2. Basic equations

Let us consider a one-dimensional collisional unmagnetized dusty plasma consisting of a mixture of warm ion fluid, Boltzmann-distributed electrons and stationary dust grains. We assume that (i) the size of the particles is much smaller than the electron Debye length and the distance between grains; (ii) the dust particles are stationary in the ion acoustic time scale; (iii) the heat-transfer process which might affect the propagation of ion-acoustic waves is neglected; (iv) the number densities of electrons and ions are assumed to be constant due to the competition of the processes of electron and ion recombination on the dust particles in the absence of perturbation; (v) generation of ions from dust particles in the ionization source model is neglected (Samsonov & Goree Reference Samsonov and Goree1999). In such a dusty plasma system, the dynamics of a low-frequency electrostatic perturbation can be governed by the following set of normalized equations (Goree et al. Reference Goree, Morfill, Tsytovich and Vladimirov1999; Vladimirov, Ostrikov & Yu Reference Vladimirov, Ostrikov and Yu1999; Cramer & Vladimirov Reference Cramer and Vladimirov2001; Popel et al. Reference Popel, Golub, Losseva, Ivlev, Khrapak and Morfill2003):

where $n_{i}$ is the ion number density normalized by its equilibrium value $n_{i0}$ , $u_{i}$ is the ion fluid speed normalized to the ion-acoustic speed $C_{i}=(k_{B}T_{e}/m_{i})^{1/2}$ and ${\it\phi}$ is the electrostatic wave potential normalized to $k_{B}T_{e}/e$ . Here ${\it\sigma}=T_{i}/T_{e}$ , with $T_{i}$ $(T_{e})$ being the ion (electron) temperature and ${\it\mu}=n_{e0}/n_{i0}=1-Z_{d}n_{d0}/n_{i0}$ stands for the fractional number density of the electrons relative to that of the ion number density. Also, ${\it\upsilon}_{r}$ is the frequency of the ion recombination on dust particles, ${\it\upsilon}_{i}$ is the plasma ionization frequency which increases exponentially with electron temperature $T_{e}$ and ${\it\upsilon}_{i}^{eff}$ is the effective frequency characterizing a loss in the ion momentum due to recombination on dust particles and Coulomb elastic collisions between ions and dust grains. The time and space variables are given in the units of the ion-plasma period ${\it\omega}_{pi}^{-1}=({\it\varepsilon}_{0}m_{i}/n_{i0}e^{2})^{1/2}$ and the Debye length ${\it\lambda}_{De}=({\it\varepsilon}_{0}k_{B}T_{e}/n_{i0}e^{2})^{1/2}$ , respectively. Equation (2.1) shows that the continuity equation for the ions in the plasma model under consideration takes into account plasma production where the main mechanism of ionization is electron impact ionization, as shown in Cramer & Vladimirov (Reference Cramer and Vladimirov2001). In laboratory and space dusty plasma an important traditional approach is the case that the source of ionization is due to the conventional electron impact ionization. Thus, the ionization rate is proportional to the electron density (Goree et al. Reference Goree, Morfill, Tsytovich and Vladimirov1999; Vladimirov et al. Reference Vladimirov, Ostrikov and Yu1999; Popel et al. Reference Popel, Golub, Losseva, Ivlev, Khrapak and Morfill2003). On the other hand, in the experimental plasma, when the electron mean free paths are so long that the neutrals are ionized presumably in collisions with the wall, the ionization source term in the evolutionary equation for the ion density should be dependent on the ion density. In the experiments of Nakamura et al. (Reference Nakamura, Bailung and Shukla1999) and Nakamura & Sarma (Reference Nakamura and Sarma2001) with the partial pressure of a neutral argon gas $(3{-}6)\times 10^{-4}$ torr and the electron temperature $T_{e}=0.1~\text{eV}$ , the electron–neutral collisions are on the order of $10^{4}~\text{cm}$ , which is much larger than the length of the device (90 cm) and its diameter (40 cm). Hence, the ionization term is independent of the electron density.

