Further physical study about solution structures for nonlinear q-deformed Sinh–Gordon equation along with bifurcation and chaotic behaviors

Abstract

The main novelty of this paper lies in five aspects: (1) To our best knowledge, the modified $\left( \frac{G'}{G^2}\right)$-expansion method was firstly applied in nonlinear q-deformed Sinh–Gordon equation (NQSGE). (2) The effects of wave obliqueness about NQSGE are firstly discussed in this paper which did not happen in previous papers. (3) Phase portraits and bifurcation behaviors about NQSGE are also firstly investigated in Hamiltonian system that did not appear in previous studies. (4) Sensitive analysis to initial value and chaotic behavior are also firstly studied in NQSGE. (5) The modified Riemann–Liouville, Beta, Conformable and M-truncated fractional derivatives are tested for accuracy in Fig. 18a, b. To our best knowledge, we seem firstly compare the relations and distinctions among different fractional-order derivatives in NQSGE model. The generalizations (1)–(5) indicated that the wave propagation of solitions about NQSGE model is mastered by the changed fraction, changed wave obliqueness angle and other physical factors.

Appendices

A Appendix of Equation (13)

By (18), we deduce

$$\begin{aligned} \begin{aligned} {u}'({\xi })&= \left( {{a_1}+2{a_2}\left( {{{G'}\over {{G^2}}}}\right) -{\beta _1} {\left( {{{{G'}\over {{G^2}}}}}\right) ^{-2}}-2{\beta _2}\left( {{{{G'}\over {{G^2}}}}}\right) ^{-3}}\right) \left( {\sigma +\mu \left( {{{G'}\over {{G^2}}}}\right) +R{\left( {{{G'}\over {{G^2}}}}\right) ^2}}\right) , \end{aligned} \end{aligned}$$

(50)

$$\begin{aligned} {u}'^2({\xi })= & {} 4{a_{2}}^{2}{R}^{2}\left( \frac{G'}{G^2}\right) ^6+\left( 4{R}^{2}a_{1}a_{2}+8R{a_{2}}^{2}\mu \right) \left( \frac{G'}{G^2}\right) ^5+\bigg [{R}^{2}{a_{1}}^{2}+8a_{1}a_{2}\mu \,R+4{a_{2}}^{2}\\{} & {} \left( 2R\sigma +{\mu }^{2}\right) \bigg ]\left( \frac{G'}{G^2}\right) ^4+\left( -4\beta _{1}{R}^{2}a_{2}+2{a_{1}}^{2}\mu \,R+4a_{1}a_{2}\left( 2R\sigma +{\mu }^{2}\right) +8{a_{2}}^{2}\sigma \,\mu \right) \\{} & {} \left( \frac{G'}{G^2}\right) ^3+\bigg [{a_{1}}^{2}\left( 2R\sigma +{\mu }^{2}\right) +8a_{1}a_{2}\sigma \mu -\left( 2a_{1}\beta _{1}+8a_{2}\beta _{2}\right) {R}^{2}-8a_{2}\beta _{1}\mu \,R+4{a_{2}}^{2} \\ {}{} & {} {\sigma }^{2}\bigg ]\left( \frac{G'}{G^2}\right) ^2+\bigg [2{a_{1}}^{2}\sigma \mu -4{R}^{2}\beta _{2}a_{1}-2\left( 2a_{1}\beta _{1}+8a_{2}\beta _{2}\right) \mu \,R-4a_{2}\beta _{1}\left( 2R\sigma +{\mu }^{2}\right) \\ {}{} & {} +4a_{1}a_{2}{\sigma }^{2}\bigg ]\left( \frac{G'}{G^2}\right) +{R}^{2}{\beta _{1}}^{2}-8a_{1}\beta _{2}\mu \,R-8a_{2}\beta _{1}\sigma \mu +{a_{1}}^{2}{\sigma }^{2}-\bigg (2a_{1}\beta _{1} \\ {}{} & {} +8a_{2}\beta _{2}\bigg )\left( 2R\sigma +{\mu }^{2}\right) +\bigg [4{R}^{2}\beta _{1}\beta _{2}+2{\beta _{1}}^{2}\mu \,R-4a_{1}\beta _{2}\left( 2R\sigma +{\mu }^{2}\right) -2\bigg (2a_{1}\beta _{1} \\ {}{} & {} +8a_{2}\beta _{2}\bigg )\sigma \mu -4a_{2}\beta _{1}{\sigma }^{2}\bigg ]\left( \frac{G'}{G^2}\right) ^{-1}+[4{R}^{2}{\beta _{2}}^{2}+8\beta _{2}\beta _{1}\mu \,R+{\beta _{1}}^{2}\left( 2R\sigma +{\mu }^{2}\right) \end{aligned}$$

