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Effects of exchange symmetry and quantum diffraction on amplitude-modulated electrostatic waves in quantum magnetoplasma

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Abstract

In this paper we have made use of reductive perturbation technique (RPT) and carried out homotopy analysis method (HAM) to investigate the effect of exchange correlation and quantum diffraction on the electrostatic waves in quantum magnetoplasma. We have derived a nonlinear Schrödinger equation (NLSE) by using RPT that describes the spatiotemporal evolution of an initial waveform. Apart from this technique, we have made use of HAM to second our initial findings. It has been shown that both quantum diffraction H and parameter streaming velocity \(u_0\) have significant effects in determining the stability criteria and the growth or decay of any instability created therein. The stable parametric regimes are crucial from the experimental point of view as well.

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Acknowledgements

The authors would like to thank the unknown reviewers for their time and effort in providing valuable input and pointing out some issues that improved the quality of this paper. The authors also would like to thank Institute of Natural Sciences and Applied Technology as well as the Physics Departments of Jadavpur University and Government General Degree College at Kushmandi for providing facilities to carry out this work.

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Authors and Affiliations

  1. Department of Mathematics, Government General Degree College at Kushmandi, Dakshin Dinajpur, 733 121, India

    Chinmay Das

  2. Department of Physics, Government General Degree College at Kushmandi, Dakshin Dinajpur, 733 121, India

    Swarniv Chandra

  3. Department of Physics, Jadavpur University, Kolkata, 700 032, India

    Chinmay Das, Swarniv Chandra & Basudev Ghosh

  4. Institute of Natural Sciences and Applied Technology, Kolkata, 700 032, India

    Chinmay Das, Swarniv Chandra & Basudev Ghosh

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  1. Chinmay Das
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  2. Swarniv Chandra
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Correspondence to Swarniv Chandra.

Appendix

Appendix

1.1 Appendix I(A)

1.1.1 \(n=\mathrm{1}\), \(l=\mathrm{1}\) perturbation relations

$$\begin{aligned}&-i\omega \gamma _1 n_{e,1}^{(1)} +ik\gamma _1 v_{e,1}^{(1)} + ik \gamma _1 u_0 n_{e,1}^{(1)}=0,\nonumber \\&\quad -i\omega n_{i,1}^{(1)} + ik v_{i,1}^{(1)}=0,\nonumber \\&\quad - ik\varphi _{1}^{(1)}- \frac{ {v}_{e\perp ,1 }^{(1)}}{\rho _s} +\frac{i k \lambda _1}{3} n_{e,1}^{(1)} -\frac{2 i k \lambda _2 }{3} n_{e,1}^{(1)}\nonumber \\&\quad \quad \quad \quad \quad + i\frac{H^2}{4\gamma _3 }k^3 n_{e,1}^{(1)}=0 ,\nonumber \\&\quad -i\omega v_{i,1}^{(1)} + ik\varphi _{1}^{(1)} +i k n_{i,1}^{(1)} =0,\nonumber \\&\quad -k^2 \varphi _{1}^{(1)} - n_{e,1}^{(1)} + n_{i,1}^{(1)} = 0 . \end{aligned}$$
(39)

1.1.2 \(n=1\), \(l=1\) perturbation coefficients

$$\begin{aligned}&A_{11}=\frac{k^4+k^2-\omega ^2 k^2}{\omega ^2-k^2} , \nonumber \\&B_{11}=\frac{k^2}{\omega ^2-k^2}, \nonumber \\&C_{11}=\left( \frac{\omega -ku_0}{k }\right) \left( \frac{k^4+k^2-\omega ^2 k^2}{\omega ^2-k^2}\right) ,\nonumber \\&D_{11}=\frac{\omega k}{\omega ^2-k^2}. \end{aligned}$$
(40)

