# Effects of Landau damping on finite amplitude low-frequency nonlinear waves in a dusty plasma

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## Abstract

The effect of linear ion Landau damping on weakly nonlinear as well as weakly dispersive low-frequency waves in a dusty plasma is investigated. The standard perturbative approach leads to the Korteweg–de Vries (KdV) equation with a linear Landau damping term for the dynamics of the low-frequency nonlinear wave. Landau damping causes the wave amplitude to decay with time and the dust charge variation enhances the damping rate.

### Keywords

Landau damping Dusty plasma Korteweg–de Vries equation Dust acoustic wave## Introduction

The Landau damping is a physical phenomenon which is related to the resonant particles (the particles whose velocity is nearly equal to the wave phase velocity) [1, 2]. The resonant particles may include both trapped and un-trapped particles. The usual ion acoustic wave in electron-ion plasma suffers Landau damping due to these resonant particles [1, 2, 3]. However, the presence of charged dust grains in a plasma gives rise to very low-frequency new mode (~10–15 Hz), called dust acoustic wave (DAW) [4, 5, 6, 7, 8], where the inertia is provided by the charged and massive dust grains. In the linear theory, it has already been seen that this mode also suffers Landau damping due to the resonant wave–particle interactions [9, 10]. Another well known non-Landau damping mechanism in a dusty plasma is due to the dust charge variations in the presence of waves [11, 20]. Actually, dust grains immersed in a plasma can exhibit self-consistent charge variations in response to the surrounding plasma oscillations and thus become a time-dependent dynamical variable which causes an anomalous dissipation in a dusty plasma.

In the nonlinear theory, the linear electron Landau damping effects on ion acoustic solitary wave have been investigated in an electron-ion plasma neglecting the particle trapping effect under the assumption that the particle trapping time is much larger than the Landau damping time [1, 13]. It has been shown that the solitary wave amplitude decays with time due to the linear electron Landau damping [13]. Later, theoretical [14, 15] and experimental [16] results show similar behavior. The wave–particle interactions also cause the oscillations in the tail of the solitary waves in which the shape of the tail depends on the strength of the Landau damping [17]. Recent experimental observation also predicts the formation of ion acoustic shock wave due to the Landau damping induced dissipation [18]. However, no study of nonlinear DAW is carried out including Landau damping in a dusty plasma. In this paper, the effect of linear ion Landau damping on dust acoustic solitary wave has been investigated neglecting the particle trapping effect. The instantaneous dust charge variation effects are also incorporated.

The manuscript is organized in the following manner. Formulation of the problem including the physical assumptions and basic equations is described in Sect. 2. The Korteweg–de Vries (KdV) equation with linear damping is derived using the reductive perturbation technique in Sect. 3. The analytical solution and the effect of Landau damping on the solitary wave solution are investigated in Sect. 4. The results of the present investigation are summarized in Sect. 5.

## Formulation of the problem and the basic equations

A fully ionized, un-magnetized plasma consisting of electrons, ions, and negatively charged dust grains are considered. The plasma is assumed to be in its equilibrium state at \(-\infty\), where electrostatic potential \(\phi =0\), electron number density \(n_{\text{e}}=n_{\text{e0}}\), ion density \(n_{\text{i}}=n_{\text{i}0}\), dust density \(n_{\text{d}}=n_{\text{d}0}\), and dust charge \(q_{\text{d}}=-z_{\text{d}e}\), so that the quasi-neutrality condition \(n_{\text{e}0}+z_{\text{d}}n_{\text{d}0}=n_{\text{i}0}\) is satisfied, where \(z_{\text{d}}\) is the number of electrons residing on the dust grains, \(n_{{ j }}\) is the number density of the \(j{\text{th }}\, (e=\text{ electron }, i=\text{ ion }\; \text{ and }\; d=\text{ dust } \text{ grain})\) species, and \(n_{\text{ j }0}\) be its equilibrium value.