$$\begin{aligned} \begin{aligned}&-8a_{1}\beta _{2}\sigma \mu -\left( 2a_{1}\beta _{1}+8a_{2}\beta _{2}\right) {\sigma }^{2}]\left( \frac{G'}{G^2}\right) ^{-2}+[8{\beta _{2}}^{2}\mu \,R+4\beta _{2}\beta _{1}\left( 2R\sigma +{\mu }^{2}\right) \\ {}&+2{\beta _{1}}^{2}\sigma \mu -4a_{1}\beta _{2}{\sigma }^{2}]\left( \frac{G'}{G^2}\right) ^{-3}+\left[ 4{\beta _{2}}^{2}\left( 2R\sigma +{\mu }^{2}\right) +8\beta _{2}\beta _{1}\sigma \mu +{\beta _{1}}^{2}{\sigma }^{2}\right] \\&\left( \frac{G'}{G^2}\right) ^{-4}+\left( 4\beta _{1}\beta _{2}{\sigma }^{2}+8{\beta _{2}}^{2}\mu \sigma \right) \left( \frac{G'}{G^2}\right) ^{-5}+4{\beta _{2}}^{2}{\sigma }^{2}\left( \frac{G'}{G^2}\right) ^{-6}, \end{aligned} \end{aligned}$$

(51)

$$\begin{aligned} \begin{aligned} {u}^2({\xi })&= \left( a_0^2+2a_1\beta _1+2a_2\beta _2\right) +\left( 2a_0a_1+2a_2\beta _1\right) \left( \frac{G'}{G^2}\right) +\left( a^2_1+2a_0a_2\right) \left( \frac{G'}{G^2}\right) ^2 \\ {}&\quad +2a_1a_2\left( \frac{G'}{G^2}\right) ^3+a_2^2\left( \frac{G'}{G^2}\right) ^4+\left( 2a_0\beta _1+2a_1\beta _2\right) \left( \frac{G'}{G^2}\right) ^{-1}-\left( \beta ^2_1+2a_0\beta _2\right) \left( \frac{G'}{G^2}\right) ^{-2} \\ {}&\quad +2\beta _1\beta _2\left( \frac{G'}{G^2}\right) ^{-3}+\beta _2^2\left( \frac{G'}{G^2}\right) ^{-4}, \end{aligned} \end{aligned}$$

(52)

