Dust-acoustic Rossby waves in magnetized plasma

Abstract

The proliferation of dust-acoustic nonlinear electrostatic waves in dusty plasma bound to nonthermal electrons and trapped nonthermal ions have been studied. The fluid hydrodynamic equations for dust fluid combined with appropriate electrons and ions distributions have been used to derive an evolution equation called a nonlinear Schrödinger equation. The rogue wave solution of the latter is used to describe a bright envelope soliton that can transform to Rossby structure at certain magnetic field strength and directional cosine. This work provides a theoretical aspects of Rossby waves, bright envelope soliton, and rogue waves on different spatial in the context of Earth magnetosphere.

Appendix

$$\begin{aligned} h_{1}&= \frac{3k^{2}\left( -R_{1}+1\right) }{5k^{2}\sigma -3\omega ^{2}},h_{2}=\frac{3k\omega \left( -R_{1}+1\right) }{5k^{2}\sigma -3\omega ^{2} },h_{3}=\frac{1}{16}(16b-189\Gamma +105b\Gamma -16),\\ h_{4}&= 1-\beta _{e},s_{1}=16\delta \left( \beta _{e}-1\right) -16k^{2}+(16(b-1)+21(5b-9)\Gamma )\mu ,\\ h_{5}&= \frac{i}{5k^{2}\sigma -3\omega ^{2}}(k(3R_{1}-3-3h_{2}V_{g}+5h_{1} \sigma )+3\omega (h_{2}-h_{1}V_{g}),\\ h_{6}&= \frac{i}{5k^{2}\sigma -3\omega ^{2}}(3\omega (R_{1}-1)+5h_{1}\sigma (-kV_{g}+\omega )+h_{2}(5k\sigma -3V_{g}\omega ),\\ s_{01}&= 8+16b+135\Gamma ,s_{2}=\frac{s_{01}}{8},s_{3} =1-b+\frac{189\Gamma }{16}-\frac{105b\Gamma }{16},h_{7}=\frac{-h_{71} }{h_{70}}, \end{aligned}$$

$$\begin{aligned} h_{71}&=k(9k(8\delta -8R_{1}\delta +4h_{3}R_{1}(4k^{2}+\delta -\beta _{e}\delta )+4h_{3}^{2}R_{1}\mu +s_{01}\mu (R_{1}-1)+8h_{2}^{2}(4k^{2}\\ &\quad +\delta -\beta _{e}\delta +h_{3}\mu ))-40h_{1}^{2}k(4k^{2}+\delta -\beta _{e}\delta +h_{3}\mu )\sigma +144h_{1}h_{2}(4k^{2}+\delta -\beta _{e}\delta +h_{3}\mu )\omega ), \end{aligned}$$

$$\begin{aligned} h_{70}=48(k^{2}(3-3R_{1}+5(4k^{2}+\delta -\beta _{e}\delta +h_{3}\mu )\sigma )-3(4k^{2}+\delta -\beta _{e}\delta +h_{3}\mu )\omega ^{2}). \end{aligned}$$

$$\begin{aligned} h_{8}&=\frac{h_{81}}{h_{70}},h_{81}=k(-48h_{1}h_{2}k(3-3R_{1} +5(4k^{2}+\delta -\beta _{e}\delta +h_{3}\mu )\sigma )-9(8\delta -8R_{1}\delta \\&\quad+4h_{3}R_{1}(4k^{2}+\delta -\beta _{e}\delta )+4h_{3}^{2}R_{1}\mu +s_{01} \mu (R_{1}-1)+8h_{2}^{2}(4k^{2}+\delta -\beta _{e}\delta +h_{3}\mu ))\omega \\&\quad+40h_{1}^{2}(4k^{2}+\delta -\beta _{e}\delta +h_{3}\mu )\sigma \omega ),\\ h_{9}&=\frac{-h_{91}}{h_{70}},h_{91}=k^{2}(36h_{3}(2h_{2} ^{2}+h_{3}R_{1})+9s_{01}(R_{1}-1)-5(8h_{1}^{2}h_{3}+12k^{2}s_{01}+3(8h_{3}\\&\quad+s_{01}-s_{01}\beta _{e})\delta )\sigma )+144h_{1}h_{2}h_{3}k\omega +9(4k^{2}s_{01}+(8h_{3}+s_{01}-s_{01}\beta _{e})\delta )\omega ^{2},\\ h_{10}&=\frac{h_{101}}{h_{70}},h_{101}=k^{2}(72-36R_{1} (2+h_{3}(\beta _{e}-1))-72h_{2}^{2}(\beta _{e}-1)+5(96k^{2}+8h_{1}^{2}(\beta _{e}-1)+3\\& \quad \times 3(8h_{3}+s_{01}-s_{01}\beta _{e})\mu )\sigma )-144h_{1}h_{2}k(\beta _{e}-1)\omega -9(32k^{2}+(8h_{3}+s_{01}-s_{01}\beta _{e})\mu )\omega ^{2},\\ h_{11}&=\frac{h_{111}}{h_{70}},h_{111}=k^{2}(72h_{2}^{2} +36h_{3}R_{1}-5(8h_{1}^{2}+24\delta -3\mu s_{01})\sigma +144h_{1}h_{2}k\omega \\&\quad+9(8\delta -s_{01}\mu )\omega ^{2}), \end{aligned}$$

