Mathematical Collapsible and Spiky Wave for Dust Acoustic Waves in Dusty Plasmas

In multicomponent dusty plasma, the Sagdeev Potential (SP) approach is employed to formulate the Energy Equation for arbitrary amplitude dust acoustic waves (DAWs), where an amount of electrons is trapped in potential well. The dependence of amplitude and width of the solitons of Sagdeev Potential on plasma parameters is widely discussed. The range of Mach number has determined for solitary waves (SWs) with the help of critical Mach number. The solution of the Energy Equation obtained, has been discussed by expanding the expression for SP in the higher terms of φ . The different solutions of Energy Equation give us SWs, breakable waves, collapsible waves and SWs with spiky and explosive nature. The role of temperature ratio on the transformation of SWs to collapsible waves is discussed. With the help of standard values of plasma parameters relevant to such plasma environment, the results so obtained, are discussed. These results may help us to explain the nature of SWs in different astrophysical situations.


Introduction
The research on SWs with small amplitude in the dynamics of dusty plasma is gradually becoming an emerging field due to the omnipresence of the different micron sized interplanetary dust particles (IDPs) (Messenger, 2000) in the astrophysical atmosphere. In initial days of studies about the different features of nonlinear waves with small amplitudes, (Washimi and Taniuti 1966) had formulated Korteweg-deVries (KdV) equation in ion acoustic waves in unmagnetized plasma, which afterward became a milestone in nonlinear dynamics. The KdV equation and its modified versions mKdV, KdV-Burgers and mKdV-Burger equations, etc. have been studies in a widespread way by many authors (Kato et. al. 1972), (Tappert 1972), (Tagare 1973), (Schamel 1973), (Das et. al. 1998),  to reduce inconsistency between theoretical anticipations and laboratory observations. The KdV family of equations have some limitations; especially it is lacking behind (Das et. al. 1998) in explain arbitrary amplitude of SWs. Contemporary researcher (Sagdeev 1966) looked into the characteristics of nonlinear waves having arbitrary amplitudes in unmagnetized isothermal plasma where T T e i  with cold ions by time independent approach. In subsequent developments this approach is known as Sagdeev Potential approach. Many authors (Nejoh 1987, Banerjee andMaitra 2016) extensively used this approach to derive several nonlinear equations to show the possibilities of formation of SWs, double layer (DL), spiky and explosive mode etc. In the solar system, presence of the charged IDPs in the plasmas, the action and reaction between particles and outwardly applied magnetics and electric field is altered by the existence of the plasmas (Goertz 1989). The dust particles increase (decrease) the angular momentum of the Saturn's ring outside (inside) of the synchronous orbit, when the dust particles are reabsorbed by the ring (Goertz et. al. 1986). In another study (Rao et. al. 1990) had noticed that, if the dust is charged, a new dust acoustic mode for small amplitude SWs has to be generated. This theoretical observation had got boost by the experimental work of (Barkan et. al. 1995), which later encouraged the anticipation of several findings. (Ma and Liu 1997) had formulated dust-acoustic rarefactive solitons by injecting the simple charge equation I I e i + =0 . In a theoretical study (Mamun et. al. 1996) investigated the propagation of solitons in dusty plasma considering negatively charge dust particles through the SP approach. (Banerjee and Maitra 2016) investigated the role of dust particles and nonthermal effects on the SWs and DL in the plasma which contains electron, positron, ion and dust particles with the nonthermal electron and positron and reported that the nonthermal characteristic parameter, concentration of dust grain, temperatures of ion and positron have the capacity to modify the characteristics of wave propagation. The dynamical system in plasma produces density cavity in which some electrons to be abundant as trapped electron. So the influences of trapped electrons in pp.82 shaping and spreading of the nonlinear wave in plasma have a great importance. The role of trapped particle in collisionless plasma was discussed by many authors (Schamel 1972, 1980, 1986, Alinejad 2011, Dev et. al. 2015 where they found that the trapped particles have significant influences on the propagation of waves like SWs and DL.  studies DAWs in warm dusty plasma with Boltzmann electron and ion distribution with constant dust charge with the help of the family of KdV equations and reported about the formation of compressive and rarefactive solitons and shock waves. (El-Taibany and Wadati 2007) tried to explain the dust-plasma interaction near the Mars for positively charged dust particles in nonthermal dusty plasma by means of the SP approach and also studied the role of nonthermal ions. They had reported that the nonthermal ions-charging currents desperately change the features of the dust acoustic SWs. In a report (Gosh et. al. 2008) has described that when the plasma system comes near to a single-ion population, extraordinarily narrow and spiky form of waves are obtained showing a different character of SWs. In a study about spiky SWs and explosive mode in presence of the trapped electron, (Nejoh 1990) reported that in a position where nonlinear ion-acoustic waves spreads in the interplanetary spaces, the spiky together with explosive (bursting) modes of SWs are found in the satellite observations (Gurnen et. al.1979, Temerin 1982, Kintner 1983. Generally spiky SWs are nearly associated with the energetic precursor effects in astrophysical situation (Nejoh 1988(Nejoh , 1992(Nejoh , 1994 and formation of these SWs breaks the chance of occurrence of the DL. The emission of solar-energetic-particles type of explosive events can be explained by the exclusive solution. In a recent report (Deka et. al. 2018) investigated the features of SWs in a relativistic degenerate ion beam driven magneto plasma and they have sought to explain the salient characteristics of the SWs of relativistic-degenerate plasma.
The primary aim of this study is to highlight the interactions of plasma parameters compatible for this plasma system by deriving the Energy Equation (Sagdeev Potential equation) for collapsible waves, breakable waves and waves in spiky and explosive mode. We have studied theoretically about the features of spiky and explosive waves along with collapsible waves in the different astrophysical situation. Studies of these phenomena on the SWs with arbitrary amplitudes still have necessity to explain the nonlinear waves of arbitrary amplitude in astrophysical situations.