$$\begin{aligned} \begin{aligned} {u}^3({\xi })&=a_2^3\left( \dfrac{G'}{G^2}\right) ^6+3a_1\left( \dfrac{G'}{G^2}\right) ^5a_2^2+\left( a_0a_2^2+2a_1^2a_2+a_2\left( 2a_0a_2+a_1^2\right) \right) \left( \dfrac{G'}{G^2}\right) ^4+\bigg (\beta _1a_2^2 \\ {}&\quad +2a_0a_1a_2+a_1\left( 2a_0a_2+a_1^2\right) +a_2\left( 2a_0a_1+2a_2\beta _1\right) \bigg )\left( \frac{G'}{G^2}\right) ^3+(\beta _{_2}a_{_2}^{2}+2\beta _{_1}a_{_1}a_{_2} \\ {}&\quad +a_{_0}\left( 2a_{_0}a_{_2}+a_{_1}^{2}\right) +a_1\left( 2a_oa_1+2a_2\beta _1\right) +a_2\left( a_o^2+2a_1\beta _1+2a_2\beta _2\right) \left( \frac{G'}{G^2}\right) ^2 \\ {}&\quad +(2\beta _{2}a_{1}a_{2}+\beta _{1}\left( 2a_{0}a_{2}+{a_{1}}^{2}\right) +a_{0}\left( 2a_{0}a_{1}+2a_{2}\beta _{1}\right) +a_{1}\bigg ({a_{0}}^{2}+2a_{1}\beta _{1} \\ {}&\quad +2a_{2}\beta _{2}\bigg )+a_{2}\left( 2a_{0}\beta _{1}+2a_{1}\beta _{2}\right) \left( \frac{G'}{G^2}\right) +\beta _{2}\left( 2a_{0}a_{2}+{a_{1}}^{2})+\beta _{1}(2a_{0}a_{1}+2a_{2}\beta _{1}\right) \\ {}&\quad +a_{0}\left( {a_{0}}^{2}+2a_{1}\beta _{1}+2a_{2}\beta _{2}\right) +a_{1}\left( 2a_{0}\beta _{1}+2a_{1}\beta _{2}\right) +a_{2}(2a_{0}\beta _{2}+{\beta _{1}}^{2})+(\beta _{2} \\ {}&\quad \left( 2a_{0}a_{1}+2a_{2}\beta _{1}\right) +\beta _{1}\left( {a_{0}}^{2}+2a_{1}\beta _{1}+2a_{2}\beta _{2}\right) +a_{0}(2a_{0}\beta _{1}+2a_{1}\beta _{2})+a_{1}(2a_{0}\beta _{2} \\ {}&\quad +{\beta _{1}}^{2})+2a_{2}\beta _{2}\beta _{1})\left( \frac{G'}{G^2}\right) ^{-1}+(\beta _{2}\left( {a_{0}}^{2}+2a_{1}\beta _{1}+2a_{2}\beta _{2}\right) +\beta _{1}\left( 2a_{0}\beta _{1}+2a_{1}\beta _{2}\right) \\ {}&\quad +a_{0}\left( 2a_{0}\,\beta _{2}+{\beta _{1}}^{2}\right) +2a_{1}\beta _{2}\beta _{1}+a_{2}{\beta _{2}}^{2})\left( \frac{G'}{G^2}\right) ^{-2}+(8{\beta _{2}}^{2}\mu \,R+4\beta _{2}\beta _{1} \\ {}&\quad \left( 2R\sigma +{\mu }^{2}\right) +2{\beta _{1}}^{2}\sigma \mu -4a_{1}\beta _{2}{\sigma }^{2}\left( \frac{G'}{G^2}\right) ^{-3})+(\beta _{2}(2a_{0}\beta _{2}+{\beta _{1}}^{2}) \\ {}&\quad +2{\beta _{1}}^{2}\beta _{2}+a_{0}{\beta _{2}}^{2}\left( \frac{G'}{G^2}\right) ^{-4}))+\left( 3\beta _{1}{\beta _{2}}^{2}\right) \left( \frac{G'}{G^2}\right) ^{-5})+\left( {\beta _{2}}^{3}\right) \left( \frac{G'}{G^2}\right) ^{-6}). \end{aligned} \end{aligned}$$

(53)

Next, we substitute Eqs. (18), (50), (51), (52) and (53) into Eq. (13), and rearrange the terms with the same power of $\left( \frac{G'}{G^2}\right)$ together. We then isolate the undetermined coefficients and set them to zero, resulting in the following equations:

$$\begin{aligned} \left\{ \begin{aligned}&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{6}:4({\left( k^2\sin ^2\theta \right) -\left( \rho ^2\cos ^2\theta \right) }){a_{2}}^{2}{R}^{2}+{a_{2}}^{3}=0,\quad \left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{-6}:4({\left( k^2\sin ^2\theta \right) -\left( \rho ^2\cos ^2\theta \right) }){\beta _{2}}^{2}{\sigma }^{2}+{\beta _{2}}^{3}=0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{5}:\left( {\left( k^2\sin ^2\theta \right) -\left( \rho ^2\cos ^2\theta \right) }^{2}\right) (4{R}^{2}a_{1}a_{2}+8{a_{2}}^{2}\mu \,R) +3a_{1}{a_{2}}^{2}=0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{-5}: \left( {\left( k^2\sin ^2\theta \right) -\left( \rho ^2\cos ^2\theta \right) }\right) \left( 4\beta _{2}\beta _{1}{\sigma }^{2}+8{\beta _{2}}^{2}\sigma \mu \right) +3\beta _{1}{\beta _{2}}^{2}=0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{4}: \left( {\left( k^2\sin ^2\theta \right) -\left( \rho ^2\cos ^2\theta \right) }\right) \left( {R}^{2}{a_{1}}^{2}+8a_{1}a_{2}\mu \,R+4{a_{2}}^{2}\left( 2\sigma \,R+{\mu }^{2}\right) \right) +2{a_{2}}^{2}K+a_{0}{a_{2}}^{2}+2{a_{1}}^{2}a_{2}+a_{2}\left( 2a_{0}a_{2}+{a_{1}}^{2}\right) =0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{-4}:\left( {\left( k^2\sin ^2\theta \right) -\left( \rho ^2\cos ^2\theta \right) }\right) \left( 4{\beta _{2}}^{2}\left( 2\sigma \,R+{\mu }^{2}\right) +8\beta _{2}\beta _{1}\sigma \mu +{\beta _{1}}^{2}{\sigma }^{2}\right) +2{\beta _{2}}^{2}K+\beta _{2}\left( 2a_{0}\beta _{2}+{\beta _{1}}^{2}\right) +2{\beta _{1}}^{2}\beta _{2}+a_{0}{\beta _{2}}^{2}=0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{3}:\left( {\left( k^2\sin ^2\theta \right) -\left( \rho ^2\cos ^2\theta \right) }\right) \left( -4\beta _{1}{R}^{2}a_{2}+2{a_{1}}^{2}\mu \,R+4a_{1}a_{2}(2\sigma \,R+{\mu }^{2}\right) \\ {}&+8{a_{2}}^{2}\sigma \mu )+4a_{1}a_{2}K+\beta _{1}{a_{2}}^{2}+2a_{0}a_{1}a_{2}+a_{1}(2a_{0}a_{2}+{a_{1}}^{2})+a_{2}(2a_{0}a_{1}+2a_{2}\beta _{1}) =0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{-3}:\left( {\left( k^2\sin ^2\theta \right) -\left( \rho ^2\cos ^2\theta \right) }\right) \left( 8{\beta _{2}}^{2}\mu \,R+4\beta _{2}\beta _{1}\bigg (2\sigma \,R+{\mu }^{2}\right) +2{\beta _{1}}^{2}\sigma \mu \\ {}&-4a_{1}\beta _{2}{\sigma }^{2}\bigg )+4\beta _{2}\beta _{1}K+\beta _{2}\left( 2a_{0}\beta _{1}+2a_{1}\beta _{2}\right) +\beta _{1}\left( 2a_{0}\beta _{2}+{\beta _{1}}^{2}\right) +2a_{0}\beta _{2}\beta _{1}+a_{1}{\beta _{2}}^{2} =0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{2}:\left( {\left( k^2\sin ^2\theta \right) -\left( \rho ^2\cos ^2\theta \right) }\right) (\left( -2\,a_{1}\beta _{1}-8a_{2}\beta _{2}\right) {R}^{2}-8a_{2}\beta _{1}\mu \,R+{a_{1}}^{2} \\ {}&\left( 2\sigma \,R+{\mu }^{2}\right) +8a_{1}a_{2}\sigma \mu +4{a_{2}}^{2}{\sigma }^{2})+2K\left( 2a_{0}a_{2}+{a_{1}}^{2}\right) +a_{2}q+\beta _{2}{a_{2}}^{2}+2\beta _{1}a_{1}a_{2} \\ {}&+a_{0}\left( 2a_{0}a_{2}+{a_{1}}^{2}\right) +a_{1}\left( 2a_{0}a_{1}+2a_{2}\beta _{1}\right) +a_{2}\left( {a_{0}}^{2}+2a_{1}\beta _{1}+2a_{2}\beta _{2}\right) =0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{-2}:\left( {\left( k^2\sin ^2\theta \right) -\left( \rho ^2\cos ^2\theta \right) }\right) (4{R}^{2}{\beta _{2}}^{2}+8\beta _{2}\beta _{1}\mu \,R+{\beta _{1}}^{2}\left( 2\sigma \,R+{\mu }^{2}\right) \\ {}&-8a_{1}\beta _{2}\sigma \mu -\left( 2a_{1}\beta _{1}+8a_{2}\beta _{2}\right) {\sigma }^{2})+2K\left( 2a_{0}\beta _{2}+{\beta _{1}}^{2}\right) +2a_{1}\beta _{2}\beta _{1}+a_{2}{\beta _{2}}^{2}+\beta _{2}q \\ {}&+\beta _{1}\left( 2a_{0}\beta _{1}+2a_{1}\beta _{2}\right) +a_{0}(2a_{0}\beta _{2}+{\beta _{1}}^{2})+\beta _{2}\left( {a_{0}}^{2}+2a_{1}\beta _{1}+2a_{2}\beta _{2}\right) =0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) :\left( {\left( k^2\sin ^2\theta \right) -\left( \rho ^2\cos ^2\theta \right) }\right) \left( -4{R}^{2}\beta _{2}a_{1}+2(-2a_{1}\beta _{1}-8a_{2}\beta _{2}\right) \mu \,R-4a_{2}\beta _{1} \\ {}&\left( 2\sigma \,R+{\mu }^{2})+2{a_{1}}^{2}\sigma \mu +4a_{1}a_{2}{\sigma }^{2}\right) +2K\left( 2a_{0}a_{1}+2a_{2}\beta _{1}\right) +a_{1}q+2\beta _{2}a_{1}a_{2} \\ {}&+\beta _{1}\left( 2a_{0}a_{2}+{a_{1}}^{2}\right) +a_{0}\left( 2a_{0}a_{1}+2a_{2}\beta _{1}\right) +a_{1}\left( {a_{0}}^{2}+2a_{1}\beta _{1}+2a_{2}\beta _{2}\right) +a_{2}(2a_{0}\beta _{1}+2a_{1}\beta _{2})=0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{-1}:\left( {\left( k^2\sin ^2\theta \right) -\left( \rho ^2\cos ^2\theta \right) }\right) (4{R}^{2}\beta _{1}\beta _{2}+2{\beta _{1}}^{2}\mu \,R-4a_{1}\beta _{2}\left( 2\sigma \,R+{\mu }^{2}\right) \\ {}&-2\left( 2a_{1}\beta _{1}+8a_{2}\beta _{2}\right) \sigma \mu -4a_{2}\beta _{1}{\sigma }^{2})+2K\left( 2a_{0}\beta _{1}+2a_{1}\beta _{2}\right) +q\beta _{1}+\beta _{2}\bigg (2a_{0}a_{1} \\ {}&+2a_{2}\beta _{1}\bigg )+\beta _{1}\left( {a_{0}}^{2}+2a_{1}\beta _{1}+2a_{2}\beta _{2}\right) +a_{0}\left( 2a_{0}\beta _{1}+2a_{1}\beta _{2}\right) +a_{1}\left( 2a_{0}\beta _{2}+{\beta _{1}}^{2}\right) +2a_{2}\beta _{2}\beta _{1}=0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{0}:\left( {\left( k^2\sin ^2\theta \right) -\left( \rho ^2\cos ^2\theta \right) }\right) ({R}^{2}{\beta _{1}}^{2}-8a_{1}\beta _{2}\mu \,R-\left( 2a_{1}\beta _{1}+8a_{2}\beta _{2}\right) \bigg (2\sigma \,R \\ {}&+{\mu }^{2}\bigg )-8a_{2}\beta _{1}\sigma \mu +{a_{1}}^{2}{\sigma }^{2})+2K\left( {a_{0}}^{2}+2a_{1}\beta _{1}+2a_{2}\beta _{2}\right) +qa_{0}+\beta _{2}(2a_{0}a_{2}+{a_{1}}^{2}) \\ {}&+\beta _{1}\left( 2a_{0}a_{1}+2a_{2}\beta _{1}\right) +a_{0}\left( {a_{0}}^{2}+2a_{1}\beta _{1}+2a_{2}\beta _{2}\right) +a_{1}\left( 2a_{0}\beta _{1}+2a_{1}\beta _{2}\right) +a_{2}\bigg (2a_{0}\beta _{2}+{\beta _{1}}^{2}\bigg )=0. \end{aligned} \right. \end{aligned}$$

