}\eta _{30}+u_{00}\eta _{40})_{\xi }-(c+2\Omega )u_{50,\xi }\\+ & {} u_{40,\tau }+(u_{00}u_{40}+u_{10}u_{30}+\frac{1}{2}u^{2}_{20})_{\xi }=0. \end{aligned}$$

Thanks to (4.3) and (4.6), we obtain

$$\begin{aligned} 2(\Omega+ & {} c)\eta _{40,\tau }+3c^{2}(\eta _{00}\eta _{40}+\eta _{10}\eta _{30}+\frac{1}{2}\eta ^{2}_{20})_{\xi }-2(3c+2c_{1})(c+c_{1})(\eta ^{2}_{00}\eta _{30}\\+ & {} 2\eta _{00}\eta _{10}\eta _{20}+\frac{1}{3}\eta ^{3}_{10})_{\xi }-\frac{64cc_{1}+24c_{1}^{2}+45c^{2}-15}{3(\Omega +c)}(c+c_{1})(\eta ^{3}_{00}\eta _{20}+\frac{3}{2}\eta ^{2}_{00}\eta ^{2}_{10})_{\xi }\\- & {} 5B_{2}(\eta ^{4}_{00}\eta _{10})_{\xi }-B_{5}(\eta ^{6}_{00})_{\xi }=0, \end{aligned}$$

with

$$\begin{aligned} B_{5}= & {} \frac{1}{6}B_{4}-\frac{64 cc_{1}+24c_{1}^{2}+45c^{2}+24\Omega ^{2}-3}{24(\Omega +c)^{2}}(c+c_{1})^{2}-\frac{(2c_{1}-3\Omega )^{2}(c+c_{1})^{2}}{18(\Omega +c)^{2}}+2cB_{3}\\= & {} -\frac{c^{2}(c^{2}-2)(17c^{14}-112c^{12}+281c^{10}-434c^{8}+367c^{6}-224c^{4}+63c^{2}-14)}{8(c^{2}+1)^{8}}, \end{aligned}$$

which leads to

$$\begin{aligned} \eta _{40,\tau }= & {} 2c_{1}(\eta _{00}\eta _{40}+\eta _{10}\eta _{30}+\frac{1}{2}\eta ^{2}_{20})_{\xi }+\frac{3c+2c_{1}}{\Omega +c}(c+c_{1})(\eta ^{2}_{00}\eta _{30}+2\eta _{00}\eta _{10}\eta _{20}+\frac{1}{3}\eta ^{3}_{10})_{\xi }\nonumber \\{} & {} +\frac{64cc_{1}+24c_{1}^{2}+45c^{2}-15}{6(\Omega +c)^{2}}(c+c_{1})(\eta ^{3}_{00}\eta _{20}+\frac{3}{2}\eta ^{2}_{00}\eta ^{2}_{10})_{\xi }+\frac{5B_{2}}{2(\Omega +c)}(\eta ^{4}_{00}\eta _{10})_{\xi }\nonumber \\{} & {} +\frac{B_{5}}{2(\Omega +c)}(\eta ^{6}_{00})_{\xi }. \end{aligned}$$

(4.9)

Therefore, we have

$$\begin{aligned} u_{50,\xi }= & {} c\eta _{50,\xi }-2(c+c_{1})(\eta _{00}\eta _{40}+\eta _{10}\eta _{30}+\frac{1}{2}\eta ^{2}_{20})_{\xi }-\frac{2c_{1}-3\Omega }{\Omega +c}(c+c_{1})(\eta ^{2}_{00}\eta _{30}+2\eta _{00}\eta _{10}\eta _{20}\\{} & {} +\frac{1}{3}\eta ^{3}_{10})_{\xi }-\frac{64cc_{1}+24c_{1}^{2}+45c^{2}+24\Omega ^{2}-3}{6(\Omega +c)^{2}}(c+c_{1})(\eta ^{3}_{00}\eta _{20}+\frac{3}{2}\eta ^{2}_{00}\eta ^{2}_{10})_{\xi }\\{} & {} -5B_{3}(\eta ^{4}_{00}\eta _{10})_{\xi }-(\frac{B_{5}}{2(\Omega +c)}-B_{3})(\eta ^{6}_{00})_{\xi }, \end{aligned}$$

which along with the far field conditions $\eta _{00},\eta _{10},\eta _{20},\eta _{30},\eta _{40}\rightarrow 0$ as $|\xi |\rightarrow \infty$ gives

$$\begin{aligned} u_{50}= & {} c\eta _{50}-2(c+c_{1})(\eta _{00}\eta _{40}+\eta _{10}\eta _{30}+\frac{1}{2}\eta ^{2}_{20})-\frac{2c_{1}-3\Omega }{\Omega +c}(c+c_{1})(\eta ^{2}_{00}\eta _{30}+2\eta _{00}\eta _{10}\eta _{20}+\frac{1}{3}\eta ^{3}_{10})\nonumber \\{} & {} -\frac{64cc_{1}+24c_{1}^{2}+45c^{2}+24\Omega ^{2}-3}{6(\Omega +c)^{2}}(c+c_{1})(\eta ^{3}_{00}\eta _{20}+\frac{3}{2}\eta ^{2}_{00}\eta ^{2}_{10})-5B_{3}\eta ^{4}_{00}\eta _{10}\nonumber \\{} & {} -(\frac{B_{5}}{2(\Omega +c)}-B_{3})\eta ^{6}_{00}. \end{aligned}$$

(4.10)

