Modulational instability of ion acoustic excitations in a plasma with a $\kappa$-deformed Kaniadakis electron distribution

Abstract

In this paper, we make use of the $\kappa$-deformed Kaniadakis distribution in the context of modulational instability (MI) of ion acoustic waves (IAWs) in a plasma assumed to be collisionless and unmagnetized, consisting of inertial ions and non-Maxwellian electrons. Through the multiscale perturbation method, the basic model equations are reduced to a nonlinear Schrödinger equation, based on which the MI and the growth rate of envelope excitations are calculated. It is shown that the deformation parameter $\kappa$, which develops the Kaniadakis entropy, affects the conditions for MI and thus modifies the associated threshold as well as the growth rate. The deviation from Maxwellian distribution, i.e., the presence of Kaniadakis distributed electrons, reinforces the instability. The critical wave number shows diminishing behavior with higher values of $\kappa$, suggesting that the MI sets in for lower values of wave number.

Appendix

Here the coefficients for the second-harmonic components are provided. $\Lambda {_{2}^{0}}_{1/2/3}$ are associated with the second-order, zeroth-harmonic components of the ion density, speed and potential, respectively, while $\Lambda {_{2}^{2}}_{1/2/3}$ stand for the second-order, second-harmonic components.

$$\begin{aligned} \Lambda _{2_{1}}^{0}&{=}\frac{-2k^{2}v_{g}\omega \left( k^{2}+1\right) ^{2}+k^{3}v_{g}^{2}-k\omega ^{2}\left( k^{2}+1\right) ^{2}}{ k^{3}\left( 1-v_{g}^{2}\right) } \nonumber \\ \Lambda _{2_{3}}^{0}&=\Lambda _{2_{1}}^{0}+1 \nonumber \\ \Lambda _{2_{2}}^{0}&=-2\frac{\omega }{k}\left( k^{2}+1\right) ^{2}+v_{g}\Lambda _{2_{3}}^{0} \end{aligned}$$

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$$\begin{aligned} \Lambda _{2_{3}}^{2}&=\frac{\omega ^{2}\left[ -3\left( k^{2}+1\right) ^{2}+1 \right] }{2\left[ k^{2}-4k^{2}\omega ^{2}-\omega ^{2}\right] } \nonumber \\ \Lambda _{2_{1}}^{2}&=\left( 4k^{2}+1\right) \Lambda _{2_{3}}^{2}+1/2 \nonumber \\ \Lambda _{2_{2}}^{2}&=\frac{\omega }{k}\Lambda _{2_{1}}^{2}-\frac{\omega }{k }\left( k^{2}+1\right) ^{2}. \end{aligned}$$

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