(54)

B Appendix of Equation (17)

By (18), we deduce

$$\begin{aligned} {u}''({\xi })= & {} 6a_{2}\left( \frac{G'}{G^2}\right) ^{4}{R}^{2}+\left( 2a_{1}{R}^{2}+10a_{2}\mu \,R\right) \left( \frac{G'}{G^2}\right) ^{3}+\left( 3a_{1}\mu \,R+4a_{2}\left( 2R\sigma +{\mu }^{2}\right) \right) \nonumber \\ {}{} & {} \left( \frac{G'}{G^2}\right) ^{2}+\left( a_{1}\left( 2R\sigma +{\mu }^{2}\right) +6a_{2}\mu \sigma \right) \left( \frac{G'}{G^2}\right) +2\beta _{2}{R}^{2}+\beta _{1}\mu \,R+a_{1}\mu \sigma +2a_{2}{\sigma }^{2}\nonumber \\{} & {} +\left( 6\beta _{2}\mu \,R+\beta _{1}\left( 2R\sigma +{\mu }^{2}\right) \right) \left( {\frac{G'}{G^2}}\right) ^{-1}+\left( 4\beta _{2}\left( 2R\sigma +{\mu }^{2}\right) +3\beta _{1}\mu \sigma \right) \left( \frac{G'}{G^2}\right) ^{-2}+\left( 2\beta _{1}{\sigma }^{2}+10\beta _{2}\mu \sigma \right) \left( \frac{G'}{G^2}\right) ^{-3}+6\left( \beta _{2}{\sigma }^{2}\right) \left( \frac{G'}{G^2}\right) ^{-4}. \end{aligned}$$

(55)