Thanks to (4.1) and (4.9), we deduce that

$$\begin{aligned} u_{50,\tau }= & {} c\eta _{50,\tau }-4c_{1}(c+c_{1})(\eta ^{2}_{00}\eta _{40}+\eta ^{2}_{10}\eta _{20}+\eta _{00}\eta ^{2}_{20}+2\eta _{00}\eta _{10}\eta _{30})_{\xi }\nonumber \\{} & {} -\frac{2(3c^{2}+5c c_{1}+4c_{1}^{2}-3\Omega c_{1})}{\Omega +c}(c+c_{1})(\eta ^{3}_{00}\eta _{30}+\eta _{00}\eta ^{3}_{10}+3\eta ^{2}_{00}\eta _{10}\eta _{20})_{\xi }\nonumber \\{} & {} -B_{1}(\eta ^{4}_{00}\eta _{20}+2\eta ^{3}_{00}\eta ^{2}_{10})_{\xi }-B_{4}(\eta ^{5}_{00}\eta _{10})_{\xi }-B_{6}\eta ^{6}_{00}\eta _{00,\xi }, \end{aligned}$$

(4.11)

with

$$\begin{aligned}{} & {} B_{6}=\frac{6c_{1}B_{5}}{\Omega +c}-12c_{1}B_{3}+\frac{6B_{5}(c+c_{1})}{\Omega +c}+\frac{5(2c_{1}-3\Omega )(c+c_{1})B_{2}}{2(\Omega +c)^{2}} +\frac{5(2c_{1}+3c)B_{3}}{\Omega +c}(c+c_{1})\\{} & {} +\frac{(64 cc_{1}+24c_{1}^{2}+45c^{2}-15)(64 cc_{1}+24c_{1}^{2}+45c^{2}+24\Omega ^{2}-3)}{36(\Omega +c)^{4}}(c+c_{1})^{2}\\= & {} \frac{7c^{4}(c^{2}-2)(99c^{16}-540c^{14}+1587c^{12}-2540c^{10}+3109c^{8}-2084c^{6}+1297c^{4}-308c^{2}+100)}{8(c^{2}+1)^{10}}. \end{aligned}$$

For the order $O(\varepsilon ^{6}\mu ^{0})$ terms of the governing equations (2.2), we deduce from the Taylor expansion (2.3) that

$$\begin{aligned} {\left\{ \begin{array}{ll} -cu_{60,\xi }+u_{50,\tau }+u_{50}u_{00,\xi }+u_{40}u_{10,\xi }+u_{30}u_{20,\xi }+u_{20}u_{30,\xi }+u_{10}u_{40,\xi }\\ +u_{00}u_{50,\xi }+2\Omega W_{60}=-p_{60,\xi }, &{}{ in} \ 0<z<1,\\ 2\Omega u_{60}=p_{60,z}, &{}{ in} \ 0<z<1,\\ u_{60,\xi }+W_{60,z}=0, &{}{ in} \ 0<z<1,\\ u_{60,z}=0, &{}{ in} \ 0<z<1,\\ p_{60}+p_{50,z}\eta _{00}+p_{40,z}\eta _{10}+p_{30,z}\eta _{20}+p_{20,z}\eta _{30}+p_{10,z}\eta _{40}+p_{00,z}\eta _{50}=\eta _{60}, &{}{ on} \ z=1,\\ W_{60}+W_{50,z}\eta _{00}+W_{40,z}\eta _{10}+W_{30,z}\eta _{20}+W_{20,z}\eta _{30}+W_{10,z}\eta _{40}+W_{00,z}\eta _{50}\\ =-c\eta _{60,\xi }+\eta _{50,\tau }+u_{50}\eta _{00,\xi }+u_{40}\eta _{10,\xi }+u_{30}\eta _{20,\xi }+u_{20}\eta _{30,\xi }+u_{10}\eta _{40,\xi }\\ +u_{00}\eta _{50,\xi }, &{}{ on} \ z=1,\\ W_{60}=0, &{}{ on} \ z=0.\\ \end{array}\right. } \end{aligned}$$

(4.12)

From the fourth equation in (4.12), $u_{60}$ is independent of z, that is, $u_{60}=u_{60}(\tau ,\xi ).$ Thanks to the third equation in (4.12) and the boundary condition of $W_{60}$ on $z=0$ and $z=1$, we get

$$\begin{aligned} W_{60}=W_{60}|_{z=0}+\int _{0}^{z}W_{60,z^{'}}dz^{'}=-zu_{60,\xi }, \end{aligned}$$

(4.13)

and

$$\begin{aligned} W_{60}|_{z=1}=-c\eta _{60,\xi }+\eta _{50,\tau }+(u_{50}\eta _{00}+u_{40}\eta _{10}+u_{30}\eta _{20}+u_{20}\eta _{30}+u_{10}\eta _{40}+u_{00}\eta _{50})_{\xi }. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} u_{60,\xi }=c\eta _{60,\xi }-\eta _{50,\tau }-(u_{50}\eta _{00}+u_{40}\eta _{10}+u_{30}\eta _{20}+u_{20}\eta _{30}+u_{10}\eta _{40}+u_{00}\eta _{50})_{\xi }. \end{aligned}$$

(4.14)

On the other hand, thanks to the second and the fifth equation in (4.12), we deduce that

$$\begin{aligned} p_{60}= & {} p_{60}|_{z=1}+\int _{1}^{z}p_{60,z^{'}}dz^{'}\\= & {} \eta _{60}-2\Omega (u_{50}\eta _{00}+u_{40}\eta _{10}+u_{30}\eta _{20}+u_{20}\eta _{30}+u_{10}\eta _{40}+u_{00}\eta _{50})+2\Omega (z-1)u_{60}, \end{aligned}$$

and then

$$\begin{aligned} p_{60,\xi }=\eta _{60,\xi }-2\Omega (u_{50}\eta _{00}+u_{40}\eta _{10}+u_{30}\eta _{20}+u_{20}\eta _{30}+u_{10}\eta _{40}+u_{00}\eta _{50})_{\xi }+2\Omega (z-1)u_{60,\xi }. \end{aligned}$$

