Properties of rogue, soliton, and shock-like dust-ion-acoustic waves in dusty plasma at Titan’s ionosphere

Abstract

The properties and stability of nonlinear dust-ion-acoustic waves are examined in dusty plasma environment such as Titan ionosphere. The plasma is composed of warm collisional negatively charged dust grains, positive ions contain source terms, and Maxwellian electrons. Based on the one-dimensional version of the standard reductive perturbation technique, the nonlinear waves are described by a complex Ginzburg-Landau (CGL) equation. Using a factorization method, the CGL equation is solved to obtain different nonlinear solutions in the form of a localized electrostatic potential pulse such as rogue, bright, and shock-like solutions. At certain plasma parameters, the rogue wave turns into a bright soliton or other nonlinear wave forms. The influence of various plasma configuration parameters, such as wave number k, relative temperatures $\sigma _{e}$, coefficients of plasma ionization frequency $H_{2}$, relative dust density $\delta _{d}$, and collision frequency $\nu$ on electrostatic nonlinear wave characteristics (amplitude and width) has been examined. The stability of the envelope soliton solution from the CGL equation has been investigated. Our results will be useful in understanding the dynamics of nonlinear excitations in Titan ionosphere where the dust grains have been well established in recent years by Cassini mission.

Appendix

$$\begin{aligned} s_{1}= & {} \frac{k^{2}\alpha \mu _{d}}{k^{2}\left( u_{d0}^{2}-3\mu _{d}\sigma _{d}\right) -2ku_{d0}\omega +\omega ^{2}},\,s_{2}= & \frac{s_{1}\left( \omega -ku_{d0}\right) }{k}, \\ s_{3}= & {} \frac{k^{2}}{\omega ^{2}-3k^{2}},\,s_{4}=\frac{s_{3}\omega }{k},\,b_{1}=-2ku_{d0},\,b_{3}=2ku_{d0}\left( 1+3k^{2}\right) , \\ b_{2}= & {} u_{d0}^{2}k^{2}-3k^{2}-3k^{2}\mu _{d}\sigma _{d}-1-\frac{\alpha ^{2} \delta _{d}\mu _{d}}{k^{2}},\,s_{009}\\= & {} \delta _{e}\sigma _{e}^{2}\omega ^{2}\left( 3-u_{d0}^{2}+3\mu _{d}\sigma _{d}\right) , \\ b_{4}= & {} \frac{\delta _{e}\sigma _{e}}{k^{4}}+3\alpha ^{2}\delta _{d}\mu _{d} -k^{2}\left( u_{d0}^{2}-3\mu _{d}\sigma _{d}\right) -3k^{4}\left( u_{d0} ^{2}-3\mu _{d}\sigma _{d}\right) , \\ s_{5}= & {} \frac{-2is_{1}}{k}+\frac{2is_{1}^{2}}{k^{2}\alpha \mu _{d}}\left( k\left( u_{d0}^{2}-u_{d0}V_{g}-3\mu _{d}\sigma _{d}\right) +\left( V_{g}-u_{d0}\right) \omega \right) , \\ s_{6}= & {} \left( \frac{s_{1}}{k^{2}}+\frac{6is_{1}^{2}\sigma _{d}}{\alpha k^{2} }\right) \left( kV_{g}-\omega \right) ,\,s_{7}=\frac{2i\omega k\left( kV_{g}-\omega \right) }{\left( 3k^{2}-\omega ^{2}\right) ^{2}}, \\ s_{8}= & {} \frac{i\left( -3k^{2}+\omega ^{2}\right) \left( kV_{g}-\omega \right) }{\left( 3k^{2}-\omega ^{2}\right) ^{2}},\,s_{9}\\= & {} \frac{s_{91}+\omega s_{92}+\omega ^{2}s_{93}+\omega ^{3}s_{94}+\delta _{e}\sigma _{e}^{2}\omega ^{4} }{s_{90}}, \\ s_{91}= & {} k^{4}\left( \left( -3s_{3}^{2}-s_{4}^{2}-3\delta _{e}\sigma _{e} ^{2}\right) \left( u_{d0}^{2}-3\mu _{d}\sigma _{d}\right) \right) \\{} & {} +3k^{4}\alpha \delta _{d}\left( s_{2}^{2}-2s_{1}s_{2}u_{d0}+3s_{1}^{2}\mu _{d}\sigma _{d}\right) , \\ s_{92}= & {} 2k^{3}\left( 3s_{3}^{2}u_{d0}+s_{4}u_{d0}+3s_{1}s_{2}\alpha \delta _{d}-s_{3}s_{4}\left( u_{d0}^{2}-3\mu _{d}\sigma _{d}\right) \right. \\{} & {} \left. +3u_{d0} \delta _{e}\sigma _{e}^{2}\right) , \end{aligned}$$

$$\begin{aligned} s_{93}= & {} k^{2}(3s_{3}^{2}+s_{4}-4s_{3}s_{4}u_{d0}+\alpha \delta _{d}(s_{2} ^{2}-2s_{1}s_{2}u_{d0}+3s_{1}^{2}\mu _{d}\sigma _{d})+s_{009}), \\ s_{94}= & {} -2k\left( s_{3}s_{4}+s_{1}s_{2}\alpha \delta _{d}+u_{d0}\delta _{e} \sigma _{e}^{2}\right) .\,s_{902}\\= & {} 2\left( -2k^{3}u_{d0}-24k^{5} u_{d0}-6k^{3}u_{d0}\delta _{e}\sigma _{e}\right) ,\, \\ s_{90}= & {} s_{901}+s_{902}\omega +s_{903}\omega ^{2}+s_{094}\omega ^{3}+2\left( -4k^{2}-\delta _{e}\sigma _{e}\right) \omega ^{4}, \\ s_{901}= & {} 2\left( k^{4}u_{d0}^{2}-3k^{4}\delta _{d}\mu _{d}-3k^{4}\sigma _{d} \mu _{d}+12k^{6}\left( u_{d0}^{2}-3\mu _{d}\sigma _{d}\right) \right. \\{} & {} \left. +3k^{4}\sigma _{e}\delta _{e}\left( u_{d0}^{2}-3\sigma _{d}\mu _{d}\right) \right) , \\ s_{903}= & {} 2\left( k^{2}+12k^{4}-4k^{4}u_{d0}^{2}+k^{2}\delta _{d}\mu _{d}\right. \\{} & {} \left. +12k^{4}\mu _{d}\sigma _{d}+k^{2}\delta _{e}\sigma _{e}\left( 3-u_{d0}^{2} +3\mu _{d}\sigma _{d}\right) \right) , \\ s_{904}= & {} 2\left( 8k^{3}u_{d0}+2ku_{d0}\delta _{e}\sigma _{e}\right) ,\,s_{10}\\= & {} \frac{s_{101}+s_{102}\omega +s_{103}\omega ^{2}+s_{104}\omega ^{3} -4k^{2}\sigma _{e}^{2}\omega ^{4}}{s_{90}}, \\ s_{101}=~ & {} k^{4}\sigma _{e}(\left( -3s_{3}^{2}-s_{4}^{2}\right) \left( u_{d0}^{2}-3\mu _{d}\sigma _{d}\right) \\{} & {} +3\alpha \delta _{d}(s_{2}^{2}-2s_{1} s_{2}u_{d0}+3s_{1}^{2}\mu _{d}\sigma _{d})+s_{1010}), \\ s_{1010}= ~& {} \sigma _{e}u_{d0}(1+12k^{2})-3\mu _{d}\sigma _{e}(\delta _{d}+\sigma _{d}+12k^{2}\sigma _{d}) \\ s_{102}= ~& {} 2k^{3}\sigma _{e}(3s_{3}^{2}u_{d0}+s_{4}^{2}u_{d0}+3s_{1}s_{2} \alpha \delta _{d}-s_{3}s_{4}\left( u_{d0}^{2}-3\mu _{d}\sigma _{d}\right) \\{} & {} -(1+12k^{2})u_{d0}\sigma _{e}), \end{aligned}$$

$$\begin{aligned} s_{103}= & {} k^{2}\sigma _{e}(-3s_{3}^{2}-s_{4}+4s_{3}s_{4}u_{d0}+\sigma _{e}\\{} & {} -4k^{2}\sigma _{e}\left( -3+u_{d0}^{2}-3\mu _{d}\sigma _{d}\right) +s_{1030}) \\ s_{104}= & {} k\sigma _{e}\left( -2\left( s_{3}s_{4}+s_{1}s_{2}\alpha \delta _{d}\right) +8k^{2}u_{d0}\sigma _{e}\right) ,\, \\ s_{11}= & {} \frac{s_{111}+s_{112}\omega +s_{113}\omega ^{2}+s_{114}\omega ^{3}}{s_{90}},s_{1030}\\= & {} \alpha \delta _{d}\left( -s_{2}^{2}+2s_{1}s_{2}u_{d0} -3s_{1}^{2}\mu _{d}\sigma _{d}+\alpha \mu _{d}\sigma _{e}\right) . \\ s_{111}= & {} -k^{4}(s_{2}^{2}u_{d0}\left( 1+12k^{2}+3\delta _{e}\sigma _{e}\right) \\{} & {} -6s_{1}s_{2}\mu _{d}(\delta _{d}+\sigma _{d}+12k^{2}\sigma _{d}+3\delta _{e} \sigma _{d}\sigma _{e})+s_{1110}), \\ s_{1110}= & {} u_{d0}\mu _{d}3s_{1}^{2}\sigma _{d}\left( 1+12k^{2}+3\delta _{e} \sigma _{e}\right) +(3s_{3}^{2}+s_{4}+3\delta _{e}\sigma _{e}^{2})), \\ s_{112}= & {} k^{3}(s_{2}^{2}\left( 1+12k^{2}+3\delta _{e}\sigma _{e}\right) +\mu _{d}(3s_{1}^{2}\sigma _{d}\left( 1+12k^{2}+3\delta _{e}\sigma _{e}\right) \\{} & {} +(3s_{3}^{2}+s_{4}^{2}-2s_{3}s_{4}u_{d0}+3\delta _{e}\sigma _{e}^{2}))), \\ s_{113}= & {} -k^{2}(2\alpha (s_{3}s_{4}+s_{1}s_{2}\alpha \delta _{d})\mu _{d}\\{} & {} -4k^{2}(s_{2}^{2}u_{d0}-6s_{1}s_{2}\mu _{d}\sigma _{d}+3s_{1}^{2}u_{d0}\mu _{d}\sigma _{d})\\{} & {} -s_{1130}, \\{} & {} s_{1130}\delta _{e}\left( s_{2}^{2}u_{d0}-6s_{1}s_{2}\mu _{d}\sigma _{d} +3s_{1}^{2}u_{d0}\mu _{d}\sigma _{d}\right) \sigma _{e}\\{} & {} +u_{d0}\alpha \delta _{e}\mu _{d}\sigma _{e}^{2}), \\ s_{114}= & {} -k(4k^{2}\left( s_{2}^{2}+3s_{1}^{2}\mu _{d}\sigma _{d}\right) \\{} & {} +\delta _{e}\sigma _{e}\left( s_{2}^{2}+3s_{1}^{2}\mu _{d}\sigma _{d}-\alpha \mu _{d}\sigma _{e}))\right) . \end{aligned}$$