1.2 Appendix I(B)

1.2.1 \(n=2\), \(l=1\) perturbation relations

$$\begin{aligned}&-C_g \gamma _1 \frac{\partial {n}_{e,1}^{(1)}}{\partial \xi } - i\omega \gamma _2 n_{e,1}^{(2)} + \gamma _1 \frac{\partial {v}_{e,1}^{(1)}}{\partial \xi } \nonumber \\&\qquad + ik \gamma _2 v_{e,1}^{(2)} + u_0 \gamma _1 \frac{\partial {n}_{e,1}^{(1)}}{\partial \xi } + iku_0 \gamma _2 n_{e,1}^{(2)} = 0 ,\nonumber \\&\qquad -C_g \frac{\partial {n}_{i,1}^{(1)}}{\partial \xi } - i\omega n_{i,1}^{(2)} + \frac{\partial {v}_{i,1}^{(1)}}{\partial \xi } + ik v_{i,1}^{(2)}=0 ,\nonumber \\&-\frac{\partial {\varphi }_{1}^{(1)}}{\partial \xi } - ik\varphi _{1}^{(2)}\nonumber \\&\qquad + \frac{\lambda _1}{3}\left( ik n_{e,1}^{(2)} + \frac{\partial {n}_{e,1}^{(1)}}{\partial \xi }\right) \nonumber \\&\qquad - \frac{2 \lambda _2}{3}\left( ik n_{e,1}^{(2)} + \frac{\partial {n}_{e,1}^{(1)}}{\partial \xi }\right) \nonumber \\&\qquad - \frac{H^2}{2\gamma _3 } \left( -\frac{ik^3}{2} {n}_{e,1}^{(2)} -3k^2 \frac{\partial {n}_{e,1}^{(1)}}{\partial \xi }\right) = 0,\nonumber \\&-C_g \frac{\partial {v}_{i,1}^{(1)}}{\partial \xi } - i\omega v_{i,1}^{(2)} + \frac{\partial {\varphi }_{1}^{(1)}}{\partial \xi } + ik \varphi _{1}^{(2)} \nonumber \\&\qquad + \frac{\partial {n_{i,1}}^{(1)}}{\partial \xi } + ik n_{i,1}^{(2)} =0 , \nonumber \\&2ik\frac{\partial {\varphi }_{1}^{(1)}}{\partial \xi } - k^2 \varphi _{1}^{(2)} -{n}_{e,1}^{(2)} + {n}_{i,1}^{(2)} = 0 . \end{aligned}$$
(41)

1.2.2 \(n=2\), \(l=1\) perturbation coefficients

$$\begin{aligned}&A_{21}^{\prime }=\frac{12\gamma _3 k}{4\lambda _1 \gamma _3-8\lambda _2 \gamma _3+3H^2k^2},\nonumber \\&A_{21}^{\prime \prime }=2iA_{11}\frac{\gamma _3\left( -6+2\lambda _1 -4\lambda _2 \right) +9H^2k^2}{k\left( 4\gamma _3\lambda _1 -8\gamma _3\lambda _2+3H^2k^2 \right) }, \end{aligned}$$
(42)
$$\begin{aligned}&B_{21}^{\prime }= \frac{k^2}{\omega (\omega -k)},\nonumber \\&B_{21}^{\prime \prime }=i\frac{k(1{-}C_g D_{11}{+}B_{11}){-}\left( k{-}\omega \right) (C_g B_{11}{-}D_{11})}{\omega \left( k{-}\omega \right) },\nonumber \\&C_{21}^{\prime }=\frac{\gamma _1\left( \omega -ku_0\right) A_{21}^{\prime }}{k\gamma _2},\nonumber \\&C_{21}^{\prime \prime }=\frac{k\gamma _2 A_{21}^{\prime \prime }+i\gamma _1(-C_gA_{11}+C_{11}+u_0 A_{11})}{k\gamma _2} \nonumber \\&D_{21}^{\prime }=\frac{\omega B_{21}^{\prime } }{k},\nonumber \\&D_{21}^{\prime \prime }=\frac{B_{21}^{\prime \prime }+i(C_gB_{11}-D_{11})}{k},\nonumber \\&E_{21}^{\prime }= \frac{2ik-A_{21}^{\prime \prime }+B_{21}^{\prime \prime }}{k^2+A_{21}^{\prime }-B_{21}^{\prime }},\nonumber \\&iA_{21}=A_{21}^{\prime }E_{21} +A_{21}^{\prime \prime }, \nonumber \\&iB_{21}=B_{21}^{\prime }E_{21}+B_{21}^{\prime \prime }, \nonumber \\&iC_{21}=C_{21}^{\prime }E_{21}+C_{21}^{\prime \prime }, \nonumber \\&iD_{21}=D_{21}^{\prime }E_{21}+D_{21}^{\prime \prime },\nonumber \\&iE_{21}=E_{21}^{\prime }. \end{aligned}$$
(43)