The electrons are assumed to be close to a Maxwellian equilibrium. The corresponding densities are $n_{e}={\it\mu}\text{e}^{{\it\phi}}$ . It should be noted that in dusty plasma, microparticles continuously absorb electrons and ions, and there exist plasma particle fluxes from the regions where there are no dust grains. In this case, due to the dust particle charging, the effects of ion density and ion momentum dissipation should be included. We ignore the ion-neutral collisions, which would exert a drag force on the ion species. This is justified by consideration of the ion time scales, so that the dust grains can be considered as stationary (Popel et al. Reference Popel, Golub, Losseva, Ivlev, Khrapak and Morfill2003).

3. Derivation of modified NLS equation

In order to investigate the modulation of a nonlinear dust-ion-acoustic wave and associated stability/instability profiles of this wave in the plasma model under consideration, we use the standard multiple-perturbation technique (Taniuti & Yajima Reference Taniuti and Yajima1969; Jeffery & Kawahara Reference Jeffery and Kawahara1982).

The nonlinear amplitude modulation mechanism generates other harmonics by increasing the carrier-wave amplitude, where the dynamics of the slowly varying amplitude (envelope) can be studied near the equilibrium state. Let us define $S=(n,u,{\it\phi})$ , which describes the state of the system at a given position $x$ and a time $t$ . We assume that $S$ takes the form of a modulated wave packet, i.e. the composition of a fast carrier wave with a slow variation in amplitude. In such a condition, we consider a small deviation from the equilibrium state, and expand the dynamical variables as

Here, $k$ and ${\it\omega}$ are real variables which represent the frequency and wavenumber of the electrostatic wave packet, which are normalized by ${\it\omega}_{pi}$ and ${\it\lambda}_{De}^{-1}$ , respectively, and ${\it\varepsilon}$ is a small dimensionless parameter measuring the weakness of nonlinearity. The variable $S^{(0)}$ denotes the equilibrium state, which is given by $n^{(0)}=1$ and $u^{(0)}={\it\varphi}^{(0)}=0$ . All the perturbed states are related to the fast scales due to the phase term $(kx-{\it\omega}t)$ , whereas the slow scales only enter the $l$ th-harmonic perturbation $S_{l}^{(n)}$ . The latter states satisfy the reality condition $S_{-l}^{(m)}=S_{l}^{(m)^{\ast }}$ , where the asterisk denotes the complex conjugate. Analysis of modes by the multiple-time scale implies that the wave amplitudes are allowed to depend on the new stretched coordinates as $X_{m}={\it\varepsilon}^{m}x$ and $T_{m}={\it\varepsilon}^{m}t$ , where $m=1,2,\ldots$  . It should be noted that the above stretched variables and the expanded states in (3.2) introduce two kinds of coordinate systems. These are $(x,t)$ , in which a rapid variation of the carrier wave exists, and $(X_{m},T_{m})$ , a coordinate system with a slowly modulated wave packet. Assuming that the ion recombination frequency, the plasma ionization frequency and the effective frequency are smaller than the ion-plasma frequency, i.e. weak dissipation, we assume that ${\it\upsilon}_{r}\approx {\it\varepsilon}^{2}{\it\upsilon}_{r0}$ , ${\it\upsilon}_{i}\approx {\it\varepsilon}^{2}{\it\upsilon}_{i0}$ and ${\it\upsilon}_{i}^{eff}\approx {\it\varepsilon}^{2}{\it\upsilon}_{i0}^{eff}$ , and where subscript 0 denotes unperturbed value and is considered to be constant. For a typical dusty argon plasma (as shown in Vladimirov & Ostrikov (Reference Vladimirov and Ostrikov2004)), when $T_{e}\sim 10~\text{eV}$ , $T_{i}\sim 1~\text{eV}$ , $a\sim 5~{\rm\mu}\text{m}$ ( $a$ is the average dust radius), $n_{e}\sim 5\times 10^{5}~\text{cm}^{-3}$ and ${\it\mu}=0.1$ , the inequality ${\it\upsilon}\ll 1$ is satisfied, where ${\it\upsilon}=({\it\upsilon}_{i},{\it\upsilon}_{r},{\it\upsilon}_{i}^{eff})$ indicate the dissipative frequencies. In fact, the dissipative frequencies were obtained as ${\it\upsilon}\sim 3\times 10^{7}{-}5\times 10^{18}~\text{s}^{-1}$ and ${\it\omega}\sim 10^{10}$ , which validate our assumption.

