# Feedback based Mobility Control Algorithm for Maximizing Node Coverage by
Drone Base Stations

Aniq Ur Rahman, Agnivesh Adhikari
National Institute of Technology Durgapur, India

###### Abstract

Drone base stations (DBSs) have recently gained wide popularity as a possible
solution to provide wireless connectivity in a variety of scenarios, for
example, in inaccessible terrains such as connectivity over vast areas of a
water body or in rural areas where the physical deployment of base stations is
not feasible at the moment, also in the case of terrestrial infrastructure
failure where DBSs can be rapidly deployed to re-establish communication
channel. In this paper we propose an algorithm for controlling the motion of
the DBSs which maximizes the number of DBS to mobile ground node connections.
The overlap extent between the drones is limited to reduce the count of
redundant connections. The overall approach aims at minimizing the number of
drones required to be deployed in a given region by maximizing connectivity
per drone.

###### Index Terms:

Drone Base Station, UAV, mobility, macro hotspot, deployment

##  I Introduction

Unmanned Aerial Vehicles are widely used for reconnaissance operations [1] in
disaster struck regions. In such cases, emergency base stations are required
to serve as temporary replacement of the damaged communication infrastructure.
A prospective solution for establishing a wireless recovery network in
disaster affected zones is by utilizing airborne base stations. With
developments in communication technology, providing adequate network coverage
in rural areas becomes a topic of key interest as most of the areas in
developing nations are rural and do not have proper network coverage.
Communication over water bodies or dense forests is not possible due to the
non feasibility of deployment of terrestrial base stations. This is the
motivation for our work, that is, to provide an efficient, scalable and robust
solution for the deployment of Drone Base Stations (DBSs) as a means to
provide wireless connectivity in inaccessible terrains. The word drone is used
synonymously with UAV throughout the paper.

Figure 1: The depiction of considered scenario

The utility of UAVs is undoubtedly high. This is established by the fact that
the Tech Giants are using drones for their upcoming technologies. Amazon is
using drones [2] to deliver packages to its customers under the service name
Amazon Prime Air. Facebook’s Aquila [3] [4] is an experimental drone to
function as an atmospheric satellite and provide internet access to remote
areas. The drones are solar powered to account for the high energy
requirement. Similar to Facebook’s aim for Aquila, Google is developing its
Project Loon [3] [5] [6] which aims at providing high speed internet to rural
and remote areas with the help of a network of balloons equipped with solar-
powered transceivers.

Fotouhi et al. [7] proposed a mobility control algorithm for the drone base
stations. The algorithm tries to decrease the distance between the ground
nodes and DBS in order to increase the probably of a line of sight connection.
Lei Wang et al. [8] find the optimum height of the DBS to minimize the average
transmission power to save the precious resource of the drone which is energy.
The authors in [9] assumed the DBSs transmits at full power and formulated the
3D DBSs placement problem as a quadratically-constrained mixed integer non-
linear problem. Kalantari et al. [10], developed a heuristic algorithm based
on particle swarm optimization. The algorithm suboptimally finds the minimum
number of DBSs and their locations to serve a particular region. Mozaffari et
al. [11] optimized the DBS altitude that results in maximum coverage region
and minimum transmit power for two cases, a single DBS and two DBSs. In [12],
they developed a method to deploy multiple DBSs based on circle packing.

In this paper we propose a feedback-based mobility control algorithm for drone
base stations to maximize connectivity with the mobile ground nodes exhibiting
random walk. The change in the count of ground nodes a DBS is connected to is
the feedback in our algorithm. The drone alters its position so as to maintain
connectivity if it falls below a certain threshold. We define performance
metrics based on the number of ground nodes a DBS serves. These performance
metrics are compared with the random walk mobility of DBSs.

The paper is organized as follows. In Section 2, we discuss the equations
governing the functions which are used in the algorithm described in Section 3
followed by Results and Analysis in Section 4. Section 5 presents the
concluding remarks and some possible future works.

##  II System Model

In this work, the macro hotspot scenario has been considered, similar to [7],
in which both the drones and the ground nodes are mobile. Due to the mobility
of the ground nodes, we can never estimate the positions of all the ground
nodes which makes the problem even more realistic, in which we do not have a
map depicting the density of ground nodes spread out in the considered region.

Figure 2: Top-view of the system environment

We assume that $n_{G}$ users (ground nodes) are uniformly distributed in the
Region of Interest $(L\times L)$. The drones have been placed at a fixed
altitude above the ground and are allowed movement only in the two dimensional
plane. Each mobile node connects to its nearest drone base station. Figure 2
shows the topview of the DBS and user setup. Each DBS has been assumed to have
a coverage area or a detection region equivalent to circular geographical
region having radius $R$.