1.3 Appendix I(C)

1.3.1 \(n=2\), \(l=2\) perturbation relations

$$\begin{aligned}&-C_g \gamma _1\frac{\partial {n}_{e,2}^{(1)}}{\partial \xi } - 2i\omega \gamma _2 n_{e,2}^{(2)} \nonumber \\&\quad + \gamma _1 \frac{\partial {v}_{e,2}^{(1)}}{\partial \xi } + 2ik \gamma _2 v_{e,2}^{(2)} \nonumber \\&\quad + u_0\gamma _1 \frac{\partial {n}_{e,2}^{(1)}}{\partial \xi } + 2i ku_0 \gamma _2 n_{e,2}^{(2)}\nonumber \\&\quad +2ik{\gamma _1}^2{n}_{e,1}^{(1)}{v}_{e,1}^{(1)} {\,=\,} 0 ,\nonumber \\&-C_g \frac{\partial {n}_{i,2}^{(1)}}{\partial \xi } - 2i\omega n_{i,2}^{(2)} \nonumber \\&\quad + \frac{\partial {v}_{i,2}^{(1)}}{\partial \xi } + 2ik v_{i,2}^{(2)}+2ik{n}_{i,1}^{(1)}{v}_{i,1}^{(1)}=0 ,\nonumber \\&- \frac{\partial {\varphi }_{2}^{(1)}}{\partial \xi } - 2ik\varphi _{2}^{(2)} + \frac{\lambda _1}{3} \left( 2ik n_{e,2}^{(2)} + \frac{\partial {n}_{e,2}^{(1)}}{\partial \xi }\right) \nonumber \\&\quad - \frac{2 \lambda _2}{3} \left( 2ik n_{e,2}^{(2)} + \frac{\partial {n}_{e,2}^{(1)}}{\partial \xi }\right) \nonumber \\&\quad - \frac{H^2}{2\gamma _3 } \left( -4ik^3 {n}_{e,2}^{(2)} -4k^2 \frac{\partial {n}_{e,2}^{(1)}}{\partial \xi }\right) =0,\nonumber \\&-C_g \frac{\partial {v}_{i,2}^{(1)}}{\partial \xi } - 2i\omega v_{i,2}^{(2)} + \frac{\partial {\varphi }_{2}^{(1)}}{\partial \xi } + 2ik \varphi _{2}^{(2)} \nonumber \\&\quad + \frac{\partial {n_{i,2}^{(1)}}}{\partial \xi } + 2ik n_{i,2}^{(2)}+ 2ik{v}_{i,1}^{(1)}{v}_{i,1}^{(1)} =0, \nonumber \\&4ik\frac{\partial {\varphi }_{2}^{(1)}}{\partial \xi } - 4k^2 \varphi _{2}^{(2)} -{n}_{e,2}^{(2)}\nonumber \\&\quad + {n}_{i,2}^{(2)}-\frac{1}{2}\varphi _{1}^{(1)}.\varphi _{1}^{(1)} = 0 . \end{aligned}$$
(44)