We now proceed by applying the Taylor series expansion to electron and ion densities in (2.3) and then substituting the expressions (3.1) and (3.2) along with stretching coordinates into the basic equations (2.1)–(2.3). Then, by collecting the terms in the different powers of ${\it\varepsilon}$ , we can obtain $n$ th-order reduced equations with $l$ th-harmonic contributions. The first-order $(m=1)$ equation with $l=1$ can lead to the dispersion relation for the first-harmonic amplitudes. Because of considering the modulation of a plane wave, the dependent quantities are all set to zero except for $l=\pm 1$ . In this case, the dispersion relation for the DIA wave can be found as

which clearly shows the dependence of the phase velocity $(V_{ph}={\it\omega}/k)$ of DIA waves on the temperature parameter (via ${\it\sigma}$ ) and on the fractional number density of the electrons relative to that of the ion number density (via ${\it\mu}$ ). We find that the dispersion relation (3.3) has a similar form to the one derived by Kako & Hasegawa (Reference Kako and Hasegawa1976) for ordinary plasmas with cold ions and isothermal electrons if one replaces ${\it\mu}=1$ and ignores the ion temperature. This also coincides exactly with the equation derived in Kourakis & Shukla (Reference Kourakis and Shukla2003, Reference Kourakis and Shukla2004) upon omitting the ion temperature. It is obvious that the phase velocity of the wave increases with the increase in the ion temperature and decreases with equilibrium electron density. Also, we can see from the dispersion relation (3.5) that the phase velocity is not affected by the ionization, the recombination and the ion loss. For all parameter values of interest, ${\it\omega}$ is positive. Thus, the linear DIA wave is always stable. Besides the dispersion relation, the set of equations for $m=1$ and $l=1$ gives us the first-harmonic quantities of density and fluid velocity in terms of ${\it\varphi}_{1}^{(1)}$ as

From the second-order expressions for the first harmonic, i.e. for $m=2$ and $l=1$ , we obtain an equation in which the coefficient of ${\it\varphi}_{1}^{(2)}$ vanishes by the dispersion equation (3.3), and the coefficient of $\partial {\it\varphi}_{1}^{(1)}/\partial X_{1}$ , after equating to zero, provides the following solvability condition:

Here, $V_{g}$ is the group velocity of DIA waves, and it is easy to verify that $V_{g}=\partial {\it\omega}/\partial k$ . The compatibility constraint displays that all the first-order harmonic amplitudes should be functions of the moving coordinate ${\it\xi}=X_{1}-V_{g}T_{1}$ , which represents an envelope wave moving at the group velocity. The second-order harmonic modes of the carrier wave can also be given by

Using the same procedure, the second-harmonic modes $(m=2,l=2)$ arising from the nonlinear self-interaction of the carrier waves are found to be proportional to the square of the corresponding first-order elements, e.g. in terms of $[{\it\varphi}_{1}^{(1)}]^{2}$ as where

The zeroth-harmonic modes appear due to the self-interaction of the modulated carrier wave. Its expression cannot be determined completely within the second order, and we should consider the third-order equation. Thus, the $l=0$ component of the third-order $(m=3)$ part of the reduced equations determines the following second-order quantities in the zeroth harmonic:

where

Finally, we obtain the $l=1$ component of the third-order $(m=3)$ part of the reduced equations,