(4.15)

Taking account of the first equation in (4.12), we get

$$\begin{aligned} -p_{60,\xi }=-cu_{60,\xi }+u_{50,\tau }+(u_{00}u_{50}+u_{10}u_{40}+u_{20}u_{30})_{\xi }-2\Omega zu_{60,\xi }. \end{aligned}$$

(4.16)

Combining (4.15) with (4.16), we have

$$\begin{aligned} \eta _{60,\xi }- & {} 2\Omega (u_{50}\eta _{00}+u_{40}\eta _{10}+u_{30}\eta _{20}+u_{20}\eta _{30}+u_{10}\eta _{40}+u_{00}\eta _{50})_{\xi }-(c+2\Omega )u_{60,\xi }\\+ & {} u_{50,\tau }+(u_{00}u_{50}+u_{10}u_{40}+u_{20}u_{30})_{\xi }=0. \end{aligned}$$

Substituting (4.11) and (4.14) into the above formula, we obtain

$$\begin{aligned} 2(\Omega+ & {} c)\eta _{50,\tau }+3c^{2}(\eta _{00}\eta _{50}+\eta _{10}\eta _{40}+\eta _{20}\eta _{30})_{\xi }-2(3c+2c_{1})(c+c_{1})(\eta ^{2}_{00}\eta _{40}+\eta ^{2}_{10}\eta _{20}\nonumber \\+ & {} \eta _{00}\eta ^{2}_{20}+2\eta _{00}\eta _{10}\eta _{30})_{\xi }-\frac{64cc_{1}+24c_{1}^{2}+45c^{2}-15}{3(\Omega +c)}(c+c_{1})(\eta ^{3}_{00}\eta _{30}+\eta _{00}\eta ^{3}_{10}+3\eta ^{2}_{00}\eta _{10}\eta _{20})_{\xi }\nonumber \\- & {} 5B_{2}(\eta ^{4}_{00}\eta _{20}+2\eta ^{3}_{00}\eta ^{2}_{10})_{\xi }-6B_{5}(\eta ^{5}_{00}\eta _{10})_{\xi }-B_{7}(\eta ^{7}_{00})_{\xi }=0, \end{aligned}$$

(4.17)

with

$$\begin{aligned}{} & {} B_{7}=-\frac{(2c_{1}-3\Omega )(64cc_{1}+24c_{1}^{2}+45c^{2}+24\Omega ^{2}-3)}{72(\Omega +c)^{3}}(c+c_{1})^{2} -(3c+c_{1})B_{3}+\frac{cB_{5}}{\Omega +c}+\frac{1}{7}B_{6}\\= & {} \frac{c^{2}(c^{2}-2)(25c^{18}-190c^{16}+617c^{14}-1206c^{12}+1483c^{10}-1274c^{8}+699c^{6}-274c^{4}+56c^{2}-8)}{4(c^{2}+1)^{10}}. \end{aligned}$$

From the sixth equality of (2.5), we get

$$\begin{aligned} \eta _{11,\tau }= & {} 2c_{1}(\eta _{00}\eta _{11}+\eta _{10}\eta _{01})_{\xi }+\frac{(3c+2c_{1})(c+c_{1})}{\Omega +c}(\eta ^{2}_{00}\eta _{01})_{\xi } +\frac{2}{9}c_{1}\eta _{10,\xi \xi \xi }+(-\frac{c_{1}}{9}\nonumber \\{} & {} +\frac{5c c_{1}}{9(\Omega +c)}+\frac{c_{1}^{2}}{9(\Omega +c)})(\eta ^{2}_{00,\xi })_{\xi }+(-\frac{2c_{1}}{9}+\frac{10 cc_{1}}{9(\Omega +c)}+\frac{4c_{1}^{2}}{9(\Omega +c)})(\eta _{00}\eta _{00,\xi \xi })_{\xi }. \end{aligned}$$

(4.18)

In view of the calculation of the order $O(\varepsilon ^{2}\mu ^{1})$ in [20], we have

$$\begin{aligned} u_{21,\xi }=c\eta _{21,\xi }-\eta _{11,\tau }+(\frac{z^{2}}{2}-\frac{1}{6})H_{1,\xi }-H_{2,\xi }|_{z=1}, \end{aligned}$$

with $H_{1,\xi }=2(c+c_{1})(\eta ^{2}_{00,\xi }+\eta _{00}\eta _{00,\xi \xi })_{\xi }-c\eta _{10,\xi \xi \xi },$ $H_{2,\xi }|_{z=1}=2c(\eta _{00}\eta _{11}+\eta _{01}\eta _{10})_{\xi }-(\frac{c}{3}+\frac{2c_{1}}{9}) (\eta _{00}\eta _{00,\xi \xi })_{\xi }-3(c+c_{1})(\eta ^{2}_{00}\eta _{01})_{\xi }.$ Hence,