Basic Equations and Formulation of Energy Equation
To investigate the properties of dust acoustic solitary wave with arbitrary amplitude in different form, we consider an unmagnetized plasma system consisting micron size dust particles having trapped electrons contaminated with dust-charge grain and Boltzmann ion distribution with negative dust charge fluctuation. To describe the basic equations, the ions are supposed to obey the Boltzmann relation (Das et. al. 1989). In fluid description, the nondimensional equations for dust charge dynamics can be written in the form given by: The equation (1) and (2) are accompanied by the Poissons' equation given by Although the electrons naturally move freely in the medium but a small fraction of them comes into the potential well and reduces energy continuously. As a result, this phenomenon, the electrons come back and kept inside the potential wells and finally are trapped therein. So the electron density can be considered as a sequence of free and trapped electrons can be described by the following distribution where n d e i α α = ( ) , , are the density numbers of the dust, electron and ion particles. They are normalized by n d0 (number density of unperturbed dust particles), n e 0 (number density of unperturbed electrons), n i0 (number density of unperturbed ions) respectively. The In isothermal plasmas, it is evident to deduce the density of electrons by considering b = 0 and c = 0 but in the case of nonisothermal plasmas, b and c are taken as 0 < b <1/ π and 0 1 < < c / π . The Boltzmann ion distribution  is given by The amount of the dust charge is calculated by the currents gathered in the dust grain. The charging process of dust particles have been investigated by several authors (Northrop 1992, Barnes et. al. 1992. For present plasma system, we consider a simple model for dust grains charging as Where I e and I i are the infinitesimal electron and ion currents charging the dust particles and are expressed (Wang et. al. 2016) as The plasma model is simplified to discuss the role of varying parameters and other relevant small effect such as dissipation and dispersion has neglected in the dynamical system. To derive the Energy Equation, the coordinate systems are transformed to time independent coordinate system through the stretching coordinate ξ = - x Mt , here M represents Mach number for the plasma wave. With the help of this coordinate, equations (1) and (2) can be written as and Mu u d d For simplicity, we have used a dimensionless function (Ma and Liu 1997) (10) and (11) which formulated the expression for dust number density as After using Equation (9), the dimensionless function can be expressed as , where the dust surface floating- ,where the principal value of the integral is taken potential (El-Labany and El-Taibany 2003) in an infinite place compared to the unperturbed plasma potential is given by ω 0 . The quantity ω 0 can be obtained from the equation (9) by using φ =0 and is evaluated as Further, imposing the boundary condition d d The Equation (15) is known as Energy Equation and the Sagdeev Potential V φ ( ) is written by the following expression    (17) which is also essential for creation of SWs.