TABLE I: Nomenclature Notation | Description
---|---
$n_{D}$ | Number of DBSs deployed
$n_{G}$ | Number of ground nodes
$L$ | Side length of the area of interest
$R$ | Radius of the DBS connectivity range
$\bar{X},\bar{Y}$ | Centroid of all the groundnodes connected to a DBS
$w_{x},w_{y}$ | Destination coordinates for a DBS
$R_{o}$ | Radius set for limiting overlap
$M$ | Instantaneous connectivity
$M^{\prime}$ | Previous connectivity
$M_{max}$ | Personal best Connectivity
$\phi$ | Threshold

Simulations have been performed and results have been obtained considering
that the mobile users or nodes are in a state of random walk mobility. The
drone changes its position depending on centroids of the connected nodes and
the change in the number of nodes it is connected to in the current state
w.r.t. the previous state (feedback) and the threshold.

##  III Algorithm

The drones first maximize the distance between them, if its falls below
minimum value of $R_{o}$, by each of them moving away through the line of
sight. For a pair of drones with coordinates $(x_{i},y_{i})$ and
$(x_{j},y_{j})$ we first find the centroid $(c_{x},c_{y})$ such that
$c_{x}=(x_{i}+x_{j})/2,c_{y}=(y_{i}+y_{j})/2$ and then shift both drones a
distance $R_{o}/2$ away from the centroid.

$\displaystyle s_{j}\leftarrow
c_{s}+\dfrac{s_{j}-c_{s}}{\sqrt{(x_{j}-c_{x})^{2}+(y_{j}-c_{y})^{2}}}R_{o}/2;\quad
s\in\\{x,y\\}$ (1)

Then the drones check how many ground nodes they are serving instantaneously.
For each drone, we have a set $\mathcal{C}=\\{g_{1},g_{2},g_{3},...g_{M}\\}$
of ground nodes $g_{i}(x_{i},y_{i})$ which fall within its communication range
$R$. Based on the xy coordinates of all the ground nodes connected to a drone
base station, it calculates the centroid $(\bar{X},\bar{Y})$ as

$\bar{X}=\dfrac{\sum_{i=1}^{M}x_{i}}{M};\quad\bar{Y}=\dfrac{\sum_{i=1}^{M}y_{i}}{M};\quad
g_{i}(x_{i},y_{i})\in\mathcal{C},\quad|\mathcal{C}|=M$ (2)

and then calculates the destination $(w_{x},w_{y})$ as

$\displaystyle w_{x}=\bar{X}+(-1)^{p}R/2;\quad w_{y}=\bar{Y}+(-1)^{p}R/2;$ (3)

where $R$ is the connectivity range of each drone and $p\in\\{0,1\\}$ having
uniform probability of occurrence. We basically try to increase the distance
between the drone and the centroid of connected ground nodes so that it may
get connected to ground notes lying outside. This kind of approach is helpful
because in order to maximize connections, we try to retain the current number
of connections and then by maintaining a safe distance so as not to lose
connection, we fetch more on the way.

After this, the drone proceeds to move towards its destination and after a
certain time step performs a check. The number of ground nodes being served
$(M)$ is counted and checked against the pervious value $(M^{\prime})$ and
threshold $\phi$. If $M^{\prime}$ exceeds $M$ and the threshold or when $M$
exceeds $M^{\prime}$ by a defined vale called $Loss$ then the drone base
station reverts its position.

Algorithm 1 Drone Mobility Algorithm

1: Start

2: Minimize Overlap between drones

3: Find Connections

4: Set Centroid

5: Set Destination

6: Set Velocity

7: Update Position

8: if $M^{\prime}>\phi$ & $M<M^{\prime}$ then

9: Revert Position

10: Find Connections

11: if $M<M_{max}$ then

12: $M_{max}=\max(M,\phi)$

13: end if

14: else

15: if $M<M^{\prime}-Loss$ then

16: Revert Position

17: end if

18: end if

19: Go to Step 2

##  IV Results and Analysis

The main focus of this algorithm is the maximization of connectivity (M) of
each DBS. To show the performance of our algorithm as claimed, we plot
connectivity in figure 3 The solid black line represents the mean connectivity
of all the drones and the thin color lines are the connectivities of
individual drones. It is evident, the value of $M$ increases with time. The
deep crest around $t=400s$ highlights the feedback mechanism present in the
algorithm based on the values of $M$ and $M^{\prime}$.

Figure 3: Connectivity (M) of all DBS plotted against time. (Simulation
Parameters: $n_{D}=15,n_{G}=10^{4},T=10^{3}s,\phi=25$)

Figure 4 is a different visual representation of the data present in figure 3.
The bar graphs depict the average connectivity, the error bar show the minimum
and maximum values of M and the star indicates the value of M for the drone at
completion time.. Major observations to be made from this plot are that some
peaks shoot to very high values $(>\phi)$ and most of the connectivities
towards completion of the simulation (star marked) are very close to the
maximum connectivity achieved throughout the runtime.

Figure 4: The Various values of connectivity (M) plotted for different drones.
(Simulation Parameters: $n_{D}=15,n_{G}=10^{4},T=10^{3}s,\phi=25$)

A metric, called the Average Connectivity $\bar{M}$ is defined as

$\bar{M}=\dfrac{1}{T}\int_{t=0}^{T}\sum_{i=0}^{n_{D}}M_{i}(t)$ (4)

Since the simulation is done in time steps, the integral can be converted to
summation.