$$\begin{aligned} s_{12}= & {} \frac{s_{121}+s_{122}\omega +s_{123}\omega ^{2}+s_{124}\omega ^{3}}{s_{90}+s_{126}\omega +s_{127}\omega ^{2}+s_{128}\omega ^{3}+s_{129}\omega ^{3} },s_{1210}\\= & {} u_{d0}+3s_{1}^{2}\mu _{d}\sigma _{d})\sigma _{e}+3\alpha \delta _{e} \mu _{d}\sigma _{e}^{2}), \\ s_{121}= & {} -k^{4}(\alpha \mu _{d}(3s_{3}^{2}+s_{4}^{2})-(1+12k^{2})(s_{2} ^{2}-2s_{1}s_{2}u_{d0}+3s_{1}^{2}\mu _{d}\sigma _{d})\\{} & {} -3\delta _{e}(s_{2} ^{2}-2s_{1}s_{2}+s_{1210}, \\ s_{122}= & {} 2k^{3}\left( -s_{3}s_{4}\alpha \mu _{d}+s_{1}s_{2}\left( 1+12k^{2}+3\delta _{e}\sigma _{e}\right) \right) , \\ s_{123}= & {} -k^{2}(4k^{2}\left( s_{2}^{2}-2s_{1}s_{2}u_{d0}+3s_{1}^{2}\mu _{d}\sigma _{d}\right) \\{} & {} +\delta _{e}\sigma _{e}(s_{2}^{2}-2s_{1}s_{2} u_{d0}+3s_{1}^{2}\mu _{d}\sigma _{d}-\alpha \mu _{d}\sigma _{e}))), \\ s_{124}= & {} -2ks_{1}s_{2}\left( 4k^{2}+\delta _{e}\sigma _{e}\right) ,s_{126}=2-2k^{3}u_{d0}(1+12k^{2}+3\delta _{e}\sigma _{e}), \\ s_{125}= & {} 2k^{4}(u_{d0}^{2}\left( 1+12k^{2}+3\delta _{e}\sigma _{e}\right) \\{} & {} -3\mu _{d}\left( \delta _{d}+\sigma _{d}+12k^{2}\sigma _{d}+3\delta _{e}\sigma _{e}\sigma _{d}\right) ), \\ s_{127}= & {} 2k^{2}(1+\delta _{d}\mu _{d}-4k^{2}\left( u_{d0}^{2}-3-3\mu _{d} \sigma _{d}\right) \\{} & {} +\delta _{e}\sigma _{e}\left( 3-u_{d0}^{2}+3\mu _{d} \sigma _{d}\right) ), \\ s_{128}= & {} -2u_{d0}s_{129},\,s_{129}=-2\left( 4k^{2}+\delta _{e}\sigma _{e}\right) .\,s_{13}\\= & {} \frac{s_{131}+s_{132}\omega +s_{133}\omega ^{2}+s_{134}\omega ^{3}}{s_{90}}, \\ s_{131}= & {} -2k^{4}s_{3}s_{4}(u_{d0}^{2}\left( 1+12k^{2}+3\delta _{e}\sigma _{e}\right) \\{} & {} -3\mu _{d}\left( \delta _{d}+\sigma _{d}+12k^{2}\sigma _{d} +3\delta _{e}\sigma _{d}\sigma _{e})\right) , \end{aligned}$$

$$\begin{aligned} s_{132}= & {} -k^{3}(4k^{2}s_{4}^{2}(u_{d0}^{2}-3\mu _{d}\sigma _{d})+\alpha \delta _{d}(s_{2}^{2}\\{} & {} -2s_{1}s_{2}u_{d0}-s_{4}^{2}\alpha \mu _{d}+3s_{1}^{2}\mu _{d}\sigma _{d})+s_{1320+}s_{13201}, \\ s_{1320}= & {} \left( s_{4}^{2}-\sigma _{e}\right) \delta _{e}\left( u_{d0} ^{2}-3\mu _{d}\sigma _{d}\right) \sigma _{e}\\{} & {} -4s_{3}s_{4}u_{d0}\left( 1+12k^{2}+3\delta _{e}\sigma _{e}\right) + \\ s_{13201}= & {} 3s_{3}^{2}(-\delta _{d}\mu _{d}+\left( u_{d0}^{2}-3\mu _{d}\sigma _{d}\right) \left( 4k^{2}+\delta _{e}\sigma _{e}\right) )), \\ s_{133}= & {} -2k^{2}(-4k^{2}s_{4}^{2}u_{d0}+s_{1}s_{2}\alpha \delta _{d}+u_{d0} \delta _{e}\sigma _{e}\left( \sigma _{e}-s_{4}^{2}\right) \\ {}{} & {} -3s_{3}^{2} u_{d0}\left( 4k^{2}+\delta _{e}\sigma _{e}\right) +s_{1330}, \\ s_{134}= & {} -k\left( 4k^{2}\left( 3s_{3}^{2}+s_{4}\right) +\delta _{e}\left( 3s_{3}^{2}+s_{4}^{2}-\sigma _{e}\right) \sigma _{e}\right) ,\\{} & {} s_{1330}s_{3}s_{4}\left( 1+12k^{2}+3\delta _{e}\sigma _{e}\right) ),. \\ s_{14}= & {} \frac{s_{141}+s_{142}\omega +s_{143}\omega ^{2}+s_{144}\omega ^{3}}{s_{90}}, \\ s_{141}= & {} -k^{4}(4k^{2}\left( 3s_{3}+s_{4}^{2}\right) \left( u_{d0}^{2} -3\mu _{d}\sigma _{d}\right) \\{} & {} +\alpha \delta _{d}(s_{2}^{2}-2s_{1}s_{2} u_{do0}-\left( 3s_{3}+s_{4}^{2}\right) \alpha \mu _{d}+s_{1410}, \\ s_{1410}= & {} 3s_{1}^{2}\mu _{d}\sigma _{d})+\left( 3s_{3}+s_{4}^{2}\right) \delta _{e}\left( u_{d0}^{2}-3\mu _{d}\sigma _{d}\right) \sigma _{e}\\{} & {} -\delta _{e}\left( u_{d0}^{2}-3\mu _{d}\sigma _{d}\right) \sigma _{e}^{2}) \\ s_{142}= & {} -2k^{3}(\alpha \delta _{d}\left( s_{1}s_{2}-s_{3}s_{4}\alpha \mu _{d0}\right) \\{} & {} -4k^{2}(u_{d0}(3s_{3}^{2}+s_{4}^{2}-s_{3}s_{4}\mu _{d0} )+s_{1420}, \end{aligned}$$

$$\begin{aligned} s_{1420}= & {} 3s_{3}s_{4}\mu _{d}\sigma _{d})-\delta _{e}(u_{d0}(3s_{3}^{2}+s_{4} ^{2}-s_{3}s_{4}\mu _{d0})\\{} & {} +3s_{3}s_{4}\mu _{d}\sigma _{d})\sigma _{e}+u_{d0} \delta _{e}\sigma _{e}^{2}), \\ s_{143}= & {} -k(4k^{3}(3s_{3}^{2}-4s_{4}s_{3}u_{d0})+k\delta _{e}(3s_{3}^{2} +s_{4}-4s_{4}s_{3}u_{d0}-\sigma _{e}), \\ s_{134}= & {} -k\left( 8k^{2}s_{3}s_{4}+2s_{3}s_{4}\delta _{e}\sigma _{e}\right) .\,s_{15}=\frac{-s_{151}}{s_{152}}, \\ s_{151}= & {} \left( 3s_{3}^{2}+s_{4}^{2}+2s_{3}s_{4}V_{g}\right) \left( \left( u_{d0}-V_{g}\right) ^{2}-3\mu _{d}\sigma _{d}\right) \\{} & {} +\left( V_{g}-3\right) (\alpha \delta _{d}(s_{2}(s_{2}-s_{1510}, \\ s_{1510}= & {} 2s_{1}u_{d0}+2s_{1}V_{g})+3s_{1}^{2}\mu _{d}\sigma _{d})\\{} & {} -\delta _{e}\left( \left( u_{d0}-V_{g}\right) ^{2}-3\mu _{d}\sigma _{d}\right) \sigma _{e}^{2}), \\ s_{152}= & {} u_{d0}^{2}-2u_{d0}V_{g}+V_{g}^{2}-3\delta _{d}\mu _{d}+V_{g}^{2} \delta _{d}\mu _{d}\\{} & {} -3\mu _{d}\sigma _{d}-\left( V_{g}^{2}-3\right) s_{1520}, \\ s_{1520}= & {} \left( \left( u_{d0}-V_{g}\right) ^{2}-3\mu _{d}\sigma _{d}\right) \delta _{e}\sigma _{e}.