Substituting the above reduced expressions (3.4a )–(3.10) into (3.11a–c ), and eliminating the term ${\it\phi}_{1}^{(2)}$ , we can obtain the following modified NLS equation for the propagation of modulated DIA wave packets in the weakly dissipative dusty plasma:

where ${\it\phi}={\it\varphi}_{1}^{(1)}$ and the slow variables ${\it\xi}=X_{1}-V_{g}T_{1}$ and ${\it\tau}=T_{2}$ . In the above equation, the coefficients $P$ , $Q$ and $R$ are given by

In the aforementioned equations, the dispersion coefficient $P$ (which is related to the curvature of the dispersion curve) and the nonlinearity coefficient $Q$ are functions of the wavenumber $k$ , ion temperature and electron concentration. The additional term $-\text{i}R{\it\phi}$ in the right-hand side of (3.12) exists in the modified NLS equation due to the effects of ionization, recombination and ion loss (weak dissipation). It is worth noting that if the particle loss and ionization are ignored in our analysis, i.e.  $R=0$ (in collisionless dusty plasma), the modified NLS equation will be reduced to the ordinary NLS equation. We can see from (2.13a,b ) that the nonlinear term $Q$ and the dispersion term $P$ are not affected by ion–dust collisions. From the expression of the coefficient $R$ , we observe that the dissipative term can strongly affect the modulational instability. In the next section, we will show that the collisions, the ion loss and the ionization are responsible for the modification of the stability and instability regions.

4. Modulational instability region

Now, we focus on the existence region of the modulational instability and the stability/instability solutions of the modified NLS equation (3.12). In order to do that, let us first find a constraint on the occurrence of the modulation instability of (3.12). Following the standard instability analysis, we separate the perturbed amplitude ${\it\phi}$ into two parts as

where ${\it\phi}_{0}$ is the amplitude of the pump carrier waves, which is much larger than the perturbation, i.e.  ${\it\phi}_{0}\gg |{\it\delta}{\it\phi}|$ , and ${\rm\Delta}({\it\tau})=-Q{\it\phi}_{0}^{2}\exp (-2R{\it\tau})$ is the nonlinear frequency shift. Substituting (4.1) into (3.12), after linearizing, we obtain

Now, separating the real and imaginary parts of the amplitude disturbance, as shown in Xue (Reference Xue2003), we obtain the following nonlinear dispersion relation for the small-amplitude modulation of the DIA wave packet:

where $K_{c}^{2}({\it\tau})=(2Q|{\it\phi}_{0}|^{2}/P)\exp (-2R{\it\tau})$ is the critical modulational wavenumber, and $K$ and ${\it\Omega}$ are the perturbation wavenumber and frequency, respectively. If one neglects the ion–dust collisions, i.e.  $R=0$ , the dispersion relation reduces to ${\it\Omega}^{2}=P^{2}K^{2}(K^{2}-2Q|{\it\phi}_{0}|^{2}/P)$ , which is the well-known modulational instability of the electrostatic wave packets in plasma. It is obvious that the dissipative term $R$ strongly affects the nonlinear dispersion relation and critical wavenumber. Also, the coefficients of the dispersion term $P$ , nonlinear term $Q$ and damping term $R$ are related to a number of physical parameters such as ion temperature, population of dust particles and ion–dust collisions. Thus, these parameters are expected to significantly change the stability characteristics of modulated DIA waves. On the other hand, (4.3) shows that the modulation instability sets in for a wavenumber satisfying $K<K_{c}$ , or for all wavelengths above the threshold ${\it\lambda}_{c}=2{\rm\pi}/K_{c}$ , provided the frequency of the perturbation is negative. If the nonlinear saturation occurs, the formation of a nonlinear envelope localized structure may be expected. Otherwise, i.e.  $K>K_{c}$ , the wave is stable under the modulation. In the case that the right-hand side of (4.3) is negative, the local instability growth rate is given by ${\it\Gamma}=PK^{2}(K_{c}^{2}({\it\tau})/K^{2}-1)^{1/2}$ . Contrary to what has been found in the collisionless dusty plasma (Saini & Kourakis Reference Saini and Kourakis2008), as $K_{c}$ decreases exponentially with time ${\it\tau}$ , the instability growth rate decreases with time when the instability condition is satisfied. Therefore, the conditions based on the negative sign of $PQ$ , i.e.  $K^{2}<K_{c}^{2}({\it\tau})$ , yield a period ${\it\tau}>{\it\tau}_{max}=1/2R\ln 2Q|{\it\phi}_{0}|^{2}/K^{2}P$ , when the modulational instability will cease. This means that when the time ${\it\tau}$ exceeds a maximum one ( ${\it\tau}_{max}$ ), the relation $K^{2}<K_{c}^{2}$ may be destroyed even if the inequality $PQ>0$ is satisfied. Thus, the permitted maximum time of the modulational instability may be estimated by ${\it\tau}_{max}<1/R$ . This modulational instability period arises due to the combined effects of ionization, recombination and ion loss. The critical wavenumber and modulation instability region can also be affected strongly due to the collisional effects between particles. If we neglect the dissipative term, $R$ , the critical modulation wavenumber will be independent of time and then there is no constraint on time for the modulational instability region. Physically, in this case, there are two types of stationary envelope soliton from the NLS equation. These are: bright envelope soliton which is modulationally unstable and dark envelope soliton which is always stable. For the two cases of envelope solitons, if $R\rightarrow 0$ , the width is proportional to $|P|$ and the soliton amplitude is inversely proportional to $|Q|$ (Fedele & Schamel Reference Fedele and Schamel2002; Fedele, Schamel & Shukla Reference Fedele, Schamel and Shukla2002; Kourakis & Shukla Reference Kourakis and Shukla2005). In our cases, the stability and instability regions are dependent on time ${\it\tau}$ because of the existence of the dissipative term.

Figure 1.

Stability/Instability region in ${\it\upsilon}_{i0}^{eff}{-}{\it\sigma}$ plane at fixed values of ${\it\phi}_{0}=0.01$ , ${\it\tau}=5$ , ${\it\upsilon}_{i0}=0.05$ , ${\it\mu}=0.2$ , $K=0.01$ and $k=1$ for different values of the frequency of the ion recombination on dust particles with (a) ${\it\upsilon}_{r0}=0.5$ and (b) ${\it\upsilon}_{r0}=0.7$ .

Figure 2.

Stability/Instability region in ${\it\upsilon}_{r0}{-}{\it\sigma}$ plane at fixed values of ${\it\phi}_{0}=0.01$ , ${\it\tau}=4$ , ${\it\upsilon}_{i0}=0.05$ , ${\it\mu}=0.2$ , $K=0.01$ and $k=1$ for different values of the effective frequency characterizing a loss in the ion momentum with (a) ${\it\upsilon}_{i0}^{eff}=0.8$ and (b) ${\it\upsilon}_{i0}^{eff}=0.9$ .

Figure 3.

Stability/Instability region in ${\it\upsilon}_{i0}^{eff}{-}{\it\sigma}$ plane at fixed values of ${\it\phi}_{0}=0.01$ , ${\it\tau}=5$ , ${\it\upsilon}_{r0}=0.4$ , ${\it\mu}=0.3$ , $K=0.01$ and $k=1$ for different values of the plasma ionization frequency with (a) ${\it\upsilon}_{i0}=0.02$ and (b) ${\it\upsilon}_{i0}=0.09$ .