The charge on the dust grain varies continuously in space (*x*) and time (*t*). The temperature of dust grain is very low compared to that of electrons \((T_{\text{e}})\) and ions (\(T_{\text{i}})\). Therefore, the dust grains are effectively cold with respect to the electrons and ions. The dust grains are moving with fluid velocity *U*. It is convenient to express all the variables in non-dimensional form before going to the details of the basic formalism of the problem. For this purpose, let us introduce the following normalization: \(\Phi =e\phi /T_{\text{i}}\), \(N=n_{\text{d}}/n_{\text{d}0}\), \(N_{\text{e}}=n_{\text{e}}/n_{\text{e}0}\), \(N_{\text{i}}=n_{\text{i}}/n_{\text{i}0}\), \(\bar{x}=x/\lambda _{\text{D}}\), \(\bar{t}=\omega _{\text{ pd }}t\), \(\bar{U}=U/C_{\text{ s }}\) and \(\bar{q_{\text{d}}}=q_{\text{d}}/z_{\text{ de }}=-1+\Delta Q\), and \(\Delta Q\) is the fluctuating dust charge. Here, \(\omega _{\text{ pd }}=(z_{\text{d}}^2e^2 n_{\text{d}0}/\epsilon _0 m_{\text{d}})^{1/2}\) is the dust plasma frequency, \(\lambda _{\text{D}}=(\epsilon _0 T_{\text{i}}/n_{\text{i}0}e^2)^{1/2}\) is the plasma Debye length, \(C_{\text{ s }}=(z_{\text{d}}T_{\text{i}}/m_{\text{d}})^{1/2}\) is the dust acoustic speed, \(\delta _{\text{i}}=n_{\text{i}0}/n_{\text{e}0}\), \(\sigma =T_{\text{i}}/T_{\text{e}}\), and \(z=z_{\text{ de }}^2/4\pi \epsilon _0 r_{\text{d}} T_{\text{e}}\) are the dimensionless dusty plasma parameters (the ratio of the electrostatic energy of a dust grain of radius \(r_{\text{d}}\) to the electron thermal energy). Hereafter, we will be using these new variables and remove all the bars for simplicity of notations.

We assume that \(\delta _{\text{i}}\left( m_{\text{i}}/m_{\text{d}}\right) ^{1/2} \gg \sigma ^{3/2} \left( m_{\text{e}}/m_{\text{i}}\right) ^{1/2}\), so that the electron Landau damping effect is neglected. The dust Landau damping effect is also neglected as the dust thermal velocity is much smaller than the wave phase velocity. Moreover, we are interested to study the low-frequency nonlinear DAW and, therefore, we neglect the inertia of the electrons compared to the dust grains. On this slow time scale, the electrons are in local thermodynamic equilibrium and their densities are modeled by the Boltzmann distribution: \(n_{\text{e}}=n_{\text{e}0}\exp (\sigma \Phi ).\)

*f*(normalized) satisfies the following Vlasov–Boltzmann equation:

*V*is normalized in units of ion thermal velocity \(V_{\text{ ti }}=\left( T_{\text{i}}/m_{\text{i}}\right) ^{1/2}\) and velocity distribution function

*f*is normalized by \(V_{\text{ ti }}/n_{\text{ i0 }}\). It is to be noted that when the plasma is in thermodynamical equilibrium, the velocity distribution of the ion is given by the following Maxwellian distribution:

## Korteweg–de Vries equation with Landau damping

## Landau damping effect on dust acoustic solitary wave

*A*is the amplitude of the solitary wave, \(3A/\alpha\) is the solitary wave velocity, and \(\left( 12\beta /A\alpha \right) ^{1/2}\) is the spatial width of the solitary wave.

## Conclusions

In this paper, we have investigated the effects of ion Landau damping on nonlinear dust acoustic wave. It is shown that the nonlinear wave is governed by a modified form of KdV equation [see Eq. (28)]. In the presence of Landau damping, approximate analytical solutions reveal that the wave amplitude decays algebraically with time. To understand the feature, the wave amplitude modulations with time \(\tau\) [see Eqs. (38) and (39)] are shown graphically for different \(\sigma\) and \(\delta _{\text{i}}\) in Figs. 1 and 2 for \(K^+\) ion and electron plasma. Figures 1 and 2 show that wave amplitude decreases with time \(\tau\) for any fixed value of \(\delta _{\text{i}}\) and \(\sigma.\) However for any fixed time \(\tau,\) the amplitude increases with the increase of ion-electron temperature ratio (\(\sigma\)) and ion-electron density ratio (\(\delta _{\text{i}}\)). In addition, the Landau damping rate (\(\gamma _{\text{L}}\)) decreases with the increase of ion-electron temperature ratio (\(\sigma\)), as shown in Fig. 3.

## Notes

### Acknowledgements

The author (A.S) would like to thank Prof. Samiran Ghosh of Department of Applied Mathematics, University of Calcutta for fruitful discussion.

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