$$\begin{aligned} u_{21,\xi }= & {} c\eta _{21,\xi }-2(c+c_{1})(\eta _{00}\eta _{11}+\eta _{01}\eta _{10})_{\xi }-\frac{2c_{1}-3\Omega }{\Omega +c}(c+c_{1})(\eta ^{2}_{00}\eta _{01})_{\xi }\\{} & {} +(\frac{c}{6}-\frac{2}{9}c_{1}-\frac{c}{2}z^{2})\eta _{10,\xi \xi \xi }+(z^{2}(c+c_{1})-\frac{c}{3}-\frac{2c_{1}}{9}-\frac{5c c_{1}}{9(\Omega +c)}-\frac{c_{1}^{2}}{9(\Omega +c)})(\eta ^{2}_{00,\xi })_{\xi }\\{} & {} +(z^{2}(c+c_{1})+\frac{c_{1}}{9}-\frac{10c c_{1}}{9(\Omega +c)}-\frac{4c_{1}^{2}}{9(\Omega +c)})(\eta _{00}\eta _{00,\xi \xi })_{\xi }, \end{aligned}$$

which yields

$$\begin{aligned} u_{21}= & {} c\eta _{21}-2(c+c_{1})(\eta _{00}\eta _{11}+\eta _{10}\eta _{01})-\frac{2c_{1}-3\Omega }{\Omega +c}(c+c_{1})\eta _{00}^{2}\eta _{01}\nonumber \\{} & {} +(\frac{c}{6}-\frac{2c_{1}}{9}-\frac{c}{2}z^{2})\eta _{10,\xi \xi }+(z^{2}(c+c_{1})-\frac{c}{3}-\frac{2c_{1}}{9}-\frac{5c c_{1}}{9(\Omega +c)}-\frac{c_{1}^{2}}{9(\Omega +c)})\eta ^{2}_{00,\xi }\nonumber \\{} & {} +(z^{2}(c+c_{1})+\frac{c_{1}}{9}-\frac{10c c_{1}}{9(\Omega +c)}-\frac{4c_{1}^{2}}{9(\Omega +c)})\eta _{00}\eta _{00,\xi \xi }. \end{aligned}$$

(4.19)

Taking the $\tau$ derivative of the above equality, it then follows from (2.5) that

$$\begin{aligned} u_{21,\tau }= & {} c\eta _{21,\tau }-4c_{1}(c+c_{1})(\eta ^{2}_{00}\eta _{11}+2\eta _{00}\eta _{10}\eta _{01})_{\xi }-\frac{2(3c^{2}+5c c_{1}+4c_{1}^{2}-3\Omega c_{1})}{\Omega +c}(c+c_{1})(\eta _{00}^{3}\eta _{01})_{\xi }\nonumber \\{} & {} +(\frac{cc_{1}}{6}-\frac{2c_{1}^{2}}{9}-\frac{cc_{1}z^{2}}{2})(2\eta _{00}\eta _{10})_{\xi \xi \xi }-\frac{4}{9}c_{1}(c+c_{1})(\eta _{00}\eta _{10,\xi \xi \xi }+\eta _{10}\eta _{00,\xi \xi \xi })\nonumber \\{} & {} +\{-2(c+c_{1})(-\frac{4c_{1}}{9}+\frac{20cc_{1}}{9(\Omega +c)}+\frac{6c_{1}^{2}}{9(\Omega +c)})+\frac{6(2c_{1}+3c)(c+c_{1})}{\Omega +c}(\frac{c}{6}-\frac{2c_{1}}{9}-\frac{c}{2}z^{2})\nonumber \\{} & {} +4c_{1}(z^{2}(c+c_{1})-\frac{c}{3}-\frac{2c_{1}}{9}-\frac{5cc_{1}}{9(\Omega +c)}-\frac{c_{1}^{2}}{9(\Omega +c)})+8c_{1}(z^{2}(c+c_{1})+\frac{c_{1}}{9}-\frac{10cc_{1}}{9(\Omega +c)}\nonumber \\{} & {} -\frac{4c_{1}^{2}}{9(\Omega +c)})\}\eta _{00}\eta _{00,\xi }\eta _{00,\xi \xi }+\{-2(c+c_{1})(-\frac{2c_{1}}{9}+\frac{10cc_{1}}{9(\Omega +c)}+\frac{4c_{1}^{2}}{9(\Omega +c)})\nonumber \\{} & {} -\frac{2c_{1}}{9}\frac{(2c_{1}-3\Omega )(c+c_{1})}{\Omega +c}+\frac{(2c_{1}+3c)(c+c_{1})}{\Omega +c}(\frac{c}{6}-\frac{2c_{1}}{9}-\frac{c}{2}z^{2})+2c_{1}(z^{2}(c+c_{1})+\frac{c_{1}}{9}\nonumber \\{} & {} -\frac{10cc_{1}}{9(\Omega +c)}-\frac{4c_{1}^{2}}{9(\Omega +c)})\}\eta ^{2}_{00}\eta _{00,\xi \xi \xi }+\{\frac{2(2c_{1}+3c)(c+c_{1})}{\Omega +c}(\frac{c}{6}-\frac{2c_{1}}{9}-\frac{c}{2}z^{2})\nonumber \\{} & {} +4c_{1}(z^{2}(c+c_{1})-\frac{c}{3}-\frac{2c_{1}}{9}-\frac{5cc_{1}}{9(\Omega +c)}-\frac{c_{1}^{2}}{9(\Omega +c)})\}\eta ^{3}_{00,\xi }. \end{aligned}$$

(4.20)