1.3.2 \(n=2\), \(l=2\) perturbation coefficients

$$\begin{aligned}&E_{22}=\frac{2kD_{11}(\omega B_{11}{+}kD_{11}){-}1}{2(\omega ^2 {-} k^2)\left( 4k^2{+}\frac{3\gamma _3}{\gamma _3(\lambda _1{-}2\lambda _2){+}3H^2k^2}{-}\frac{k^2}{\omega ^2{-}k^2}\right) },\nonumber \\&A_{22}=\frac{3\gamma _3 E_{22}}{\gamma _3(\lambda _1-2\lambda _2)+3H^2k^2},\nonumber \\&B_{22}=\frac{kD_{11}(\omega B_{11}+kD_{11})+k^2E_{22}}{\omega ^2-k^2}, \nonumber \\&C_{22}=\frac{A_{22}(\omega -ku_0)+k A_{11}C_{11}}{k},\nonumber \\&D_{22}=\frac{\omega B_{22}}{k} - B_{11}D_{11}. \end{aligned}$$
(45)

1.4 Appendix I(D)

1.4.1 \(n=3\), \(l=0\) perturbation relations

$$\begin{aligned}&\gamma _1\frac{\partial {n}_{e,0}^{(1)}}{\partial \tau }-C_g \gamma _2\frac{\partial {n}_{e,0}^{(2)}}{\partial \xi } + \gamma _2 \frac{\partial {v}_{e,0}^{(2)}}{\partial \xi } + u_0 \gamma _2\frac{\partial {n}_{e,0}^{(2)}}{\partial \xi } = 0 ,\nonumber \\&\frac{\partial {n}_{i,0}^{(1)}}{\partial \tau } - C_g \frac{\partial {n}_{i,0}^{(2)}}{\partial \xi } + \frac{\partial {v}_{i,0}^{(2)}}{\partial \xi } =0 ,\nonumber \\&- \frac{\partial {\varphi }_{0}^{(2)}}{\partial \xi } +\frac{\lambda _1}{3} \left( \frac{\partial {n}_{e,0}^{(2)}}{\partial \xi } \right) - \frac{2 \lambda _2}{3}\left( \frac{\partial {n}_{e,0}^{(2)}}{\partial \xi } \right) =0 ,\nonumber \\&\frac{\partial {v}_{i,0}^{(1)}}{\partial \tau } - C_g \frac{\partial {v}_{i,0}^{(2)}}{\partial \xi } + \frac{\partial {\varphi }_{0}^{(2)}}{\partial \xi } + \frac{\partial {n}_{i,0}^{(2)}}{\partial \xi } =0,\nonumber \\&\frac{\partial ^2 {\varphi }_{0}^{(1)}}{\partial \xi ^2} -{n}_{e,0}^{(3)} + {n}_{i,0}^{(3)}-| \varphi _{1}^{(1)}| ^2 = 0 . \end{aligned}$$
(46)

1.4.2 \(n=2\), \(l=0\) perturbation coefficients

$$\begin{aligned} E_{20}= & {} \frac{(C_g^2-1)\left( \lambda _1-2\lambda _2\right) }{\left( \lambda _1-2\lambda _2\right) -3(C_g^2-1)},\nonumber \\ A_{20}= & {} \frac{3 E_{20}}{\lambda _1-2\lambda _2},\nonumber \\ B_{20}= & {} \frac{E_{20}}{C_g^2-1}, \end{aligned}$$
(47)
$$\begin{aligned} C_{20}= & {} \frac{3(C_g-u_0)E_{20}}{ \left( \lambda _1-2\lambda _2\right) },\nonumber \\ D_{20}= & {} \frac{C_g E_{20}}{C_g^2-1}. \end{aligned}$$
(48)