Figure 5: Average connectivity ($\bar{M}$) plotted against the number of
drones ($n_{D}$). (Simulation Parameters: $n_{G}=10^{4},T=10^{3}s,\phi=25$)

In figure 5 the plot of $\bar{M}$ vs. $n_{D}$ is shown. The error bar depicts
the maximum and minimum connectivity values and the star marks the value of
average connectivity towards completion of the simulation. The number of users
served increases linearly with linear increase in $n_{D}$.

###  IV-A Comparison with Random Walk

In this part of the results, we show how the performance of the proposed
mobility algorithm is superior to random walk motion of DBSs.

|
---|---
Random Walk | Mobility Algorithm
Figure 6: Mean connectivity vs. time. (Simulation Parameters:
$n_{G}=10^{4},T=10^{3}s,\phi=25$)

In figure 6 the scatter plot is the simulated data and the solid lines are the
polynomial fits of the scattered plots. It is clearly evident that over the
course of time, the drones are successful in maximizing the number of
connections but the random walk mobility provides nearly constant connectivity
throughout.

In figure 7 the proposed mobility algorithm outperforms random walk and the
difference in performance increases with increase in number of drones.

Figure 7: Average connectivity ($\bar{M}$) plotted against the number of
drones ($n_{D}$). (Simulation Parameters: $n_{G}=10^{4},T=10^{3}s,\phi=25$)

##  V Conclusions

In this paper, we presented a drone mobility algorithm which tends to maximize
the number of ground nodes it serves. The superiority of our algorithm over
random walk of drones is established in section IV. When implemented, it
increases the total number of users served for a scenario thereby minimizing
the amount of drones needed which in simple words is minimizing the resources
required. The mobility control algorithm also provides a comparatively stable
connection by dividing the region into exclusive coverage zone for each drone.
For future work we plan to study the effect of threshold $\phi$ value on
connectivity $M$ and find the optimum value of $\phi$ as a function of number
of ground nodes and $L$. In this paper, we considered the ground nodes to be
uniformly distributed in the confined region, for extension we plan to
distribute the ground nodes as dynamic clusters and test our proposed
algorithm in that case.

## References

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  * [3] Rushing S. Analyzing ”global access to the internet for all” projects. Network Architectures and Services. 2016 Jul.
  * [4] Hern A. Facebook launches Aquila solar-powered drone for internet access. The Guardian. 2015 Jul;30.
  * [5] Kamnani K, Suratkar C. A review paper on Google Loon technique. International Journal of Research In Science & Engineering. 2015;1(1):167-71.
  * [6] Lardinois F. Google X Announces Project Loon: Balloon-Powered Internet For Rural, Remote And Underserved Areas [Electronic resource]/Frederic Lardinois. TechCrunch/AOL Inc.–Jun. 2013;14.
  * [7] Fotouhi A, Ding M, Hassan M. Flying Drone Base Stations for Macro Hotspots. IEEE Access. 2018 Mar 21.
  * [8] Wang L, Hu B, Chen S. Energy Efficient Placement of a Drone Base Station for Minimum Required Transmit Power. IEEE Wireless Communications Letters. 2018 Feb 23.
  * [9] I. Bor-Yaliniz, A. El-Keyi, and H. Yanikomeroglu, ”Efficient 3-D placement of an aerial base station in next generation cellular networks,” in Proc. IEEE Int. Conf. Commun. (ICC), pp. 1-5, Kuala Lumpur, Malaysia, May 2016.
  * [10] Kalantari E, Yanikomeroglu H, Yongacoglu A. On the number and 3D placement of drone base stations in wireless cellular networks. InVehicular Technology Conference (VTC-Fall), 2016 IEEE 84th 2016 Sep 18 (pp. 1-6). IEEE.
  * [11] Mozaffari M, Saad W, Bennis M, Debbah M. Drone small cells in the clouds: Design, deployment and performance analysis. InGlobal Communications Conference (GLOBECOM), 2015 IEEE 2015 Dec 6 (pp. 1-6). IEEE.
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# Dust-acoustic rogue waves in four component plasmas

S. Jahan∗, A. Mannan, N.A. Chowdhury, and A.A. Mamun  Department of Physics,
Jahangirnagar University, Savar, Dhaka-1342, Bangladesh
Email<EMAIL_ADDRESS>
###### Abstract

A theoretical investigation has been made on modulational instability (MI) and
dust-acoustic (DA) rogue waves (DARWs) in a four dusty plasma medium
containing inertial negatively charged massive heavy (light) cold (hot) dust
grains as well as super-thermal electrons and non-thermal ions. The reductive
perturbation method is used to derive the nonlinear Schrödinger equation, and
two types of modes, namely fast and slow DA modes, have been observed. The
conditions for the MI and the formation of associated DARWs are found to be
significantly modified by the effects of non-thermality of ions ($\alpha$),
super-thermality of electrons ($\kappa$), density-ratio of non-thermal ion to
cold dust ($\mu_{i}$), and mass-ratio of cold dust to hot dust ($\sigma$),
etc. The implications of our current investigation in space and laboratory
plasmas are briefly discussed.

###### keywords:

Dust-acoustic waves , modulational instability , rogue waves.

