Dissipative high-frequency envelope soliton modes in nonthermal plasmas

The linear and nonlinear properties of modulated high-frequency (electron-acoustic) electrostatic wave packets are investigated via a fluid-dynamical approach. A three-component plasma is considered, composed of two types of electrons at different temperatures (“cold” and “hot” electrons) evolving against a cold stationary ion background. A weak dissipative effect is assumed, due to electron-neutral collisions. While the cold electrons are treated as an inertial fluid, the hot electrons are assumed to be in a non-Maxwellian state, described by a kappa ($\kappa$) type distribution. The linear characteristics of electron-acoustic waves are analyzed in detail, and a linear dispersion relation is obtained. Weakly damped electrostatic waves are shown to propagate above a wave number $k$ threshold, whose value is related to dissipation (and reduces to zero in its absence). Long-wavelength values (i.e., for $k$ below that threshold) are heavily damped and no propagation occurs. The nonlinear dynamics (modulational self-interaction) of wave packets in the propagating region is modeled via a dissipative nonlinear Schrödinger type equation, derived via a multiscale perturbation technique for the wave envelope, which includes a dissipative term associated with the finite imaginary part of the nonlinearity term. The dynamical and structural characteristics (speed, amplitude, width) of dissipative localized modes representing the amplitude of modulated electron-acoustic wave packets in a collisional plasma are thus investigated for various values of relevant plasma (configuration) parameters, namely the superthermality index $\kappa$, the cold-to-hot electron density ratio, and collisionality (strength). Our analytical predictions are tested by computer simulations. A quasilinear perturbation method for near-integrable systems leads to a theoretical prediction for the wave amplitude decay, which is shown to match our numerical result. The results presented in this paper should be useful in understanding the dynamics of localized electrostatic disturbances in space plasmas, and also in laboratory plasmas, where the combined effect(s) of excess energetic (suprathermal) electrons and (weak) electron-neutral collisions may be relevant.

Figure 1

The variation of the $\kappa$-dependent Debye screening length ${\lambda }_{D,\mathrm{eff}}^{\left(\kappa \right)}$, from (10), is depicted versus the hot electron spectral (superthermality) index $\kappa$ and the cold-to-hot electron number density ratio $\beta$.

Figure 2

Dispersion relation: The wave frequency $\omega$ is depicted versus the wave number $k$. Here, we have considered a strong superthermality index value $\kappa =3$, along with a cold-to-hot electron number density ratio $\beta =0.5$. The positive upper part denotes the real part of the frequency, ${\omega }_r$, while the constant negative part at the bottom of the graph denotes the imaginary part ${\omega }_i$.

Figure 3

The wave number threshold $k_c$ is shown versus the collisional parameter $\nu$: (a) for different cold-to-hot electron population ratio $\beta$ (for $\kappa =3$, here), and (b) for different $\kappa$ (with $\beta =0.5$).

Figure 4

The variation of the wave frequency ${\omega }_r$ (real part), as given in Eq. (11), is shown (a) versus the wave number $k$ and the cold-to-hot electron number density ratio $\beta$, for $\kappa =3$ and $\nu =0.5$, and (b) versus the wave number $k$ and the collisional parameter $\nu$, for $\kappa =3$ and $\beta =0.5$.

Figure 5

We depict the effect of superthermality (via $\kappa$) on (a) the dispersive term $P$, (b) the real part of nonlinear term $Q_r$, and (c) the imaginary part of nonlinear term $Q_i$, versus the wave number $k$ for $\beta =0.5,\nu =0.05$. The inset figures show a zoom-in view of regions of interest.

Figure 6

Variation of the wave number threshold $k_0$ [in which $Q_r\left(k_0\right)=0$] versus the superthermality index $\kappa$ for different values of cold-to-hot electron number density ratio $\beta$, where $\nu =0.05$. The inset figures show a zoom-in view of regions of interest.

Figure 7

The damping factor $D$, given in Eq. (17), is depicted versus the wave number $k$ (a) for different superthermality $\kappa$ for $\beta =0.5,\nu =0.05$, and (b) for different cold-to-hot electron density ratio $\beta$ for $\kappa =3,\nu =0.05$.

Figure 8

The evolution of the normalized envelope amplitude $\rho$ (in units of electric potential), as given in Eq. (25), is depicted versus time $T$ (in units of ${\omega }_{ph}^{-1}$). We have considered different values of (a) the carrier wave number $k$ (in units of ${\lambda }_{Dh}^{-1}$), for collision frequency $\nu =0.05$, and (b) the dissipation factor $\nu$ for $k=4$. The other parameters are fixed at $\kappa =3,\beta =0.5$. The inset plots show a zoom-in view of regions of interest.

Figure 9

The evolution of the normalized envelope amplitude $\rho$ (in units of electric potential), as given in Eq. (25), is depicted versus time $T$ (in units of ${\omega }_{ph}^{-1}$). We have considered different values of (a) $\beta$ for $\kappa =3$, and (b) $\kappa$ for $\beta =0.5$. The other parameters are fixed at $k=4,\nu =0.05$. The inset plots show a zoom-in view of regions of interest.

Figure 10

The evolution in time of the (normalized) bright envelope amplitude, given by (20), is depicted. For the initial condition, based on Eq. (20), we have considered a set of representative parameter values: $k=4, \kappa =3, \beta =0.5$, and $\nu =0.05$ (implying $P=-0.0054,Q_r=-5.99$, and $Q_i=-66.78$). The wave amplitude $\psi$ is shown versus position $\xi$, in two successive time intervals.

Figure 11

The decay of the envelope soliton (pulse) amplitude with time is illustrated, as observed numerically (rectangles) and analytically (dashed line). Different values of the wave number $k$ are considered: the red (lower) curve is for wave number $k=4$, while the blue (upper) curve is for $k=5$. The other parameters are taken as $\kappa =3, \beta =0.5$, and $\nu =0.05$. A zoom-in aspect is shown in the panel (b).

Figure 12

The decay of the envelope soliton (pulse) amplitude with time is illustrated, as observed numerically (rectangles) and analytically (dashed line). Different values of the cold-to-hot electron density ratio $\beta$ are considered: the red (upper) curve is for $\beta =0.5$, while the blue (lower) curve is for $\beta =1$. The other parameters are taken as $k=4, \kappa =3$, and $\nu =0.05$. A zoom-in aspect is shown in panel (b).

Figure 13

The decay of the envelope soliton (pulse) amplitude with time is illustrated, as observed numerically (rectangles) and analytically (dashed line). The red (lower) curve is for $\kappa =3$ and the blue (upper) curve is for $\kappa =100$ (quasi-Maxwellian). The other parameters are $k=4, \beta =0.5$, and $\nu =0.05$. A zoom-in aspect is shown in panel (b).