For the order $O(\varepsilon ^{3}\mu ^{1})$ terms of the governing equations (2.2), we obtain from the Taylor expansion (2.3) that

$$\begin{aligned} {\left\{ \begin{array}{ll} -cu_{31,\xi }+u_{21,\tau }+u_{21}u_{00,\xi }+u_{11}u_{10,\xi }+u_{01}u_{20,\xi }+u_{20}u_{01,\xi }+u_{10}u_{11,\xi }\\ +u_{00}u_{21,\xi }+W_{10}u_{11,z}+W_{00}u_{21,z}+2\Omega W_{31}=-p_{31,\xi }, &{}{ in} \ 0<z<1,\\ -cW_{20,\xi }+W_{10,\tau }+u_{10}W_{00,\xi }+u_{00}W_{10,\xi }+W_{10}W_{00,z}+W_{00}W_{10,z}\\ -2\Omega u_{31}=-p_{31,z}, &{}{ in} \ 0<z<1,\\ u_{31,\xi }+W_{31,z}=0, &{}{ in} \ 0<z<1,\\ u_{31,z}-W_{20,\xi }=0, &{}{ in} \ 0<z<1,\\ p_{31}+p_{21,z}\eta _{00}+p_{11,z}\eta _{10}+p_{01,z}\eta _{20}+p_{20,z}\eta _{01}+p_{10,z}\eta _{11}+p_{00,z}\eta _{21}\\ +p_{11,zz}\frac{1}{2}\eta ^{2}_{00}=\eta _{31}, &{}{ on} \ z=1,\\ W_{31}+W_{21,z}\eta _{00}+W_{11,z}\eta _{10}+W_{01,z}\eta _{20}+W_{20,z}\eta _{01}+W_{10,z}\eta _{11}\\ +W_{00,z}\eta _{21}+W_{11,zz}\frac{1}{2}\eta ^{2}_{00} =-c\eta _{31,\xi }+\eta _{21,\tau }+u_{21}\eta _{00,\xi }+u_{11}\eta _{10,\xi }\\ +u_{01}\eta _{20,\xi }+u_{20}\eta _{01,\xi }+u_{10}\eta _{11,\xi }+u_{00}\eta _{21,\xi }+u_{11,z}\eta _{00}\eta _{00,\xi }, &{}{ on} \ z=1,\\ W_{31}=0, &{}{ on} \ z=0. \end{array}\right. } \end{aligned}$$

(4.21)

It follows from the fourth equation of (4.21) and fifth equation of (2.4) that

$$\begin{aligned} u_{31,z}=W_{20,\xi }=-zu_{20,\xi \xi }=-z(c\eta _{20,\xi \xi }-2(c+c_{1})(\eta _{00}\eta _{10})_{\xi \xi }-\frac{2c_{1}-3\Omega }{3(\Omega +c)}(c+c_{1})(\eta ^{3}_{00})_{\xi \xi }), \end{aligned}$$

which gives

$$\begin{aligned} u_{31}= & {} \frac{z^{2}}{2}(2(c+c_{1})(\eta _{00}\eta _{10})_{\xi \xi }+\frac{2c_{1}-3\Omega }{3(\Omega +c)}(c+c_{1})(\eta ^{3}_{00})_{\xi \xi }-c\eta _{20,\xi \xi })+\Phi _{31}(\tau ,\xi )\\= & {} \frac{z^{2}}{2}H_{3}+\Phi _{31}(\tau ,\xi ), \end{aligned}$$

for some smooth function $\Phi _{31}(\tau ,\xi )$ independent of z, where we denote

$$\begin{aligned} H_{3}=2(c+c_{1})(\eta _{00}\eta _{10})_{\xi \xi }+\frac{2c_{1}-3\Omega }{3(\Omega +c)}(c+c_{1})(\eta ^{3}_{00})_{\xi \xi }-c\eta _{20,\xi \xi }. \end{aligned}$$

Hence, we have

$$\begin{aligned} u_{31,\xi }=\frac{1}{2}z^{2}H_{3,\xi }+\Phi _{31,\xi }. \end{aligned}$$

On the other hand, thanks to the third equation in (4.21) and the boundary condition of $W_{31}$ on $z=0$, we get

$$\begin{aligned} W_{31}=W_{31}|_{z=0}+\int _{0}^{z}W_{31,z^{'}}dz^{'}=-\int _{0}^{z}u_{31,\xi }dz^{'}=-\frac{z^{3}}{6}H_{3,\xi }-z\Phi _{31,\xi }, \end{aligned}$$

which along with the boundary condition of $W_{31}$ on $z=1$ leads to

$$\begin{aligned} -\frac{1}{6}H_{3,\xi }-\Phi _{31,\xi }= & {} -c\eta _{31,\xi }+\eta _{21,\tau }+(u_{00}\eta _{21}+u_{10}\eta _{11}+u_{20}\eta _{01}+u_{01}\eta _{20}+u_{11}\eta _{10}+u_{21}\eta _{00})_{\xi }|_{z=1}\\{} & {} -\frac{1}{2}c\eta ^{2}_{00}\eta _{00,\xi \xi \xi }-c\eta _{00}\eta _{00,\xi }\eta _{00,\xi \xi }\\= & {} -c\eta _{31,\xi }+\eta _{21,\tau }+H_{4,\xi }|_{z=1}-\frac{1}{2}c\eta ^{2}_{00}\eta _{00,\xi \xi \xi }-c\eta _{00}\eta _{00,\xi }\eta _{00,\xi \xi }, \end{aligned}$$

where we denote

$$\begin{aligned} H_{4}=u_{00}\eta _{21}+u_{10}\eta _{11}+u_{20}\eta _{01}+u_{01}\eta _{20}+u_{11}\eta _{10}+u_{21}\eta _{00}. \end{aligned}$$