††journal: “High Energy Density Physics”

##  1 Introduction

Now-a-days, scientist are enthralled by the natural elegance of dusty plasmas
(DP) because of their existence in space (viz. Saturn F-ring [1], circum-solar
dust grains [2], and interstellar molecular clouds [3], etc.) and enormous
application in laboratory (viz. fusion device, rf discharges, and Q-machine
[3], etc.). DP can be defined as low temperature fully or partially ionized
conducting gases whose constituents are electrons, ions, charged dust grains,
and neutral atoms. Normally, the dust grains of the DP are highly charged and
massive object (billions times higher than the protons) and their size ranges
from micrometer to millimeters [3]). Dust grains acquire a negative charge by
the collection of electrons and the mechanisms of electrons collection are
photo-electric emission [3, 4], plasma currents [3], and other process, etc.
Almost 30 years ago, Rao et al. [5] first theoretically predicted the presence
of extremely low phase velocity dust-acoustic (DA) waves (DAWs) in space DP in
which the dust grains mass provide the inertia and the thermal pressures of
electrons and ions provide the restoring force to generate DAWs. A few years
later, Barkan et al. [6] have experimentally verified the DAWs in plasmas
containing negatively charged dust grains.

The formation of the nonlinear electrostatic structures in a plasma medium is
rigourously depended on the velocity distribution functions of the plasma
species. The Maxwellian velocity distribution is applicable for the plasma
medium in which the plasma species are thermally equilibrium. But, it is often
observed that the plasma species in space as well as laboratory plasmas are
not in thermally equilibrium, and their basic features may deviate from the
Maxwellian distribution. Sometimes these thermally non-equilibrium plasma
species may be governed by the non-thermal [7, 8, 9] or super-thermal
distribution [11, 10]. The non-thermal distribution function was first
introduced by Cairns et al. [7] and have shown that a plasma medium in
presence of non-thermal electrons and cold ions could support the simultaneous
existence of solitons within both positive and negative potential. Mamun et
al. [8] have considered a model composed of non-thermal ions and negatively
charged dust.

The research regarding on modulational instability (MI) of DAWs and associated
dust-acoustic rogue waves (DARWs), which are the result of unusual
disturbances, in nonlinear and dispersive mediums has been vigorously
increasing day by day. A number of authors have employed reductive
perturbation method (RPM) to obtain the nonlinear Schrödinger equation (NLSE)
to study the MI of the DAWs as well as formation of DARWs in dusty plasma
medium. Elwakil et al. [9] have examined the effects of the non-thermal
electrons on the MI of ion-acoustic waves and found that the non-thermality of
the electrons decreases the stable domain of the ion-acoustic waves. Sultana
et al. [10] have investigated the MI in four component DP in presence of
super-thermal electrons. Chowdhury et al. [11] have studied the DARWs in
presence of the super-thermal or $\kappa$-distributed electrons in space DP
and observed that within the fast DA modes, the amplitude and the width of the
DARWs are decreasing with $\kappa$. Selim et al. [2] have studied the
nonlinear DARWs in presence of two-temperature (hot and cold) highly charged
dust grains and reported that amplitude and width of the DARWs decrease with
hot dust charge state. Elwakil et al. [12] have investigated the effect of
non-thermality of ions on the nature of DAWs in two temperatures highly
charged dust grains. Akhter et al. [1] have studied the DA solitary waves in
the presence of hot and cold dust grains along with iso-thermal electron and
ions, and observed that the charge to mass ratio of each dust component plays
a vital role for the formation of the solitary structure in DP. To the best
knowledge of the authors, no attempt has been to study the MI of the DAWs and
associated DARWs in four component space DP medium containing negatively
charged hot and cold dust as well as super-thermal electrons and non-thermal
ions. The aim of the present investigation is therefore to extend the work of
Akhter et al. [1] to investigate the conditions for the MI of the DAWs as well
as the formation of the DARWs in four-component space DP medium (in which the
inertia is provided by the masses of the cold and hot dust grains and
restoring force is provided by the thermal pressure of the super-thermal
electrons and non-thermal ions).

The rest of the manuscript is organized in the following scheme: The model
equations of our considered plasma system are stated in Sec. 2. The stability
of the DAWs and the formation of rogue waves are examined in Sec. 3. Finally,
a conclusion is provided in Sec. 4.

##  2 Model Equations

we consider a collisionless, fully ionized, unmagnetized four-component DP
system composed of super-thermal electrons (charge $-e$, mass $m_{e}$), non-
thermal ions (charge $+e$; mass $m_{i}$) and inertial cold negatively charged
dust grains (charge $q_{c}=-Z_{c}e$, mass $m_{c}$) as well as hot dust grains
(charge $q_{h}=+Z_{h}e$; mass $m_{h}$); where $Z_{c}$ ($Z_{h}$) is the charge
state of the negatively charged cold (hot) dust grains. The governing equation
of our plasma model can be written as:

$\displaystyle\frac{\partial N_{c}}{\partial T}+\frac{\partial}{\partial
X}(N_{c}U_{c})=0,$ (1) $\displaystyle\frac{\partial U_{c}}{\partial
T}+U_{c}\frac{\partial U_{c}}{\partial
X}=\frac{Z_{c}e}{m_{c}}\frac{\partial\varphi}{\partial X},$ (2)
$\displaystyle\frac{\partial N_{h}}{\partial T}+\frac{\partial}{\partial
X}(N_{h}U_{h})=0,$ (3) $\displaystyle\frac{\partial U_{h}}{\partial
T}+U_{h}\frac{\partial U_{h}}{\partial
X}=\frac{Z_{h}e}{m_{h}}\frac{\partial\varphi}{\partial
X}-\frac{1}{m_{h}N_{h}}\frac{\partial P_{h}}{\partial X},\ $ (4)

where $N_{c}$ ($N_{h}$) is the number densities of the negatively charged cold
(hot) dust grains; $T$ ($X$) is the time (space) variable; $U_{c}$ ($U_{h}$)
is the fluid speed of the negatively charged cold (hot) dust species; $e$ is
the magnitude of the charge of the electron; $\varphi$ is the electrostatic
wave potential, and $P_{h}$ is the adiabatic pressure of the negatively
charged hot dust grains. The system is closed through Poisson’s equation

$\displaystyle\frac{\partial^{2}\varphi}{\partial X^{2}}=4\pi
e(N_{e}-N_{i}+Z_{c}N_{c}+Z_{h}N_{h}),$ (5)

where $N_{i}$ and $N_{e}$ are, respectively, the ion and electron number
densities. Now, in terms of normalized variables, namely, $n_{c}=N_{c}/n_{c0}$
(with $n_{c0}$ being the equilibrium number densities of the negatively
charged cold dust grains), $n_{h}=N_{h}/n_{h0}$ (with $n_{h0}$ being the
equilibrium number densities of the negatively charged hot dust grains),
$u_{c}=U_{c}/C_{dc}$ (with $C_{dc}$ being the sound speed of the negatively
charged cold dust grains), $C_{dc}=(Z_{c}k_{B}T_{i}/m_{c})^{1/2}$ (with
$T_{i}$ being the temperature of non-thermal ion and $k_{B}$ is the Boltzman
constant), $u_{h}=U_{h}/C_{dc}$; $\phi=e\varphi/k_{B}T_{i}$; $t=T\omega_{pdc}$
(with $\omega_{pdc}$ being the plasma frequency of the negatively charged cold
dust grains), $\omega_{pdc}=(4\pi e^{2}Z_{c}^{2}n_{c0}/m_{c})^{1/2}$,
$x=X/\lambda_{Ddc}$ (with $\lambda_{Ddc}$ being the Debye length of the
negatively charged cold dust species), $\lambda_{Ddc}=(k_{B}T_{i}/4\pi
e^{2}Z_{c}n_{c0})^{1/2}$; $P_{h}=P_{h0}(N_{h}/n_{h0})^{\gamma}$ [with being
$P_{h}$ ($P_{h0}$) are the adiabatic (equilibrium adiabatic) pressure of the
negatively charged hot dust grains and $\gamma=(N+2)/N$, where $N$ is the
degree of freedom, for one-dimensional case, $N=1$ so that $\gamma=3$], and
$P_{h0}=n_{h0}k_{B}T_{h}$ (with $T_{h}$ being the temperature of the
negatively charged hot dust grains). It may be noted here that for numerical
analysis we have considered that $m_{c}>m_{h}$, $Z_{c}>Z_{h}$,
$n_{c0}>n_{h0}$, and $T_{i},T_{e}>>T_{h}$ (with $T_{e}$ being the temperature
of the super-thermal electron). By normalization, (1)-(5) becomes

$\displaystyle\frac{\partial n_{c}}{\partial t}+\frac{\partial}{\partial
x}(n_{c}u_{c})=0,$ (6) $\displaystyle\frac{\partial u_{c}}{\partial
t}+u_{c}\frac{\partial u_{c}}{\partial x}=\frac{\partial\phi}{\partial x},$
(7) $\displaystyle\frac{\partial n_{h}}{\partial t}+\frac{\partial}{\partial
x}(n_{h}u_{h})=0,$ (8) $\displaystyle\frac{\partial u_{h}}{\partial
t}+u_{h}\frac{\partial u_{h}}{\partial x}+\delta n_{h}\frac{\partial
n_{h}}{\partial x}=\sigma\frac{\partial\phi}{\partial x},$ (9)
$\displaystyle\frac{\partial^{2}\phi}{\partial
x^{2}}=(\mu_{i}-\lambda-1)n_{e}-\mu_{i}n_{i}+n_{c}+\lambda n_{h},$ (10)

where $\delta=3T_{h}m_{c}/Z_{c}T_{i}m_{h}$, $\sigma=Z_{h}m_{c}/Z_{c}m_{h}$,
$\mu_{i}=n_{i0}/Z_{c}n_{c0}$ (with $n_{i0}$ being the equilibrium number
densities of the non-thermal ions), and $\lambda=Z_{h}n_{h0}/Z_{c}n_{c0}$. The
quasi-neutrality condition at equilibrium can be written as

$\displaystyle n_{i0}=n_{e0}+Z_{c}n_{c0}+Z_{h}n_{h0},$ (11)

where $n_{e0}$ is the equilibrium number densities of the super-thermal
electrons. The number densities of the super-thermal electrons and non-thermal
ions [7] can be written by the following normalized equation