Next, we substitute Eqs. (18), (51), (53) and (55) into Eq. (17), and rearrange the terms with the same power of $\left( \frac{G'}{G^2}\right)$ together. We then isolate the undetermined coefficients and set them to zero, resulting in the following equations:

$$\begin{aligned} \left\{ \begin{aligned}&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{6}:-4\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) {a_{2}}^{2}{R}^{2}+{a_{2}}^{3}=0, \left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{-6}:-4\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) {\beta _{2}}^{2}{\sigma }^{2}+{\beta _{2}}^{3}=0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{5}:2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) \left( 4{R}^{2}a_{1}a_{2}+8R{a_{2}}^{2}\mu \right) +3a_{1}{a_{2}}^{2}-2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) \left( 6{R}^{2}a_{1}a_{2}+a_{2}\left( 2{R}^{2}a_{1}+10Ra_{2}\mu \right) \right) =0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{-5}: 2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) \left( 4\beta _{2}\beta _{1}{\sigma }^{2}+8{\beta _{2}}^{2}\sigma \mu \right) +3\beta _{1}{\beta _{2}}^{2}-2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) \left( \beta _{2}(2\beta _{1}{\sigma }^{2}+10\beta _{2}\sigma \mu \right) +6\beta _{2}\beta _{1}{\sigma }^{2})=0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{4}: 2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) \left( {a_{1}}^{2}{R}^{2}+8a_{1}Ra_{2}\mu +4{a_{2}}^{2}\left( 2R\sigma +{\mu }^{2}\right) \right) \\ {}&+a_{0}{a_{2}}^{2}+2{a_{1}}^{2}a_{2}+a_{2}\left( 2a_{0}a_{2}+{a_{1}}^{2}\right) -2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) (6a_{2}a_{0}{R}^{2} +a_{1}\left( 2{R}^{2}a_{1}+10Ra_{2}\mu \right) +a_{2}\left( 3a_{1}\mu \,R+4a_{2}\left( 2R\sigma +{\mu }^{2}\right) \right) )=0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{-4}:2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) \left( 4{\beta _{2}}^{2}\left( 2R\sigma +{\mu }^{2}\right) +8\beta _{2}\beta _{1}\sigma \mu +{\beta _{1}}^{2}{\sigma }^{2}\right) +\beta _{2}\left( 2\beta _{2}a_{0}+{\beta _{1}}^{2}\right) +2{\beta _{1}}^{2}\beta _{2}+a_{0}{\beta _{2}}^{2} \\ {}&-2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) \bigg (\beta _{2}\bigg (4\beta _{2}\bigg (2R\sigma +{\mu }^{2}\bigg )+3\beta _{1}\mu \sigma \bigg )+\beta _{1}\left( 2\beta _{1}{\sigma }^{2}+10\beta _{2}\sigma \mu \right) +6\beta _{2}a_{0}{\sigma }^{2}\bigg )=0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{3}:2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) \left( -4\beta _{1}{R}^{2}a_{2}+2R{a_{1}}^{2}\mu +4a_{2}a_{1}(2R\sigma +{\mu }^{2}\right) \\ {}&+8{a_{2}}^{2}\sigma \mu )+\beta _{1}{a_{2}}^{2}+2a_{0}a_{2}a_{1}+a_{1}(2a_{0}a_{2}+{a_{1}}^{2})+a_{2}(2a_{0}a_{1}+2a_{2}\beta _{1}) -2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) (6\beta _{1}{R}^{2}a_{2}+a_{0}\left( 2{R}^{2}a_{1}+10Ra_{2}\mu \right) \\ {}&+a_{1}\left( 3a_{1}\mu \,R+4a_{2}\left( 2R\sigma +{\mu }^{2}\right) \right) +a_{2}\left( a_{1}\left( 2R\sigma +{\mu }^{2})+6a_{2}\mu \sigma \right) \right) =0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{-3}:2(\left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) )(8R{\beta _{2}}^{2}\mu +4\beta _{2}\beta _{1}(2R\sigma +{\mu }^{2})+2{\beta _{1}}^{2}\sigma \mu \\ {}&-4\beta _{2}{\sigma }^{2}a_{1})+\beta _{2}\left( 2\beta _{1}a_{0}+2a_{1}\beta _{2}\right) +\beta _{1}\left( 2\beta _{2}a_{0}+{\beta _{1}}^{2}\right) +2a_{0}\beta _{2}\beta _{1}+a_{1}{\beta _{2}}^{2} -2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) \left( \beta _{2}\bigg (6\beta _{2}\mu \,R+\beta _{1}\left( 2R\sigma +{\mu }^{2}\right) \right) \\ {}&+\beta _{1}\left( 4\beta _{2}\left( 2R\sigma +{\mu }^{2}\right) +3\beta _{1}\mu \sigma \right) +a_{0}\left( 2\beta _{1}{\sigma }^{2}+10\beta _{2}\sigma \mu \right) +6\beta _{2}{\sigma }^{2}a_{1}\bigg )=0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{2}:2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) (\left( -2a_{1}\beta _{1}-8a_{2}\beta _{2}\right) {R}^{2}-8R\beta _{1}a_{2}\mu +{a_{1}}^{2}\left( 2R\sigma +{\mu }^{2}\right) +8a_{2}a_{1}\sigma \mu +4{\sigma }^{2}{a_{2}}^{2})+{q}^{2}a_{2}+\beta _{2}{a_{2}}^{2}+2\beta _{1}a_{1}a_{2} \\ {}&+a_{0}\left( 2a_{0}a_{2}+{a_{1}}^{2}\right) +a_{1}\left( 2a_{0}a_{1}+2a_{2}\beta _{1}\right) +a_{2}\left( {a_{0}}^{2}+2a_{1}\beta _{1}+2a_{2}\beta _{2}\right) -2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) \left( 6a_{2}\beta _{2}{R}^{2}+\beta _{1}(2{R}^{2}a_{1}+10Ra_{2}\mu \right) \\ {}&+a_{0}\bigg (3a_{1}\mu \,R+4a_{2}\left( 2R\sigma +{\mu }^{2}\right) )+a_{1}(a_{1}\left( 2R\sigma +{\mu }^{2}\right) +6a_{2}\mu \sigma \bigg )+a_{2}\bigg (2{R}^{2}\beta _{2}+\beta _{1}\mu \,R+a_{1}\mu \sigma +2a_{2}{\sigma }^{2}\bigg ))=0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{-2}:2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) (4{R}^{2}{\beta _{2}}^{2}+8\beta _{1}\beta _{2}\mu \,R+{\beta _{1}}^{2}\left( 2R\sigma +{\mu }^{2}\right) -8a_{1}\beta _{2}\sigma \mu -\left( 2a_{1}\beta _{1}+8a_{2}\beta _{2}\right) {\sigma }^{2})+{q}^{2}\beta _{2} \\ {}&+\beta _{2}\left( {a_{0}}^{2}+2a_{1}\beta _{1}+2a_{2}\beta _{2}\right) +\beta _{1}\left( 2\beta _{1}a_{0}+2a_{1}\beta _{2}\right) +a_{0}\left( 2\beta _{2}a_{0}+{\beta _{1}}^{2}\right) +2a_{1}\beta _{2}\beta _{1}+a_{2}{\beta _{2}}^{2}-2\bigg (\left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \bigg ) \\ {}&\left( \beta _{2}(2{R}^{2}\beta _{2}+\beta _{1}\mu \,R+a_{1}\mu \sigma +2a_{2}{\sigma }^{2}\right) +\beta _{1}(6\beta _{2}\mu \,R+\beta _{1}\left( 2R\sigma +{\mu }^{2}\right) )+a_{0}(4\beta _{2}(2R\sigma +{\mu }^{2})3\beta _{1}\mu \sigma )+a_{1}(2\beta _{1}{\sigma }^{2}+10\beta _{2}\sigma \mu ) +6a_{2}\beta _{2}{\sigma }^{2})=0,\\&\left( \frac{G'(\xi )}{G^2(\xi )}\!\right) \!:2\left( \left( \rho ^2\cos ^2\theta \right) \!-\!\left( k^2\sin ^2\theta \right) \right) \bigg (-4{R}^{2}\beta _{2}a_{1}-2\left( 2a_{1}\beta _{1}\!+\!8a_{2}\beta _{2}\right) \mu \,R\!-\!4a_{2}\beta _{1}\left( 2R\sigma \!+\!{\mu }^{2}\right) \!+\!2{a_{1}}^{2}\mu \sigma \!+\!4{\sigma }^{2}a_{1}a_{2}\bigg )\!+\!{q}^{2}a_{1}\!+\! 2\beta _{2}a_{1}a_{2} \\ {}&+\beta _{1}\left( 2a_{0}a_{2}\!+\!{a_{1}}^{2}\right) +a_{0}(2a_{0}a_{1}+2a_{2}\beta _{1})+a_{1}({a_{0}}^{2}+2a_{1}\beta _{1}+2a_{2}\beta _{2})+a_{2}(2\beta _{1}a_{0}+2a_{1}\beta _{2}) -2(\left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) )(\beta _{2}\left( 2{R}^{2}a_{1}+10Ra_{2}\mu \right) \\ {}&+\beta _{1}\bigg (a_{1}\mu \,R+4a_{2}\bigg (2R\sigma +{\mu }^{2}\bigg )\bigg )+a_{0}\left( a_{1}\left( 2R\sigma +{\mu }^{2}\right) +6a_{2}\mu \sigma \right) +a_{1}\left( 2{R}^{2}\beta _{2}+\beta _{1}\mu \,R+a_{1}\mu \sigma +2a_{2}{\sigma }^{2}\right) +a_{2}\left( 6\beta _{2}\mu \,R+\beta _{1}\left( 2R\sigma +{\mu }^{2}\right) \right) )=0,\\ \end{aligned} \right. \end{aligned}$$