It then follows that

$$\begin{aligned} \Phi _{31,\xi }=c\eta _{31,\xi }-\eta _{21,\tau }-\frac{1}{6}H_{3,\xi }-H_{4,\xi }|_{z=1}+\frac{1}{2}c\eta ^{2}_{00}\eta _{00,\xi \xi \xi }+c\eta _{00}\eta _{00,\xi }\eta _{00,\xi \xi }, \end{aligned}$$

(4.22)

which implies

$$\begin{aligned} u_{31,\xi }=c\eta _{31,\xi }-\eta _{21,\tau }+(\frac{z^{2}}{2}-\frac{1}{6})H_{3,\xi }-H_{4,\xi }|_{z=1}+\frac{1}{2}c\eta ^{2}_{00}\eta _{00,\xi \xi \xi }+c\eta _{00}\eta _{00,\xi }\eta _{00,\xi \xi }, \end{aligned}$$

(4.23)

and

$$\begin{aligned} W_{31}=\frac{z(1-z^{2})}{6}H_{3,\xi }-cz\eta _{31,\xi }+z\eta _{21,\tau }+z(H_{4,\xi }|_{z=1})-\frac{1}{2}cz\eta ^{2}_{00}\eta _{00,\xi \xi \xi }-cz\eta _{00}\eta _{00,\xi }\eta _{00,\xi \xi }. \end{aligned}$$

(4.24)

Substituting the expression of $W_{10,\tau }$, $u_{10}$, $W_{00,\xi }$, $u_{00}$, $W_{10,\xi }$, $W_{10}$, $W_{00,z}$, $W_{00}$, $W_{10,z}$ and $W_{20,\xi }$ into the second equation in (4.21), we obtain

$$\begin{aligned} p_{31,z}= & {} 2\Omega u_{31}-c^{2}z\eta _{20,\xi \xi }+c(3c+4c_{1})z(\eta _{00}\eta _{10,\xi \xi }+\eta _{10}\eta _{00,\xi \xi })\nonumber \\{} & {} +2c(c+4c_{1})z\eta _{00,\xi }\eta _{10,\xi }+\{2z(c+c_{1})(c-4c_{1}+\frac{c(4c_{1}+3c-3\Omega )}{\Omega +c})\}\eta _{00}\eta ^{2}_{00,\xi }\nonumber \\{} & {} +\{z(c+c_{1})(-3c-4c_{1}+\frac{c(4c_{1}+3c-3\Omega )}{\Omega +c})\}\eta ^{2}_{00}\eta _{00,\xi \xi }. \end{aligned}$$

(4.25)

While from the boundary condition of $p_{31}$ on $z=1$, we have

$$\begin{aligned} p_{31}|_{z=1}= & {} \eta _{31}-2\Omega H_{4}|_{z=1}+c^{2}(\eta _{00}\eta _{10,\xi \xi }+\eta _{10}\eta _{00,\xi \xi })-c(c+4c_{1})\eta _{00}\eta ^{2}_{00,\xi }\\{} & {} +\{\frac{1}{2}(c+2\Omega c)-c(3c+4c_{1})\}\eta ^{2}_{00}\eta _{00,\xi \xi }, \end{aligned}$$

which along with (4.25) leads to

$$\begin{aligned}{} & {} p_{31}=p_{31}|_{z=1}+\int ^{z}_{1}p_{31,z^{'}}dz^{'} =\eta _{31}-2\Omega H_{4}|_{z=1}+2\Omega \int ^{z}_{1}u_{31}dz^{'}-\frac{c^{2}}{2}(z^{2}-1)\eta _{20,\xi \xi }\\{} & {} +(c^{2}+\frac{c(3c+4c_{1})}{2}(z^{2}-1))(\eta _{00}\eta _{10,\xi \xi }+\eta _{10}\eta _{00,\xi \xi })+c(c+4c_{1})(z^{2}-1)\eta _{00,\xi }\eta _{10,\xi }\\{} & {} +\{(z^{2}-1)(c+c_{1})(c-4c_{1}+\frac{c(4c_{1}+3c-3\Omega )}{\Omega +c})-c(c+4c_{1})\}\eta _{00}\eta ^{2}_{00,\xi }\\{} & {} +\{\frac{1}{2}(z^{2}-1)(c+c_{1})(-3c-4c_{1}+\frac{c(4c_{1}+3c-3\Omega )}{\Omega +c})+\frac{c^{2}+2\Omega c}{2}-c(3c+4c_{1})\}\eta ^{2}_{00}\eta _{00,\xi \xi }, \end{aligned}$$

