Propagation of different kinds of non-linear ion-acoustic waves in Earth’s magnetosphere

Abstract

The non-linear propagation of different kinds of ion-acoustic waves (IAWs) in multi-component plasma involving proton beam, positive ions and isothermal electrons has been studied. Using the derivative expansion method of basic equations, namely the hydrodynamic and Poisson equations, they are reduced to a single evolution equation of the non-linear Schrödinger (NLS)-type equation. By applying this model to plasma formed in Earth’s magnetosphere, different waves can be predicted that express the properties of the plasma. Using the separating variables method and the $G^{{\prime}}/G$-expansion method, we derived the exact analytical solutions to the evolution equation by using different solution regions in which non-linear waves are defined. A comparison has been made between the solutions describing the differential equation in each region in which the solution can appear using the data on the Earth’s magnetosphere in the study by Alotaibi et al. (2021).

Appendix

$$\begin{aligned} & s_{1}=\frac{-k^{2}}{3k^{2}-\omega ^{2}},\text{ }s_{2}= \frac{s_{1}\omega }{k},\text{ }s_{3}= \frac{k^{2}\mu _{b}}{\omega ^{2}-2ku_{b0}\omega +k^{2}\left ( u_{b0}^{2}-3\mu _{b}\sigma _{b}\right ) }, \\ & s_{4}= \frac{k\mu _{b}\left ( \omega -ku_{b0}\right ) }{\omega ^{2}-2ku_{b0}\omega +k^{2}\left ( u_{b0}^{2}-3\mu _{b}\sigma _{b}\right ) },\text{ }s_{5}= \frac{i\left ( k+3ks_{1}-ks_{2}V_{g}+\omega s_{2}-s_{1}V_{g}\omega \right ) }{3k^{2}-\omega ^{2}}, \\ & s_{6}= \frac{i\left ( 3ks_{2}-3ks_{1}V_{g}+\omega +3\omega s_{1}-s_{2}V_{g}\omega \right ) }{3k^{2}-\omega ^{2}}, \\ & s_{7}= \frac{-i\left ( k\left ( \mu _{b}-s_{4}V_{g}+s_{3}\left ( u_{b0}\left ( V_{g}-u_{b0}\right ) +3\mu _{b}\sigma _{b}\right ) \right ) +\left ( s_{4}+s_{3}\left ( u_{b0}-V_{g}\right ) \right ) \omega \right ) }{\omega ^{2}-2ku_{b0}\omega +k^{2}\left ( u_{b0}^{2}-3\mu _{b}\sigma _{b}\right ) }, \\ & s_{8}= \frac{i\left ( ku_{b0}(s_{4}\left ( u_{b0}-V_{g}\right ) +\mu _{b})-3k\left ( s_{4}-s_{3}V_{g}\right ) \mu _{b}\sigma _{b}-(s_{4}\left ( u_{b0}-V_{g}\right ) +\mu _{b}+3s_{3}\mu _{b}\sigma _{b}\right ) \omega )}{\omega ^{2}-2ku_{b0}\omega +k^{2}\left ( u_{b0}^{2}-3\mu _{b}\sigma _{b}\right ) }, \\ & s_{9}=\frac{s_{91}}{s_{90}},\text{ }s_{90}=s_{901}+s_{902},\text{ }s_{91}=s_{911}+s_{912}+s_{913}, \\ & s_{901}=2k^{4}\left ( u_{b0}^{2}\left ( 1+12k^{2}3\sigma _{e}\delta _{e} \right ) -3\mu _{b}\left ( \delta _{b}\sigma _{b}+12k^{2}\sigma _{b}+3 \sigma _{b}\sigma _{e}\delta _{e}\right ) \right ) -4k^{3}u_{b0} \omega (1+12k^{2}-3\sigma _{e}\delta _{e}), \\ & s_{902}=2k^{2}(-1-\delta _{b}\mu _{b}+(u_{b0}^{2}-3-3\mu _{b}\sigma _{b})(4k^{2}+ \sigma _{e}\delta _{e}))\omega ^{2}+2\omega ^{3}(4k^{2}+\sigma _{e} \delta _{e})(2ku_{b0}-\omega ), \\ & s_{911}=ik^{3}(a_{1}\delta _{b}-ik(2s_{3}s_{4}u_{b0}\delta _{b}+ \delta _{e}(u_{b0}^{2}-3\mu _{b}\sigma _{b})\sigma _{e}^{2})+a_{2}(- \delta _{b}\mu _{b}+(u_{b0}^{2}-3\mu _{b}\sigma _{b})(4k^{2}+\sigma _{e} \delta _{e}))), \\ & s_{912} =-2k^{2}(ia_{2}u_{b0}(4k^{2}+\sigma _{e}\delta _{e})+k(s_{3}s_{4} \delta _{b}-s_{1}s_{2}\delta _{b}\mu _{b}+4k^{2}s_{1}s_{2}(u_{b0}^{2}-3 \mu _{b}\sigma _{b})+ \\ &\hspace{1cm} s_{1}s_{2}\delta _{e}(u_{b0}^{2}-3\mu _{b}\sigma _{b})\sigma _{e}+u_{b0} \delta _{e}\sigma _{e}^{2}))\omega +\omega ^{3}(-2s_{1}ks_{2}(4k^{2}+ \sigma _{e}\delta _{e})), \\ & s_{913}=\omega ^{3}(k(ia_{2}(4k^{2}+\sigma _{e}\delta _{e})+k(16k^{2}s_{1}s_{2}u_{b0}+ \delta _{e}\sigma _{e}(4s_{1}s_{2}u_{b0}+\sigma _{e}))\omega ^{2}, \\ & s_{10}=\frac{s_{101}}{s_{90}},s_{101}=s_{102}+s_{103}+s_{104}+s_{105},s_{104}=\omega ^{3}(k\sigma _{e}^{2} \delta _{e}+ia_{2}\delta _{e}\sigma _{e}+4ia_{2}k^{2}), \\ & s_{102}=-2k^{4}s_{1}s_{2}(u_{b0}^{2}(1+12k^{2}+3\delta _{e}\sigma _{e})-3 \mu _{b}(\delta _{b}+\sigma _{b}+12k^{2}\sigma _{b}+3\delta _{e} \sigma _{e}\sigma _{e})), \\ &s_{103} =k^{2}(48k^{3}s_{1}s_{2}u_{b0}+4ia_{2}k^{2}(u_{b0}^{2}-3 \mu _{b}\sigma _{b}))\omega +k4s_{1}s_{2}u_{b0}(1+3\delta _{e}\sigma _{e}) \omega \\ &\hspace{1cm} k^{2}(i(a_{1}\delta _{b}-a_{2}\delta _{b}\mu _{b}+a_{2}\delta _{e}(u_{b0}^{2}-3 \mu _{b}\sigma _{b})\sigma _{e})\omega +k(\delta _{e}(u_{b0}^{2}-3 \mu _{b}\sigma _{b})\sigma _{e}^{2}+2s_{3}s_{4}\delta _{b}u_{b0}) \omega \\ & s_{105}=-2k\omega ^{2}(12k^{3}s_{1}s_{2}+4ia_{2}k^{2}u_{b0}+ia_{2}u_{b0} \delta _{e}\sigma _{e}+k(s_{3}s_{4}\delta _{b}+u_{b0}\delta _{e} \sigma _{e}^{2}+s_{1}(s_{2}+3s_{2}\delta _{e}\sigma _{e}))), \\ & s_{11}=\frac{s_{111}+s_{112}}{s_{90}}, \\ & s_{111}=-k(2s_{3}s_{4}(ku_{b0}-\omega )(12k^{4}-\delta _{e}\sigma _{e} \omega ^{2}+k^{2}(1+3\delta _{e}\sigma _{e}-4\omega ^{2})), \\ & s_{112}=ik(a_{1}k^{2}(1+12k^{2}+3\delta _{e}\sigma _{e})-a_{1}(4k^{2}+ \delta _{e}\sigma _{e})\omega ^{2}-ik\mu _{b}(-ia_{2}k+3k^{2}\delta _{e} \sigma _{e}^{2}+2ks_{1}s_{2}\omega -\delta _{e}\sigma _{e}^{2}\omega ^{2}))), \\ & s_{12}=\frac{s_{121}+s_{122}+s_{123}+s_{124}}{s_{90}},\text{ }s_{124}= \omega ^{3}(k\delta _{e}\mu _{b}\sigma _{e}^{2}+ia_{1}(4k^{2}+\delta _{e} \sigma _{e})), \\ & s_{121}=-i\omega