$\displaystyle
n_{e}=\Big{[}1-\frac{\eta\phi}{(\kappa-3/2)}\Big{]}^{-\kappa+1/2},$ (12)
$\displaystyle n_{i}=(1+\beta\phi+\beta\phi^{2})~{}\mbox{exp}(-\phi),$ (13)

where $\kappa$ is the super-thermality of electrons and
$\beta=4\alpha/(1+3\alpha)$ (with $\alpha$ being the non-thermality of ions)
and $\eta=T_{i}/T_{e}$ ($T_{e}>T_{i}$). Now, by substituting (11) and (12)
into (10), and extending up to third order in $\phi$, one can obtain the
following relation

$\displaystyle\frac{\partial^{2}\phi}{\partial x^{2}}=n_{c}+\lambda
n_{h}-\lambda-1+\gamma_{1}\phi+\gamma_{2}\phi^{2}+\gamma_{3}\phi^{3}+\cdot\cdot\cdot,$
(14)

where

$\displaystyle\gamma_{1}=\frac{[\eta(\mu_{i}-\lambda-1)(\kappa-1/2)-\mu_{i}(\beta-1)(\kappa-3/2)]}{(\kappa-3/2)},$
$\displaystyle\gamma_{2}=\frac{[\eta^{2}(\mu_{i}-\lambda-1)(\kappa-1/2)(\kappa+1/2)-\mu_{i}(\kappa-3/2)^{2}]}{2(\kappa-3/2)^{2}},$
$\displaystyle\gamma_{3}=\frac{[\eta^{3}(\mu_{i}-\lambda-1)(\kappa-1/2)(\kappa+1/2)(\kappa-3/2)+\prod]}{6(\kappa-3/2)^{3}},\
$

where $\prod=3\mu_{i}(\beta+3)(\kappa-3/2)^{3}$. We will employ the RPM to
derive the NLSE to study the MI of the DAWs. The independent variables are
stretched as $\xi=\epsilon(x-v_{g}t)$ and $\tau=\epsilon^{2}t$, where
$\epsilon$ is a small expansion parameter and $v_{g}$ is the group velocity of
the IAWs. The dependent variables [13] can be expressed as

$\displaystyle\Upsilon(x,t)=\Upsilon_{0}+\sum_{m=1}^{\infty}\epsilon^{(m)}\sum_{l=-\infty}^{\infty}\Upsilon_{l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il\Lambda],$
(15)

where
$\Upsilon_{l}^{(m)}=[n_{cl}^{(m)},u_{cl}^{(m)},n_{hl}^{(m)},u_{hl}^{(m)},\phi_{l}^{(m)}]$,
$\Lambda_{0}=[1,0,1,0,0]^{\tau}$, $\Lambda=(kx-\omega t)$, and $k$ ($\omega$)
can be represented as the carrier wave number (frequency), respectively. We
are following parallel mathematical steps as Chowdhury et al. [13] have done
in their work to obtain successively the IAWs dispersion relation, group
velocity, and NLSE. The dispersion relation for the DAWs can be expressed as

$\displaystyle\omega^{2}=\frac{B\pm\sqrt{B^{2}-4ME}}{2M},$ (16)

where $B=\delta(k^{2}+\gamma_{1})+\lambda\sigma+1$,
$M=(k^{2}+\gamma_{1})/k^{2}$, and $E=\delta k^{2}$. It may be noted that the
real and positive values of $\omega$ can be obtained under the satisfactory
condition of $B^{2}>4ME$. One may recognize the fast ($\omega_{f}$)/slow
($\omega_{s}$) DA modes can according to to the positive/negative sign of
(18). The group velocity $v_{g}$ of DAWs can be written as

$\displaystyle v_{g}=\frac{\lambda\sigma\omega^{5}+\delta\lambda\sigma
k^{2}\omega^{3}+2\omega A^{2}-\lambda\sigma
A\omega^{3}-2A^{2}\omega^{3}}{2k(A^{2}+\lambda\sigma\omega^{4})}.$ (17)

where $A=\delta k^{2}-\omega^{2}$. Finally, the NLSE can be written as:

$\displaystyle
i\frac{\partial\Phi}{\partial\tau}+P\frac{\partial^{2}\Phi}{\partial\xi^{2}}+Q|\Phi|^{2}\Phi=0,$
(18)

where $\Phi=\phi_{1}^{(1)}$ for simplicity and $P$ ($Q$) is the dispersion
(nonlinear) coefficient, and is written by

$\displaystyle P=\frac{C_{1}}{2\omega Ak^{2}(A^{2}+\lambda\sigma\omega^{4})}$
$\displaystyle Q=\frac{C_{2}}{2k^{2}(A^{2}+\lambda\sigma\omega^{4})}\ $

Figure 1: The variation of $P/Q$ with $k$ for different values of $\alpha$;
along with $\kappa=1.7$, $\delta=0.0006$, $\lambda=0.5$, $\mu_{i}=2.0$,
$\sigma=1.6$, $\eta=0.8$, and $\omega_{f}$.