1.5 Appendix I(D)

1.5.1 \(n=3\), \(l=1\) perturbation relations

$$\begin{aligned}&\gamma _1\frac{\partial {n}_{e,1}^{(1)}}{\partial \tau }-C_g \gamma _2 \frac{\partial {n}_{e,1}^{(2)}}{\partial \xi } - i\omega \gamma _3 n_{e,1}^{(3)} \nonumber \\&\qquad + \gamma _2 \frac{\partial {v}_{e,1}^{(2)}}{\partial \xi } + ik \gamma _3 v_{e,1}^{(3)} + u_0 \gamma _2\frac{\partial {n}_{e,1}^{(2)}}{\partial \xi } + i k u_0 \gamma _3 n_{e,1}^{(3)} \nonumber \\&\qquad +ik\gamma _1\gamma _2\left[ n_{e,1}^{(1)}.v_{e,0}^{(2)}+n_{e,-1}^{(1)}.v_{e,2}^{(2)}\right. \nonumber \\&\qquad \left. +n_{e,0}^{(2)}.v_{e,1}^{(1)} +n_{e,2}^{(2)}.v_{e,-1}^{(1)} \right] = 0 ,\nonumber \\&\quad \frac{\partial {n}_{i,1}^{(1)}}{\partial \tau } - C_g \frac{\partial {n}_{i,1}^{(2)}}{\partial \xi } - i\omega n_{i,1}^{(3)} + \frac{\partial {v}_{i,1}^{(2)}}{\partial \xi } \nonumber \\&\qquad + ik v_{i,1}^{(3)} + ik\left[ n_{i,1}^{(1)}.v_{i,0}^{(2)}+n_{i,-1}^{(1)}.v_{i,2}^{(2)}\right. \nonumber \\&\qquad \left. +n_{i,0}^{(2)}.v_{i,1}^{(1)} +n_{i,2}^{(2)}.v_{i,-1}^{(1)} \right] = 0 ,\nonumber \\&\qquad -\frac{\partial {\varphi }_{1}^{(2)}}{\partial \xi } - ik \varphi _{1}^{(3)} \nonumber \\&\qquad + \frac{\lambda _1}{3} \left( ik n_{e,1}^{(3)} + \frac{\partial {n}_{e,1}^{(2)}}{\partial \xi } \right) \nonumber \\&\qquad - \frac{2 \lambda _2}{3}\left( ik n_{e,1}^{(3)} + \frac{\partial {n}_{e,1}^{(2)}}{\partial \xi } \right) \nonumber \\&\qquad - \frac{H^2}{2\gamma _3 } \left( -\frac{ik^3}{2}{n}_{e,1}^{(3)} - 3k^2\frac{\partial {n}_{e,1}^{(2)}}{\partial \xi } +ik \frac{\partial ^2 {n}_{e,1}^{(1)}}{\partial \xi ^2}\right) =0 , \nonumber \\&\qquad \frac{\partial {v}_{i,1}^{(1)}}{\partial \tau } - C_g \frac{\partial {v}_{i,1}^{(2)}}{\partial \xi } -i\omega v_{i,1}^{(3)} \nonumber \\&\qquad + \frac{\partial {\varphi }_{1}^{(2)}}{\partial \xi } + ik \varphi _{1}^{(3)} +\frac{\partial {n}_{i,1}^{(2)}}{\partial \xi } +ik{n}_{i,1}^{(3)} \nonumber \\&\qquad +ik\left[ 2v_{i,-1}^{(1)}.v_{i,2}^{(2)} + v_{i,0}^{(2)}.v_{i,1}^{(1)}-v_{i,2}^{(2)}.v_{i,-1}^{(1)}\right] =0,\nonumber \\&\qquad \frac{\partial ^2 {\varphi }_{1}^{(1)}}{\partial \xi ^2} + 2ik\frac{\partial {\varphi }_{1}^{(2)}}{\partial \xi } - k^2 \varphi _{1}^{(3)} -{n}_{e,1}^{(3)} \nonumber \\&\qquad + {n}_{i,1}^{(3)} -\left[ \varphi _{1}^{(1)}.\varphi _{0}^{(2)} + \varphi _{-1}^{(1)}.\varphi _{2}^{(2)}\right] = 0 . \end{aligned}$$
(49)

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Das, C., Chandra, S. & Ghosh, B. Effects of exchange symmetry and quantum diffraction on amplitude-modulated electrostatic waves in quantum magnetoplasma. Pramana - J Phys 95, 78 (2021). https://doi.org/10.1007/s12043-021-02108-x

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