k^{2}(a_{1}(1+12k^{2}+3\delta _{e}\sigma _{e})-\mu _{b}(a_{2}-2iks_{1}s_{2}u_{b0}+3ik \delta _{e}\sigma _{e}^{2})), \\ & s_{122}=-k\omega ^{2}(ia_{1}u_{b0}(4k^{2}+\delta _{e}\sigma _{e})+k \mu _{b}(2s_{1}s_{2}+u_{b0}\delta _{e}\sigma _{e}^{2}+2s_{3}s_{4}( \delta _{b}+3\sigma _{b}(4k^{2}+\delta _{e}\sigma _{e})))), \\ & s_{13}=\frac{s_{131}+s_{132}+s_{133}+s_{134}}{s_{90}}+ \frac{\delta _{e}\sigma _{e}^{2}\omega ^{4}}{s_{90}},s_{131}=-2k \omega ^{3}(s_{1}s_{2}+s_{3}s_{4}\delta _{b}+u_{b0}\delta _{e}\sigma _{e}^{2}), \\ & s_{132}=k\omega ^{2}(ia_{2}+ia_{1}\delta _{b}+2ku_{b0}(2s_{1}s_{2}+s_{3}s_{4} \delta _{b})+k\delta _{e}\sigma _{e}^{2}(u_{b0}^{2}-3-3\delta _{b} \mu _{b})), \\ & s_{133}=2k^{2}\omega (-ia_{2}u_{b0}+k(3s_{3}s_{4}\delta _{b}-s_{1}s_{2}u_{b0}^{2}+3s_{1}s_{2} \mu _{b}\sigma _{b}+3u_{b0}\delta _{e}\sigma _{e}^{2})), \\ & s_{134}=-k^{3}(3ia_{1}\delta _{b}+6ks_{3}s_{4}u_{b0}\delta _{b}-ia_{2}(u_{b0}-3 \mu _{b}\sigma _{b})+3k\delta _{e}(u_{b0}^{2}-3\mu _{b}\sigma _{b}) \sigma _{e}^{2}), \\ & s_{14}=\frac{s_{141}+s_{142}+s_{143}+s_{144}}{s_{90}}- \frac{4k^{2}\sigma _{e}^{2}\omega ^{4}}{s_{90}}, \\ & s_{141}=2k\sigma _{e}(-s_{1}s_{2}-s_{3}s_{4}\delta _{b}+4k^{2}u_{b0} \sigma _{e})\omega ^{3}, \\ & s_{142}=k\sigma _{e}\omega ^{2}(-i(a_{2}+a_{1}\delta _{b}-2iku_{b0}(2s_{1}s_{2}+s_{3}s_{4} \delta _{b}))+k(4k^{2}(-3+u_{b0}^{2}-3\mu _{b}\sigma _{b})-1-\delta _{b} \mu _{b})), \\ & s_{143}=-2k^{2}\sigma _{e}\omega (ia_{2}u_{b0}+k(-3s_{3}s_{4}\delta _{b}+s_{1}s_{2}(u_{b0}^{2}-3 \mu _{b}\sigma _{b})+u_{b0}\sigma _{e}+12k^{2}u_{b0}\sigma _{e})), \\ & s_{144} =k^{3}\sigma _{e}(-3ia_{1}\delta _{b}-6ks_{3}s_{4}u_{b0} \delta _{b}+ia_{2}(u_{b0}^{2}-3\mu _{b}\sigma _{b})+k(1+12k^{2})u_{b0} \\ &\hspace{1cm} -3\mu _{b}(\delta _{b}+\sigma _{b}+12k^{2}\sigma _{b}))\sigma _{e}), \\ & s_{15}=\frac{s_{151}+s_{152}}{s_{150}},\text{ }s_{16}= \frac{s_{161}+s_{162}+s_{163}}{s_{150}},\text{ }s_{17}= \frac{s_{171}+s_{172}}{s_{150}}, \\ & s_{150}=(u_{b0}-V_{g})^{2}-\delta _{e}(V_{g}^{2}-3)((u_{b0}-V_{g})^{2}-3 \mu _{b}\sigma _{b})\sigma _{e}+\mu _{b}((V_{g}^{2}-3)\delta _{b}-3 \sigma _{b}), \\ & s_{151}=\delta _{b}(-s_{4}(s_{4}+s_{1}(V_{g}-u_{b0}))+s_{3}^{2}(u_{b0}-V_{g}-3 \mu _{b}\sigma _{b}))+\delta _{e}((u_{b0}-V_{g})^{2}-3\mu _{b}\sigma _{b}) \sigma _{e}^{2}, \\ & s_{152}=(3s_{1}^{2}+s_{2}^{2}+2s_{1}s_{2}V_{g})(-\delta _{b}\mu _{b}+ \delta _{e}((u_{b0}-V_{g})^{2}-3\mu _{b}\sigma _{b})\sigma _{e}), \\ & s_{161}=3s_{1}^{2}V_{g}(\delta _{b}\mu _{b}-\delta _{e}((u_{b0}-V_{g})^{2}-3 \mu _{b}\sigma _{b})\sigma _{e}+V_{g}(-s_{4}^{2}\delta _{b}+s_{3}^{2} \delta _{b}(u_{b0}-V_{g}-3\mu _{b}\sigma _{b})), \\ & s_{162}=V_{g}\delta _{e}((u_{b0}-V_{g})^{2}-3\mu _{b}\sigma _{b}) \sigma _{e}^{2}+s_{2}^{2}(\delta _{b}\mu _{b}-\delta _{e}((u_{b0}-V_{g})^{2}-3 \mu _{b}\sigma _{b})\sigma _{e}), \\ & s_{163}=s_{1}(s_{4}(u_{b0}-V_{g})V_{g}\delta _{b}-2s_{2}((u_{b0}-V_{g})^{2}-3 \mu _{b}(\delta _{b}+\sigma _{b})-\delta _{e}((u_{b0}-V_{g})^{2}-3 \mu _{b}\sigma _{b})\sigma _{e})), \\ & s_{171}=-\mu _{b}(3s_{1}^{2}+s_{2}^{2}+2s_{1}s_{2}V_{g})+\mu _{b} \delta _{e}\sigma _{e}^{2}(V_{g}^{2}-3)+s_{4}^{2}(1-\delta _{e} \sigma _{e}(V_{g}^{2}-3)), \\ & s_{172}=s_{1}s_{4}(u_{b0}-V_{g})(-1+\delta _{e}\sigma _{e}(V_{g}^{2}-3))+s_{3}(u_{b0}-V_{g}-3 \mu _{b}\sigma _{b})(-1+\delta _{e}\sigma _{e}(V_{g}^{2}-3)). \\ & s_{18}=\frac{s_{181}+s_{182}+s_{183}}{s_{150}},\text{ }s_{19}= \frac{s_{191}+s_{192}+s_{193}}{s_{150}},\text{ }s_{20}= \frac{s_{201}+s_{202}+s_{203}}{s_{150}}, \\ & s_{181}=s_{4}(u_{b0}-V_{g})(-1+\delta _{e}\sigma _{e}(V_{g}^{2}-3))+ \mu _{b}((2s_{1}s_{2}V_{g}+3s_{1}^{2}+s_{2}^{2})(u_{b0}-V_{g})), \\ & s_{182}=\mu _{b}(s_{3}^{2}(3\sigma _{b}(1-u_{b0}+V_{g})-(V_{g}^{2}-3))+3s_{3}^{2}(-1+u_{b0}-V_{g}) \delta _{e}\sigma _{e}\sigma _{b}(V_{g}^{2}-3)), \\ & s_{183}=\mu _{b}((-u_{b0}+V_{g})(V_{g}^{2}-3)\delta _{e}\sigma _{e}^{2})+s_{1}s_{4} \mu _{b}(-\delta _{b}(V_{g}^{2}-3)+\sigma _{b}(3-3(V_{g}^{2}-3) \delta _{e}\sigma _{e})), \\ & s_{191}=s_{1}(u_{b0}-V_{g})(2s_{2}V_{g}(-u_{b0}+V_{g})+s_{4}(-V_{g}^{2}+3)\delta _{b})+6s_{1}s_{2}V_{g}\mu _{b}\sigma _{b}+3s_{1}^{2}(3\mu _{b} \sigma _{b}-(u_{b0}-V_{g})^{2}), \\ & s_{192}=s_{2}(-(3\mu _{b}\sigma _{b}-(u_{b0}-V_{g})^{2}+(V_{g}^{2}-3)(- \delta _{b}(s_{4}^{2}+s_{3}^{2}(3\mu _{b}\sigma _{b}-u_{b0}+V_{g})), \\ & s_{193}=\delta _{e}\sigma _{e}^{2}(V_{g}^{2}-3)((u_{b0}-V_{g})^{2}-3 \mu _{b}\sigma _{b}), \\ & s_{201}=-\sigma _{e}(s_{1}(u_{b0}-V_{g})(2s_{2}(u_{b0}-V_{g})V_{g}-s_{4} \delta _{b}(V_{g}^{2}-3))-6s_{1}s_{2}V_{g}\mu _{b}\sigma _{b}), \\ & s_{202}=-\sigma _{e}((3s_{1}^{2}+s_{2}^{2})(u_{b0}-V_{g})^{2}-3\mu _{b} \sigma _{b})+\delta _{b}(V_{g}^{2}-3)(s_{4}^{2}+s_{3}^{2}(-u_{b0}+V_{g}+3 \mu _{b}\sigma _{b}))), \\ & s_{203}=\sigma _{e}^{2}(u_{b0}^{2}-2u_{b0}V_{g}+V_{g}^{2}(1+\delta _{b} \mu _{b})-3\mu _{b}(\delta _{b}+\sigma _{b})). \\ & a_{1}= \frac{ik^{3}\mu _{b}^{2}(k^{2}(u_{b0}^{2}+3_{b}\mu _{b}\sigma _{b})-2ku_{b0}\omega +\omega ^{2})}{(k^{2}(u_{b0}^{2}-3_{b}\mu _{b}\sigma _{b})-2ku_{b0}\omega +\omega ^{2})^{2}}, \text{ }a_{2}= \frac{ik^{3}(3k^{2}+\omega ^{2})}{(-3k^{2}+\omega ^{2})^{2}}. \end{aligned}$$