Modulated 3D electron-acoustic rogue waves in magnetized plasma with nonthermal electrons

Abstract

Modulational instability analysis of Electron-acoustic (EA) rogue waves is carried out in a three-dimensional magnetized plasma holding two electron populations (nonthermal hot electrons following Cairns distribution and cold electron fluid) and stationary ions. A $(3+1)$-dimensional nonlinear Schrödinger (NLS) equation that governs the evolution of the EA rogue waves in the current plasma system is derived through derivative expansion method. The existence domains for the first and second order rogue waves are investigated. Effects of the plasma parameters on the propagation of the first and second order rogue waves are examined analytically and numerically. Furthermore, the role of the different plasma parameters as well as the critical threshold of the obliqueness $\theta$ on the first and second order rogue waves for parameters regimes corresponding to dayside auroral zone plasma are emphasized.

Appendix

The coefficients $\Delta _{j}$ (i.e., $j=1,\ldots,8$) are

$$\begin{aligned} \Delta _{1} =&\bigl[ \bigl\{ \bigl( -54\alpha k^{3}\omega V_{g}-3V_{g}^{2} \bigl( 5k^{2}\sigma -3\omega ^{2} \bigr) ^{2} \\ &{}+5k^{4} \sigma \bigl( 3\alpha +25\sigma ^{2} \bigr) -3k^{2}\omega ^{2} \bigl( 9\alpha +50\sigma ^{2} \bigr) \\ &{}+45\sigma \omega ^{4} \bigr) \bigr\} \bigl\{ \bigl( 5k^{2}\sigma -3\omega ^{2} \bigr) ^{2} \bigl( 5(\beta -1)\sigma \\ &{}-3(\beta-1)V_{g}^{2}-3 \bigr) \bigr\} ^{-1}\bigr] \end{aligned}$$

(A.1)

$$\begin{aligned} \Delta _{2} =&\bigl\{ -5k^{6} \bigl( 3\alpha \sigma +25\sigma ^{3} \bigr) +9k^{4}\omega ^{2} \bigl( 9\alpha +25\sigma ^{2} \bigr) \\ &{}+27\omega ^{4} \bigl( \omega ^{2}-5k^{2}\sigma \bigr) \bigr\} \bigl\{ 2 \bigl( 5k^{2}\sigma -3\omega ^{2} \bigr) ^{2} \bigl( 3( \beta -1)\omega ^{2} \\ &{}+20k^{4}\sigma +k^{2} \bigl( -5(\beta -1)\sigma -12\omega ^{2}+3 \bigr) \bigr) \bigr\} ^{-1}, \end{aligned}$$

(A.2)

$$\begin{aligned} \Delta _{3} =&\bigl\{ 3\alpha k^{2} \bigl( 3k^{2}\omega ^{2} \bigl( 10\sigma -9\alpha \bigl( -\beta +4k^{2}+1 \bigr) \bigr) \\ &{}+5k^{4}\sigma \bigl( -\alpha \beta + \alpha +4\alpha k^{2}-5\sigma \bigr) -9\omega ^{4} \bigr) \bigr\} \\ &{}\times\bigl\{ 2 \bigl( 5k^{2}\sigma -3\omega ^{2} \bigr) ^{2} \bigl( 3(\beta -1)\omega ^{2}+20k^{4}\sigma \\ &{}+k^{2} \bigl( -5(\beta -1)\sigma -12\omega ^{2}+3 \bigr) \bigr)\bigr\} ^{-1}, \end{aligned}$$

(A.3)

$$\begin{aligned} \Delta _{4} =&-\bigl\{ 3\alpha k\omega \bigl( 3k^{2}\omega ^{2} \bigl( 3\alpha \bigl( -\beta +4k^{2}+1 \bigr) -10\sigma \bigr) \\ &{}+k^{4} \bigl( \alpha \bigl( 25\sigma \bigl( -\beta +4k^{2}+1 \bigr) +18 \bigr) +25\sigma ^{2} \bigr) +9\omega ^{4} \bigr) \bigr\} \\ &{}\times\bigl\{ 2 \bigl( 5k^{2}\sigma -3\omega ^{2} \bigr) ^{2} \bigl( 3( \beta -1)\omega ^{2}+20k^{4}\sigma \\ &{}+k^{2} \bigl( -5(\beta -1)\sigma -12\omega ^{2}+3 \bigr) \bigr) \bigr\} ^{-1}, \end{aligned}$$