Figure 2: The variation of $P/Q$ with $k$ for different values of $\delta$;
along with $\alpha=0.3$ $\kappa=1.7$, $\lambda=0.5$, $\mu_{i}=2.0$,
$\sigma=1.6$, $\eta=0.8$, and $\omega_{s}$.

where

$\displaystyle C_{1}=\lambda\sigma\omega^{4}(\omega-
kv_{g})(2kv_{g}\omega^{2}+Akv_{g}-\omega^{3}-\delta\omega k^{2})$
$\displaystyle+\lambda\sigma k\omega^{4}(\delta k-\omega v_{g})(2\omega
kv_{g}+A-\omega^{2}-\delta k^{2})$ $\displaystyle+A^{3}(\omega-
kv_{g})(\omega-3kv_{g})-A^{3}\omega^{4},$ $\displaystyle
C_{2}=3\gamma_{3}A^{2}\omega^{3}+2\gamma_{2}A^{2}\omega^{3}(C_{7}+C_{12})-2A^{2}k^{3}(C_{4}+C_{9})$
$\displaystyle-2\lambda\sigma k^{3}\omega^{4}(C_{6}+C_{11})-\omega
A^{2}k^{2}(C_{3}+C_{8})$ $\displaystyle-\sigma\lambda
k^{2}\omega^{3}(\omega^{2}+\delta k^{2})(C_{5}+C_{10}),\ $ $\displaystyle
C_{3}=\frac{3k^{4}-2C_{5}\omega^{2}k^{2}}{2\omega^{4}},$ $\displaystyle
C_{4}=\frac{C_{1}\omega^{4}-k^{4}}{k\omega^{3}},$ $\displaystyle
C_{5}=\frac{2\sigma
C_{5}A^{2}k^{2}-\delta\sigma^{2}k^{6}-3\sigma^{2}\omega^{2}k^{4}}{2A^{3}},$
$\displaystyle C_{6}=\frac{\omega C_{3}A^{2}-\omega\sigma^{2}k^{4}}{kA^{2}},$
$\displaystyle
C_{7}=\frac{3A^{3}k^{4}+2\gamma_{2}A^{3}\omega^{4}-3\lambda\sigma^{2}k^{4}\omega^{6}-\delta\lambda\sigma^{2}\omega^{4}k^{6}}{2\omega^{2}A^{2}(Ak^{2}-4A\omega^{2}k^{2}-\lambda\sigma\omega^{2}k^{2}-\gamma_{1}A\omega^{2})},$
$\displaystyle C_{8}=\frac{2v_{g}k^{3}+\omega
k^{2}-C_{10}\omega^{3}}{v_{g}^{2}\omega^{3}},$ $\displaystyle
C_{9}=\frac{C_{6}v_{g}\omega^{3}-2k^{3}}{\omega^{3}},$ $\displaystyle
C_{10}=\frac{2\omega
v_{g}\sigma^{2}k^{3}+\sigma^{2}\omega^{2}k^{2}+\delta\sigma^{2}k^{4}-\sigma
C_{10}A^{2}}{A^{2}(v_{g}^{2}-\delta)},$ $\displaystyle
C_{11}=\frac{C_{8}v_{g}A^{2}-2\omega\sigma^{2}k^{3}}{A^{2}},$ $\displaystyle
C_{12}=\frac{C_{13}}{A^{2}\omega^{3}\left\\{(v_{g}^{2}-\delta)+\lambda\sigma
v_{g}^{2}-\gamma_{1}v_{g}^{2}(v_{g}^{2}-\delta)\right\\}},$ $\displaystyle
C_{13}=2\gamma_{2}A^{2}v_{g}^{2}\omega^{3}(v_{g}^{2}-\delta)+A^{2}(v_{g}^{2}-\delta)(2v_{g}k^{3}+\omega
k^{2})$ $\displaystyle+\lambda v_{g}^{2}\omega^{3}(2\omega
v_{g}\sigma^{2}k^{3}+\delta\sigma^{2}k^{4}+\sigma^{2}\omega^{2}k^{2}).\ $

##  3 MI and rogue waves

The stability of the DAWs profile, which amplitude evolution is governed by
the NLSE, is totally depended on the the nonlinear ($P$) and dispersive ($Q$)
co-efficient [13, 14, 15, 16, 17, 18]. It should be noted that the nonlinear
and the dispersive co-efficient can be either positive or negative according
to the values of $k$, $\alpha$, $\mu_{i}$, $\delta$, $\kappa$, and $\sigma$,
etc. In our present numerical analysis, for small wave number, i.e.,
$k\rightarrow 0$, the sign of $P$ is negative, but for large wave number,
i.e., $k\rightarrow 70$, the sign of $P$ is positive (figure is not included).
Generally, the sign of $P$ is always negative without consideration of
degenerate or adiabatic pressure in the momentum equation [14], but if any one
consider the degenerate/adiabatic pressure term in their momentum equation, it
may possible for $P$ to obtain positive or negative values depending on the
$k$ and other plasma parameters [17]. On the other hand, for small wave
number, i.e., $k\rightarrow 0$, the values of $Q$ is positive, but for large
wave number, i.e., $k\rightarrow 5$, the values of $Q$ is negative.