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$$\begin{aligned} \left\{ \begin{aligned}&\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{-1}:2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) (4\beta _{1}{R}^{2}\beta _{2}+2{\beta _{1}}^{2}\mu \,R-4a_{1}\beta _{2}\left( 2R\sigma +{\mu }^{2}\right) -2(2a_{1}\beta _{1}+8a_{2}\beta _{2})\mu \sigma -4a_{2}\beta _{1}{\sigma }^{2})+{q}^{2}\beta _{1}+\beta _{2}(2a_{0}a_{1} \\ {}&+2a_{2}\beta _{1})+\beta _{1}({a_{0}}^{2}+2a_{1}\beta _{1} +2a_{2}\beta _{2})+a_{0}(2\beta _{1}a_{0}+2a_{1}\beta _{2})+a_{1}(2\beta _{2}a_{0}+{\beta _{1}}^{2})+2a_{2}\beta _{2}\beta _{1}-2(\left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) )(\beta _{2}(a_{1}\left( 2R\sigma +{\mu }^{2}\right) \\ {}&+6a_{2}\mu \sigma )+\beta _{1}(2{R}^{2}\beta _{2}+\beta _{1}\mu \,R+a_{1}\mu \sigma +2a_{2}{\sigma }^{2}) +a_{0}(6\beta _{2}\mu \,R+\beta _{1}\left( 2R\sigma +{\mu }^{2}\right) )+a_{1}(4\beta _{2}(2R\sigma +{\mu }^{2})+3\beta _{1}\mu \sigma )\\&+a_{2}(2\beta _{1}{\sigma }^{2} +10\beta _{2}\sigma \mu ))=0,\left( \frac{G'(\xi )}{G^2(\xi )}\right) ^{0}:2\left( \left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \right) \\&\times \bigg ({R}^{2}{\beta _{1}}^{2}-8\beta _{2}\mu \,a_{1}R-\left( 2a_{1}\beta _{1}+8a_{2}\beta _{2}\right) \left( 2R\sigma +{\mu }^{2}\right) -8a_{2}\sigma \beta _{1}\mu +{a_{1}}^{2}{\sigma }^{2}\bigg )+{q}^{2}a_{0}+\beta _{2}(2a_{0}a_{2}+{a_{1}}^{2})+\beta _{1}(2a_{0}a_{1}+2a_{2}\beta _{1}) \\ {}&+a_{0}\left( {a_{0}}^{2}+2a_{1}\beta _{1}+2a_{2}\beta _{2}\right) +a_{1}\left( 2\beta _{1}a_{0}+2a_{1}\beta _{2}\right) +a_{2}\left( 2\beta _{2}a_{0}+{\beta _{1}}^{2}\right) \bigg (\left( \rho ^2\cos ^2\theta \right) -\left( k^2\sin ^2\theta \right) \bigg )(\beta _{2}\left( 3a_{1}\mu \,R+4a_{2}\left( 2R\sigma +{\mu }^{2}\right) \right) \\ {}&+\beta _{1}\left( a_{1}\left( 2R\sigma +{\mu }^{2}\right) +6a_{2}\mu \sigma \right) +a_{0}\left( 2{R}^{2}\beta _{2}+\beta _{1}\mu \,R+a_{1}\mu \sigma +2a_{2}{\sigma }^{2}\right) +a_{1}\left( 6\beta _{2}\mu \,R+\beta _{1}\left( 2R\sigma +{\mu }^{2}\right) \right) +a_{2}\left( 4\beta _{2}\left( 2R\sigma +{\mu }^{2}\right) +3\beta _{1}\mu \sigma \right) )=0. \end{aligned} \right. \end{aligned}$$

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