and then

$$\begin{aligned} p_{31,\xi }= & {} \eta _{31,\xi }-2\Omega H_{4,\xi }|_{z=1}+2\Omega \int ^{z}_{1}u_{31,\xi }dz^{'}-\frac{c^{2}}{2}(z^{2}-1)\eta _{20,\xi \xi \xi }\nonumber \\{} & {} +(c^{2}+\frac{c(3c+4c_{1})}{2}(z^{2}-1))(\eta _{00}\eta _{10,\xi \xi }+\eta _{10}\eta _{00,\xi \xi })_{\xi }+c(c+4c_{1})(z^{2}-1)(\eta _{00,\xi }\eta _{10,\xi })_{\xi }\nonumber \\{} & {} +\{(z^{2}-1)(c+c_{1})(c-4c_{1}+\frac{c(4c_{1}+3c-3\Omega )}{\Omega +c})-c(c+4c_{1})\}(\eta _{00}\eta ^{2}_{00,\xi })_{\xi }\nonumber \\{} & {} +\{\frac{1}{2}(z^{2}-1)(c+c_{1})(-3c-4c_{1}+\frac{c(4c_{1}+3c-3\Omega )}{\Omega +c})\nonumber \\{} & {} +\frac{1}{2}(c^{2}+2\Omega c)-c(3c+4c_{1})\}(\eta ^{2}_{00}\eta _{00,\xi \xi })_{\xi }\nonumber \\= & {} \eta _{31,\xi }-2\Omega zH_{4,\xi }|_{z=1}+\frac{\Omega z}{3}(z^{2}-1)H_{3,\xi }+2\Omega (z-1)(c\eta _{31,\xi }-\eta _{21,\tau })\nonumber \\{} & {} +\Omega c(z-1)\eta ^{2}_{00}\eta _{00,\xi \xi \xi }+2\Omega c(z-1)\eta _{00}\eta _{00,\xi }\eta _{00,\xi \xi }-\frac{c^{2}}{2}(z^{2}-1)\eta _{20,\xi \xi \xi }\nonumber \\{} & {} +(c^{2}+\frac{c(3c+4c_{1})}{2}(z^{2}-1))(\eta _{00}\eta _{10,\xi \xi }+\eta _{10}\eta _{00,\xi \xi })_{\xi }+c(c+4c_{1})(z^{2}-1)(\eta _{00,\xi }\eta _{10,\xi })_{\xi }\nonumber \\{} & {} +\{(z^{2}-1)(c+c_{1})(c-4c_{1}+\frac{c(4c_{1}+3c-3\Omega )}{\Omega +c})-c(c+4c_{1})\}(\eta _{00}\eta ^{2}_{00,\xi })_{\xi } \nonumber \\{} & {} +\{\frac{1}{2}(z^{2}-1)(c+c_{1})(-3c-4c_{1}+\frac{c(4c_{1}+3c-3\Omega )}{\Omega +c})\nonumber \\{} & {} +\frac{1}{2}(c^{2}+2\Omega c)-c(3c+4c_{1})\}(\eta ^{2}_{00}\eta _{00,\xi \xi })_{\xi }. \end{aligned}$$

(4.26)

Thanks to the first equation in (4.21), we get

$$\begin{aligned} -p_{31,\xi }= & {} -cu_{31,\xi }+u_{21,\tau }+(u_{00}u_{21}+u_{10}u_{11}+u_{20}u_{01})_{\xi }+c^{2}z^{2}(\eta _{00,\xi }\eta _{10,\xi })_{\xi }\nonumber \\{} & {} -c(c+c_{1})z^{2}(2\eta _{00}\eta ^{2}_{00,\xi })_{\xi }+\frac{\Omega z}{3}(1-z^{2})H_{3,\xi }-2\Omega cz\eta _{31,\xi }+2\Omega z\eta _{21,\tau }\nonumber \\{} & {} +2\Omega z(H_{4,\xi }|_{z=1})-\Omega cz\eta ^{2}_{00}\eta _{00,\xi \xi \xi }-2\Omega cz\eta _{00}\eta _{00,\xi }\eta _{00,\xi \xi }. \end{aligned}$$

(4.27)

Combining (4.26) with (4.27), we get

$$\begin{aligned} 0= & {} (1-2\Omega c)\eta _{31,\xi }+2\Omega \eta _{21,\tau }-\Omega c\eta ^{2}_{00}\eta _{00,\xi \xi \xi }-2\Omega c\eta _{00}\eta _{00,\xi }\eta _{00,\xi \xi }-\frac{c^{2}}{2}(z^{2}-1)\eta _{20,\xi \xi \xi }\nonumber \\{} & {} +(c^{2}+\frac{c(3c+4c_{1})}{2}(z^{2}-1))(\eta _{00}\eta _{10,\xi \xi }+\eta _{10}\eta _{00,\xi \xi })_{\xi }+(c(c+4c_{1})(z^{2}-1)+c^{2}z^{2})(\eta _{00,\xi }\eta _{10,\xi })_{\xi }\nonumber \\{} & {} +\{(z^{2}-1)(c+c_{1})(c-4c_{1}+\frac{c(4c_{1}+3c-3\Omega )}{\Omega +c})-c(c+4c_{1})-2c(c+c_{1})z^{2}\}(\eta _{00}\eta ^{2}_{00,\xi })_{\xi }\nonumber \\{} & {} +\{\frac{1}{2}(z^{2}-1)(c+c_{1})(-3c-4c_{1}+\frac{c(4c_{1}+3c-3\Omega )}{\Omega +c})+\frac{1}{2}(c^{2}+2\Omega c)-c(3c+4c_{1})\}(\eta ^{2}_{00}\eta _{00,\xi \xi })_{\xi }\nonumber \\{} & {} -cu_{31,\xi }+u_{21,\tau }+(u_{00}u_{21}+u_{10}u_{11}+u_{20}u_{01})_{\xi }. \end{aligned}$$

(4.28)

Notice that

$$\begin{aligned} (u_{00}u_{21}+ & {} u_{10}u_{11}+u_{20}u_{01})_{\xi }\\= & {} c^{2}(\eta _{00}\eta _{21}+\eta _{10}\eta _{11}+\eta _{20}\eta _{01})_{\xi }-3c(c+c_{1})(\eta ^{2}_{00}\eta _{11}+2\eta _{00}\eta _{10}\eta _{01})_{\xi }\\{} & {} +(\frac{-4c(2c_{1}-3\Omega )}{3(\Omega +c)}(c+c_{1})+2(c+c_{1})^{2})(\eta ^{3}_{00}\eta _{01})_{\xi }+(\frac{c^{2}}{6}-\frac{2c c_{1}}{9}-\frac{c^{2}z^{2}}{2})(\eta _{00}\eta _{10,\xi \xi }\\{} & {} +\eta _{10}\eta _{00,\xi \xi })_{\xi }+c(z^{2}(c+c_{1})-\frac{c}{3}-\frac{2c_{1}}{9}-\frac{5cc_{1}}{9(\Omega +c)}-\frac{c_{1}^{2}}{9(\Omega +c)})(\eta _{00}\eta ^{2}_{00,\xi })_{\xi }\\{} & {} +\{c(z^{2}(c+c_{1})+\frac{c_{1}}{9}-\frac{10cc_{1}}{9(\Omega +c)}-\frac{4c_{1}^{2}}{9(\Omega +c)})-(c+c_{1})(\frac{c}{6}-\frac{2c_{1}}{9}-\frac{c}{2}z^{2})\}(\eta ^{2}_{00}\eta _{00,\xi \xi })_{\xi }, \end{aligned}$$