$$\begin{aligned} h_{12}&=\frac{h_{121}}{h_{120}},h_{120}=6(3R_{1}-3+(\delta (\beta _{e}-1)-s_{3}\mu )(3V_{g}^{2}-5\sigma )), \\ h_{121}&=18(R_{1}-1)(\delta -s_{2}\mu )+(\delta (\beta _{e}-1)-s_{3}\mu )(9h_{3}R_{1}+18h_{2}(h_{2}+2h_{1}V_{g})-10h_{1}^{2}\sigma ), \end{aligned}$$

$$\begin{aligned} h_{13}&=\frac{-h_{131}}{h_{120}},h_{131}=18V_{g}\delta +9(4h_{1}h_{2}(R_{1}-1)+V_{g}(R_{1}(h_{3}-2-h_{3}\beta _{e})\delta +(2(R_{1}-1)s_{2}\\&\quad+h_{3}R_{1}s_{3})\mu +2h_{2}^{2}(\delta -\beta _{e}\delta +s_{3}\mu )))+10h_{1}(h_{1}V_{g}-6h_{2})((\beta _{e}-1)\delta -s_{3}\mu )\sigma , \end{aligned}$$

$$\begin{aligned} h_{14}&=\frac{h_{141}}{h_{120}},h_{141}=9s_{3}(h_{3}R_{1} +2h_{2}(h_{2}+2h_{1}V_{g})-2V_{g}^{2}\delta )+6s_{2}(3R_{1}-3+(\beta _{e}-1)\delta \\&\qquad(3V_{g}^{2}-5\sigma ))-10s_{3}(h_{1}^{2}-3\delta )\sigma , \\ h_{15}&=\frac{h_{151}}{h_{120}},h_{151}=6(3R_{1}-3-(s_{2} +s_{3}-s_{2}\beta _{e})\mu (3V_{g}^{2}-5\sigma ))+(\beta _{e}-1)(9h_{3}R_{1}\\&\quad+18h_{2}(h_{2}+2h_{1}V_{g})-10h_{1}^{2}\sigma ), \end{aligned}$$

$$\begin{aligned} h_{16}&=\frac{h_{161}}{h_{120}},h_{161}=18h_{2}^{2}+9R_{1}h_{3} +36h_{1}h_{2}V_{g}-10h_{1}^{2}\sigma +6(\delta -s_{2}\mu )(-3V_{g}^{2}+5\sigma ).\\ s_{4}&=35\Gamma +3b(8+81\Gamma ),s_{5}=16-16b+189\Gamma -105\Gamma .\\ P_{1}&=\frac{k(5ih_{5}\sigma -3ih_{6}V_{g}-5k\sigma )+3i(h_{6}-h_{5}V_{g} )\omega +3\omega ^{2}}{3(h_{2}k+h_{1}\omega )}, \end{aligned}$$

$$\begin{aligned} P_{2}&=((3\omega ^{2}-5\sigma k^{2})(k^{2}(54(8h_{11}h_{3}-h_{3}^{2} -4h_{9})R_{1}+5(32h_{1}^{3}-48h_{1}^{2}h_{7}+9(16h_{11}\\ &\quad +15(18h_{11}-7)\Gamma +b(32h_{11}-72-729\Gamma ))\mu )\sigma )+432h_{1} h_{8}k\omega -27(16h_{11}+15\Gamma (18h_{11}-7)\\&\qquad b(h_{11}(32+42\Gamma )-9(8+81\Gamma )))\mu \omega ^{2}+432kh_{2}(h_{8} k+h_{7}\omega )+6(k^{2}(72h_{13}h_{2}+36h_{14}R_{1}\\&\quad-5(8h_{1}h_{12}+12(1+h_{11}+2h_{16}+3\beta _{e})\delta -3h_{16} (8+16b+135\Gamma +21b\Gamma )\mu )\sigma )+72(h_{1}h_{13}\\&\quad+h_{2}h_{12})\omega k+92h_{11}+2h_{16}+3\beta _{e})\delta -h_{16} (8+135\Gamma ))\mu )\omega ^{2})))(432(h_{2}k+h_{1}\omega )(5k^{2}\sigma -3\omega ^{2})))^{-1}), \end{aligned}$$

$$\begin{aligned} P_{3}=\frac{3\omega ^{4}+5k^{2}\sigma \Omega _{c}^{2}+\omega ^{2}(3R_{1} -3+5h_{1}\sigma -5k^{2}\sigma -3\Omega _{c}^{2}}{3(h_{2}k+h_{1}\omega )(\omega ^{2}-\Omega _{c}^{2})}. \end{aligned}$$