\,s_{20}=\frac{-s_{21}}{s_{22}}, \\ s_{16}= & {} \frac{-s_{161}}{s_{152}},\,s_{17}=\frac{-s_{171}}{s_{152}},\,s_{18}=\frac{s_{181}}{s_{152}},\,s_{19}=\frac{s_{191}}{s_{152}}, \\ s_{161}= & {} \sigma _{e}(\left( 3s_{3}^{2}+s_{4}^{2}+2s_{3}s_{4}V_{g}\right) (\left( \left( u_{d0}-V_{g}\right) ^{2}-3\mu _{d}\sigma _{d}\right) \\{} & {} -\left( V_{g}^{2}-3\right) \alpha \delta _{d}(s_{2}^{2}+s_{1610}, \end{aligned}$$

$$\begin{aligned} s_{1610}= & {} 2s_{1}s_{2}-\left( \left( -u_{d0}+V_{g}\right) +3s_{1}^{2}\mu _{d}\sigma _{d}\right) -u_{d0}^{2}\\{} & {} -2u_{d0}V_{g}+V_{g}^{2}\left( 1+\delta _{d}\mu _{d}\right) -3\left( \delta _{d}+\sigma _{d}\right) \sigma _{e})). \\ s_{171}= & {} 3s_{3}^{2}V_{g}(-\delta _{d}\mu _{d}+\delta _{e}\left( \left( u_{d0}-V_{g}\right) ^{2}-3\mu _{d}\sigma _{d}\right) \sigma _{e})\\{} & {} +s_{1710} +s_{1711}+s_{1712}, \\ s_{1710}= & {} 2s_{3}s_{4}(\left( \left( u_{d0}-V_{g}\right) ^{2}-3\left( \sigma _{d}+\sigma _{d}\right) \right) \\{} & {} +3\delta _{e}\sigma _{e}\left( \left( u_{d0}-V_{g}\right) ^{2}-3\mu _{d}\sigma _{d}\right) ), \\ s_{1711}= & {} V_{g}(\alpha \delta _{d}(s_{2}^{2}+2s_{1}s_{2}\left( -u_{d0} +V_{g}\right) \\{} & {} -s_{4}^{2}\alpha \mu _{d}+3s_{1}^{2}\mu _{d}\sigma _{d})+s_{4} ^{2}\delta _{e}, \\{} & {} s_{1712}\left( \left( u_{d0}-V_{g}\right) ^{2}-3\left( \mu _{d}\sigma _{d}\right) \right) \sigma _{e}\\{} & {} \quad -\delta _{e}\left( \left( u_{d0} -V_{g}\right) ^{2}-3\left( \mu _{d}\sigma _{d}\right) \right) \sigma _{e} ^{2}. \\ s_{181}=\, & \alpha \delta _{d}(-s_{2}^{2}+2s_{1}s_{2}(u_{d0}-V_{g})+\alpha \mu _{d}(3s_{3}^{2}+s_{4}^{2}V_{g})\\{} & {} -3s_{1}^{2}\mu _{d}\sigma _{d})-s_{1810}, \\ s_{1810} = \,& {} (3s_{3}^{2}+s_{4}^{2}+2s_{3}s_{4}V_{g})\delta _{e}\left( \left( u_{d0}-V_{g}\right) ^{2}-3\mu _{d}\sigma _{d}\right) \sigma _{e}\\{} & {} +\left( \left( u_{d0}-V_{g}\right) ^{2}-3\left( \mu _{d}\sigma _{d}\right) \right) \sigma _{e}^{2}\delta _{e}. \\ s_{191}=\, & s_{2}^{2}\left( u_{d0}-V_{g}\right) (-1+(V_{g}^{2}-3)\delta _{e}\sigma _{e})\\{} & {} +2s_{1}s_{2}\mu _{d}((-V_{g}^{2}+3)\delta _{d}+3s_{1910} \\ 3s_{1910}= \,& \sigma _{d}-3(V_{g}^{2}-3)\delta _{e}\sigma _{e}\sigma _{d})+\left( u_{d0}-V_{g}\right) \mu _{d}(3s_{3}^{2}\alpha \\{} & {} +2s_{3}s_{4}V_{g}\alpha +3s_{1911}, \end{aligned}$$