Figure 3: The variation of $|\Phi|$ with $\xi$ for different values of
$\kappa$; along with $\alpha=0.3$ $\delta=0.0006$, $\lambda=0.5$,
$\mu_{i}=2.0$, $\sigma=1.6$, $\eta=0.8$, and $\omega_{f}$.

Figure 4: The variation of $|\Phi|$ with $\xi$ for different values of
$\mu_{i}$; along with $\alpha=0.3$ $\delta=0.0006$, $\kappa=1.7$,
$\lambda=0.5$, $\sigma=1.6$, $\eta=0.8$, and $\omega_{f}$.

Figure 5: The variation of $|\Phi|$ with $\xi$ for different values of
$\sigma$; along with $\alpha=0.3$ $\delta=0.0006$, $\kappa=1.7$,
$\lambda=0.5$, $\mu_{i}=2.0$, $\eta=0.8$, and $\omega_{f}$.

The criteria of the MI of DAWs and the formation of the highly energetic
nonlinear structures in four component DP medium can be regulated by the sign
of $P$ and $Q$. The DAWs are modulationally stable when $P/Q<0$ ($P$ and $Q$
have the different sign). On the other hand, when $P/Q>0$ ($P$ and $Q$ have
the same sign) the DAWs are modulationally unstable. The intersecting point,
which can open a new unstable window from the stable domain for DAWs in
nonlinear and dispersive plasma medium, between $P/Q$ curve with $k$-axis
($P/Q$ vs. $k$ graph) is referred to as the critical wave number ($k_{c}$).
The nature of the $P/Q$ curve with plasma parameter $\alpha$ and $\delta$ can
be observed from Figs. 2 and 2, and it is obvious from these two figures that
(a) the modulationally stable and unstable domain for the DAWs in nonlinear
and dispersive medium can be found for the fast ($\omega_{f}$) and slow
($\omega_{s}$) DA modes according to the sign (positive) of equation (16); (b)
the value of $k_{c}$ decreases with non-thermality of the ions (via $\alpha$);
(c) so, excess non-thermality reduces the stable domain of the DAWs and this
result is a good agreement with the result of Elwakil et al. [9] work; (d) the
stable region of DAWs increases (decreases) with ion (hot dust) temperature
for constant value of cold and hot dust mass, and hot dust charge state (via
$\delta$).

The first-order rational rogue wave solution (developed by Darboux
Transformation Scheme) in the unstable region ($P/Q>0$) can be written as [19]

$\displaystyle\Phi(\xi,\tau)=\sqrt{\frac{2P}{Q}}\left[\frac{4(1+4iP\tau)}{1+16P^{2}\tau^{2}+4\xi^{2}}-1\right]\mbox{exp}(2iP\tau).$
(19)

The rogue wave solution inform us that the wave-particle interaction occurs
within a small evolution of time and space. The effects of different plasma
parameters in the formation of the DARWs can be observed from Figs. 5-5. From
Fig. 5, it can be observed that the amplitude of the DARWs decreases with the
increase in the value $\kappa$. From Literature, it can be manifested that
small value of $\kappa$ represents a hard spectrum, and when $\kappa$ is small
then the movement of the plasma particles is larger then the large value of
$\kappa$. So, the nonlinearity of the plasma medium increases with the
decrease of $\kappa$, and this result agrees with the result of Chowdhury et
al. [11] work.

The nature of the amplitude and the width of the DARWs is also governed by
plasma parameters $\mu_{i}$ and $\sigma$, and their effects can be observed
from Figs. 5 and 5. It is obvious from Figs. 5 and 5 that (a) the height and
the width of the DARWs decrease with the increase in the value of equilibrium
ion number density ($n_{i0}$) for fixed values of $Z_{c}$ and $n_{c0}$ (via
$\mu_{i}$); (b) the nonlinearity of the plasma medium increases (decreases)
with $m_{h}$ ($m_{c}$) by depicting taller (smaller) DARWs for fixed values of
$Z_{c}$ and $Z_{h}$ (via $\sigma$); (c) on the other hand, the nonlinearity of
the plasma medium increases with the increase in the value of hot dust charge
state $Z_{c}$, decreases with increase of the cold dust charge state $Z_{h}$
when their hot and cold dust mass remain constant. So, the mass and charge
state of the hot and cold dust grains provide an opposite effects in the
plasma medium to generate highly energetic DARWs.

##  4 Conclusion

In this work, we have considered an unmagnetized four-component DP system
consisting of negatively charged cold and hot dust grains, electrons following
super-thermal $\kappa$-distribution and ions following non-thermal
distribution. The standard NLSE is used to recognize the stable and unstable
domain for the DAWs according to the sign of nonlinear and dispersive
coefficients. The results of our present investigation will be useful in
understanding the propagation of DAWs, MI of DARWs, and finally the generation
of the DARWs in space plasma medium (viz. Saturn F-ring [1], circum-solar dust
grains [2], and interstellar molecular clouds [3], etc.) and also laboratory
DP (viz. fusion device, rf discharges, and Q-machine [3], etc.).

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