Now, we are in a position to carry out some numerical work to investigate the DIA envelope soliton existence domain as well as the instability region. In figures 1–3, the stable $({\it\Omega}^{2}>0)$ and unstable $({\it\Omega}^{2}<0)$ regions are shown for different values of the plasma parameters. The modulational instability regions in the space of parameters ${\it\upsilon}_{i0}^{eff}$ and ${\it\sigma}$ are illustrated in figure 1, for different values of ion recombination frequency (a) ${\it\upsilon}_{r0}=0.5$ , (b) ${\it\upsilon}_{r0}=0.7$ . We observe that the permitted instability regions decrease as the frequency of ion recombination on dust becomes large. This is reasonable because of the expression of the dissipative term $R$ in (3.13c ); the contribution of the ion recombination frequency is positive. In such a condition, the critical modulational wavenumber reduces, which leads to a wide domain of the stability, as well shrinking the modulational instability region. This is an expected behaviour because the permitted maximum time of the modulational instability, ${\it\tau}<1/R$ , decreases. Indeed, the instability growth will cease when the frequency of ion recombination on dust particles increases. In figures 2 and 3, we have investigated the effects of ${\it\upsilon}_{i0}^{eff}$ and ${\it\upsilon}_{i0}$ on the modulational instability regions. We observe that occurrence of stable and unstable regions is significantly related to the values of ${\it\upsilon}_{i0}^{eff}$ and ${\it\upsilon}_{i0}$ . The graphs display that the permitted instability regions become wider as the effective frequency and plasma ionization frequency increase. These results can also be explained by considering the expression of $R$ . It is obvious that the contributions of ${\it\upsilon}_{i0}^{eff}$ and ${\it\upsilon}_{i0}$ in (3.13c ) are negative; thus, increasing these parameters leads to an increase in the critical modulation wavenumber $K_{c}$ . Then, one can conclude that the modulational instability region will increase. However, the combined action of all the dissipative mechanisms due to the collisions between ions and dust reveals the following criterion for the maximum time for occurrence of the modulational instability:

This shows that the carrier wave amplitude instability region decreases when the ion recombination on dust particles is the dominant dissipative process. Otherwise, the stability domain decreases when the ion loss and ionization are the dominant dissipative processes. This confirms what has been found in figures 1–3.

Figure 4 displays the effect of ion temperature on the growth rate ${\it\Gamma}$ of the modulational instability. We observe that the growth rate is significantly affected by the ion temperature in the weakly dissipative plasma. An increase of ion temperature leads to a decrease in the growth rate of modulational instability. In figure 5, we study the variation of modulation instability growth rate with the electron population for a fixed value of the carrier wavenumber. We observe that as the electron concentration increases in the weakly dissipative plasmas, the growth rate of the modulational instability decreases. This means that the instability growth rate as well as the instability window appears to increase in a plasma containing a higher concentration of dust particles. This completely confirms results obtained in Kourakis & Shukla (Reference Kourakis and Shukla2004) for a dusty plasma without source and sink.

Figure 4.

Variation in growth rate of modulational instability with $K$ at fixed values of ${\it\phi}_{0}=0.01$ , $k=1$ , ${\it\mu}=0.01$ , ${\it\tau}=0.6$ , ${\it\upsilon}_{i0}=0.7$ , ${\it\upsilon}_{r0}=0.01$ and ${\it\upsilon}_{i0}^{eff}=0.7$ . The solid curve corresponds to ${\it\sigma}=0.015$ and dashed curve corresponds to ${\it\sigma}=0.02$ .

Figure 5.

Variation in growth rate of modulational instability with $K$ at fixed values of ${\it\phi}_{0}=0.01$ , $k=1$ , ${\it\sigma}=0.02$ , ${\it\tau}=0.6$ , ${\it\upsilon}_{r0}=0.01$ , ${\it\upsilon}_{i0}=0.5$ and ${\it\upsilon}_{i0}^{eff}=0.8$ . The solid curve corresponds to ${\it\mu}=0.01$ and dashed curve corresponds to ${\it\mu}=0.02$ .