and

$$\begin{aligned} H_{4,\xi }|_{z=1}= & {} 2c(\eta _{00}\eta _{21}+\eta _{10}\eta _{11}+\eta _{20}\eta _{01})_{\xi }-3(c+c_{1})(\eta ^{2}_{00}\eta _{11}+2\eta _{00}\eta _{10}\eta _{01})_{\xi }\\{} & {} -\frac{4(2c_{1}-3\Omega )}{3(\Omega +c)}(c+c_{1})(\eta ^{3}_{00}\eta _{01})_{\xi }-(\frac{c}{3}+\frac{2c_{1}}{9})(\eta _{00}\eta _{10,\xi \xi }+\eta _{10}\eta _{00,\xi \xi })_{\xi }\\{} & {} +(\frac{2c}{3}+\frac{7c_{1}}{9}-\frac{5cc_{1}}{9(\Omega +c)}-\frac{c_{1}^{2}}{9(\Omega +c)})(\eta _{00}\eta ^{2}_{00,\xi })_{\xi }\\{} & {} +(c+\frac{10c_{1}}{9}-\frac{10c c_{1}}{9(\Omega +c)}-\frac{4c_{1}^{2}}{9(\Omega +c)})(\eta ^{2}_{00}\eta _{00,\xi \xi })_{\xi }. \end{aligned}$$

Substituting (4.20) and (4.23) into (4.28), we get

$$\begin{aligned}{} & {} 2(\Omega +c)\eta _{21,\tau }+3c^{2}(\eta _{00}\eta _{21}+\eta _{10}\eta _{11}+\eta _{20}\eta _{01})_{\xi }-2(3c+2c_{1})(c+c_{1})(\eta ^{2}_{00}\eta _{11}+ 2\eta _{00}\eta _{10}\eta _{01})_{\xi } \nonumber \\{} & {} \quad +\frac{c^{3}}{3}\eta _{20,\xi \xi \xi }-(\frac{c^{2}}{6}+\frac{10 cc_{1}}{9}+\frac{2c_{1}^{2}}{9})(2\eta _{00,\xi }\eta _{10,\xi })_{\xi }-(\frac{c^{2}}{3}+\frac{20 cc_{1}}{9}+\frac{8c_{1}^{2}}{9})(\eta _{00,\xi \xi }\eta _{10}+\eta _{10,\xi \xi }\eta _{00})_{\xi } \nonumber \\{} & {} \quad -\frac{64 cc_{1}+24c_{1}^{2}+45c^{2}-15}{3(\Omega +c)}(c+c_{1})(\eta ^{3}_{00}\eta _{01})_{\xi }-B_{8}\eta ^{3}_{00,\xi }-B_{9}\eta ^{2}_{00}\eta _{00,\xi \xi \xi }-B_{10}\eta _{00}\eta _{00,\xi }\eta _{00,\xi \xi }=0,\nonumber \\ \end{aligned}$$

(4.29)

with

$$\begin{aligned} B_{8}= & {} \frac{2c(4c_{1}+3c-3\Omega )(c+c_{1})}{3(\Omega +c)}+\frac{4c_{1}(2c_{1}+3c)(c+c_{1})}{9(\Omega +c)}+\frac{c_{1}(2c+4c_{1})(5c+c_{1})}{9(\Omega +c)}\\{} & {} +\frac{5}{3}c^{2}+\frac{16}{9}c c_{1}-\frac{28}{9}c_{1}^{2}\\= & {} -\frac{c^{2}(7c^{8}+36c^{6}+34c^{4}-21c^{2}+1)}{3(c^{2}+1)^{4}},\\ B_{9}= & {} \frac{(2c_{1}+3c)(4c_{1}+3c-3\Omega )(c+c_{1})}{9(\Omega +c)}+\frac{8c_{1}(5c+2c_{1})(c+c_{1})}{9(\Omega +c)}+\frac{2}{3}c^{2}-\frac{11}{9}c c_{1}-\frac{26}{9}c_{1}^{2}\\= & {} -\frac{c^{2}(4c^{8}+9c^{6}+37c^{4}-21c^{2}+1)}{3(c^{2}+1)^{4}},\\ B_{10}= & {} \frac{2c(4c_{1}+3c-3\Omega )(c+c_{1})}{9(\Omega +c)}+\frac{8c_{1}(12c+5c_{1})(c+c_{1})}{9(\Omega +c)}+\frac{8c_{1}(5c+2c_{1})(c+2c_{1})}{9(\Omega +c)}\\{} & {} +\frac{14}{3}c^{2}-\frac{2}{9}c c_{1}-\frac{40}{3}c_{1}^{2}\\= & {} -\frac{2c^{2}(4c^{8}+8c^{6}+23c^{4}-10c^{2}-2)}{(c^{2}+1)^{4}}. \end{aligned}$$