(A.4)

$$\begin{aligned} \Delta _{5} =&\bigl\{ 3\alpha k \bigl( 3k^{2} \omega ^{2} \bigl( 6\alpha (\beta -1) \bigl( -\beta +k^{2}+1 \bigr) \\ &{}+10\sigma \bigl( -\beta +k^{2}+1 \bigr) -3 \bigr) -9\omega ^{4} \bigl( -\beta +k^{2}+1 \bigr) \\ &{}-5k^{4}\sigma \bigl( 2\alpha (\beta -1) \bigl( -\beta +k^{2}+1 \bigr) \\ &{}+5\sigma \bigl( -\beta +k^{2}+1 \bigr) -3 \bigr) \bigr) \bigr\} \bigl\{ \bigl( -\beta +k^{2}+1 \bigr) \\ &{}\times\bigl( 5k^{2}\sigma -3\omega ^{2} \bigr) ^{2} \\ &{}\times\bigl( 5(\beta -1)\sigma -3(\beta -1)V_{g}^{2}-3 \bigr) \bigr\} ^{-1}, \end{aligned}$$

(A.5)

$$\begin{aligned} \Delta _{6} =&\bigl\{ \bigl( -27\bigl(\beta (3\beta -2)+1\bigr)\omega ^{6} \\ &{}-225\bigl(\beta (3\beta -2)+1 \bigr)k^{4}\sigma ^{2}\omega ^{2} \\ &{}+135\bigl(\beta (3 \beta -2)+1\bigr)k^{2}\sigma \omega ^{4} \\ &{}+5k^{6} \sigma \bigl( 25\bigl(\beta (3\beta -2)+1\bigr)\sigma ^{2} \\ &{}+4\alpha ^{2}(\beta -1) \bigl( -\beta +k^{2}+1 \bigr) \bigr) \bigr) \bigr\} \\ &{}\times\bigl\{ 4\omega \bigl( -\beta +k^{2}+1 \bigr) \bigl( 5k^{2}\sigma -3\omega ^{2} \bigr) ^{2} \\ &{}\times\bigl( 5( \beta -1)\sigma -3(\beta -1)V_{g}^{2}-3 \bigr) \bigr\} ^{-1}, \end{aligned}$$

(A.6)

$$\begin{aligned} \Delta _{7} =&9k^{2}\omega ^{4} \bigl( 15\sigma \bigl(\beta \bigl(5(2-3\beta )\sigma +9\bigr)-5\sigma +3\bigr) \\ &{}+18\alpha ^{2}(\beta -1) \bigl( -\beta +k^{2}+1 \bigr) \\ &{}+2\alpha \bigl( 35\sigma \bigl( -\beta +k^{2}+1 \bigr) -9 \bigr) \bigr) \\ &{} -27\omega ^{6} \bigl( \beta \bigl(5(2-3\beta )\sigma +9\bigr) \\ &{}+6\alpha \bigl( -\beta +k^{2}+1 \bigr) -5\sigma +3 \bigr) \\ &{} +5k^{6}\sigma \bigl( 25\sigma ^{2}\bigl(\beta \bigl(5(2-3\beta )\sigma +9\bigr)-5\sigma +3\bigr) \\ &{}-2\alpha ^{2}\bigl(5( \beta -1)\sigma -6\bigr) \bigl( -\beta +k^{2}+1 \bigr) \\ &{}+10\alpha \sigma \bigl( 5\sigma \bigl( -\beta +k^{2}+1 \bigr) -3 \bigr) \bigr) \\ &{} -3k^{4}\omega ^{2} \bigl( 75\sigma ^{2} \bigl(\beta \bigl(5(2-3\beta )\sigma +9\bigr)-5\sigma +3\bigr) \\ &{}-12\alpha ^{2}\bigl(25(\beta -1)\sigma -18\bigr) \bigl( -\beta +k^{2}+1 \bigr) \\ &{}+10\alpha \sigma \bigl( 25\sigma \bigl( -\beta +k^{2}+1 \bigr) -12 \bigr) \bigr) , \end{aligned}$$

(A.7)

and

$$\begin{aligned} \Delta _{8} =&12\omega \bigl( -\beta +k^{2}+1 \bigr) \bigl( 5k^{2}\sigma -3\omega ^{2} \bigr) ^{2} \\ &{}\times \bigl( 5(\beta -1)\sigma -3(\beta -1)V_{g}^{2}-3 \bigr) . \end{aligned}$$

(A.8)