$$\begin{aligned} s_{1911}= & {} 3s_{1}^{2}\sigma _{d}(-1+(V_{g}^{2}-3)\delta _{e}\sigma _{e} )+\alpha (V_{g}^{2}-3)\delta _{e}\sigma _{e}^{2})). \\ s_{21}= & {} 2s_{1}s_{2}\left( u_{d0}-V_{g}\right) (1-(V_{g}^{2}-3)\delta _{e}\sigma _{e})\\{} & {} +(s_{2}^{2}+3s_{1}^{2}\mu _{d}\sigma _{d})(-1+(V_{g}^{2} -3)\delta _{e}\sigma _{e})+s_{210}, \\ s_{210}= & {} \alpha \mu _{d}(3s_{3}^{2}+s_{4}^{2}+2s_{3}s_{4}V_{g})-(V_{g} ^{2}-3)\delta _{e}\sigma _{e}^{2}). \\ s_{22}= & {} (V_{g}^{2}-3)\delta _{d}\mu _{d}+\left( u_{d0}-V_{g}\right) ^{2}(1-(V_{g}^{2}-3)\delta _{e}\sigma _{e})\\{} & {} +3\mu _{d}\sigma _{d}(-1+(V_{g} ^{2}-3)\delta _{e}\sigma _{e}). \\ P_{0}= & {} P_{01}^{-1}P_{02},\,P_{1}=P_{01}^{-1}P_{11},\,P_{2} =P_{01}^{-1}P_{21}, \\ P_{01}^{-1}= & {} \frac{s_{1}}{k^{2}\mu _{d}}(-\delta _{d}\left( k\left( s_{2} -s_{1}u_{d0}\right) +s_{1}\omega \right) +\frac{-s_{3}}{k^{2}}\left( ks_{4}+s_{3}\omega \right) ). \\ P_{02}= & {} 1+\frac{is_{3}}{k^{2}}\left( 3ks_{7}-ks_{8}V_{g}+s_{8}\omega -s_{7}V_{g}\omega \right) +\frac{is_{1}\delta _{d}}{k^{2}\mu _{d}}+P_{021}, \\ P_{021}= & {} (k(s_{6}V_{g}+s_{5})(u_{d0}^{2}-u_{d0}V_{g}-3\mu _{d}\sigma _{d}))\\{} & {} -(s_{6}+s_{5}\left( u_{d0}-V_{g}\right) )\omega ). \\ P_{11}= & {} \frac{-\delta _{e}\sigma _{e}^{2}}{2}\left( 2\left( s_{15} +s_{9}\right) +\sigma _{e}\right) +P_{12}+P_{13}, \end{aligned}$$

$$\begin{aligned} P_{12}= & {} \frac{-s_{1}}{k}(3ks_{3}\left( s_{14}+s_{18}\right) +ks_{4}\left( s_{13}+s_{17}\right) +P_{120}, \\ P_{120}= & {} \omega \left( \left( s_{13}+s_{17}\right) s_{3}+s_{4}\left( s_{14}+s_{18}\right) \right) , \\ P_{13}= & {} \frac{s_{3\delta _{d}}}{k}(k(\left( s_{11}+s_{19}\right) (s_{2} -s_{1}u_{d0})\\{} & {} -(s_{12}+s_{20})(s_{2}u_{d0}-3s_{1}\mu _{d}\sigma _{d})+P_{130}, \\ P_{130}= & {} \omega \left( s_{1}\left( s_{11}+s_{19}\right) +s_{2}\left( s_{12}+s_{20}\right) \right) . \\ P_{21}= & {} \frac{-s_{1}s_{2}\alpha _{2}\delta _{d}}{k}+\frac{s_{3}s_{4}}{k}\left( -H_{0}-H_{1}+H_{2}+k^{2}\beta +\nu \right) +P_{210}, \\ P_{210}= & {} \frac{\omega s_{3}}{k^{2}}\left( s_{3}\left( H_{0}+H_{1}\right) -H_{2}\sigma _{e}\right) . \end{aligned}$$