Studies of Arbitrary Amplitude DAWs
To solve the Energy Equation we will go forward step by step with different approximation of φ in the expansion of V φ ( ) with usual boundary conditions to display the nonlinear behaviour of solitons.

Expansion of the SP upto
Retaining the expansion up to 5 2 using this result in (15) we have The corresponding solution of Equation (19)  wave. This type of result has been already obtained by many authors for ion-acoustic solitary waves of compressive type in connection to the solution of mKdV (Shukla and Mamun 2002) and Sagdeev potential (Tappert 1972, Sah andGoswami 1994) equations. At the same time the constants involved are dependent on the plasma parameters as well as dust charge fluctuations and dust temperatures (Sah and Goswami 1994).

Expansion of the SP upto 3 rd degree term
To obtain better approximation in solitonic solution, the expansion of V φ ( ) may consider upto order in φ and then Equation (15) reduces to the form In order to evaluate the nonlinear wave solution from the Equation (21), a transformation (Das et. al. 1998), The equivalent solution of the type of (20) has been obtained by several authors such as Tagare and Chakrabarty (1974) for mixed nonlinearity of KdV and mKdV equations and Das et. al. (1998) for Sagdeev potential equations.

Expansion of the SP upto
To get yet better estimate of solitonic solution, we may retain terms upto 7 4 then Equation (15) reduces to The Equation (25) gives a different type of spiky solitary waves which is often found (Nejoh 1994) in interplanetary space. The equivalent solution of the solution (23) has been obtained by (El-Labany and El-Taibany 2003) for dust acoustic solitary wave and (Das et. al. 1998) for ion acoustic solitary wave for 0 0 < < φ φ s . In the region φ ξ ( )< 0 , i.e. φ < 0 , the solution for the Equation (23) is obtained as pp.86 The solution profile (26) represents an explosive profile of the SWs. This type of solution profile is similar to the results as obtained by (Nejoh 1994), (Das et. al. 1998) and Naranmandula and Wang (2005). But along with the SWs (spiky and explosive nature), there is a possibility of formation of DL in the whole region 0 0 < < φ φ and the profile of solution of this DL is obtained as The results similar to (El-Taibany and Wadati 2007) have been obtained by (Das et. al. 1998) as well as (El-Labany and El-Taibany 2003).

Result and Discussion
To study the nature of the maximum Mach number, SP and solitary wave for different parameters, we use some standard values of plasma parameters applicable to our model Mamun, 2002, Dev 2017 To determine the possibility of existence of arbitrary amplitude of the SWs, the expression (17) is numerically analysed and is displayed it in Figure 1. We have seen that for particular values of parameters related to this plasma model, the upper limit (M max ) of M, which is the M-cordinate point; the graph cross the M-axis which is similar to the report as mentioned by (H. Alinejad 2011) and it helps to find the range of Mach number. It is observed that The value of M max depend on temperature ratio σ i . The behaviour of the SP given by Equation (16) has been analysed numerically and dependence of the rarefactive solitons on plasma parameters are shown in Figure 2. It has been observed that in Figs 2(a-c) the depth and width of soliton increases with increasing values of Mach number M, temperature ratio σ i and δ. These phenomena happened perhaps increase of Mach number, ion density ratio and temperature; increase the area of rarefactive SP and increase the restoring force (Alinejad 2011) which step-up the frequency of oscillation. As a result it enhances the speed of phase velocity of the solitons for unvarying wave length and contributing to propagate the solitons with higher velocity. These observations agree with the result as shown by (El-Labany and El-Taibany 2003). In Figure  3 (d), it is observed that variation of trapping parameter β does not have effect on variation of solitons of the SP. Figure 3 (e) displays the behaviour of SD with φ and and σ i find that depth of rarefactive soliton decreases σ i as increases.  Figure 3 we have investigated the primary features (amplitude and width) of dust acoustic wave's solitons of the expression (20) for parameters variation. It has been observed that the amplitude and width of the DAWs decrease when the values of β , σ i , δ increase. But opposite phenomenon is seen when M increase. The increments in β , σ i , δ and reduction in M led to decrease the wave potentials. So we may conclude that β , σ i , δ and M are vying parameters to settle the characteristics of solitons. Comparing the four figures of Figure 3, it has observed that increments in trapping parameter β and temperature ratio σ i create solitons with high amplitudes. This phenomenon may happen perhaps large number of resonant electrons increases the nonlinear effect of the DAWs. In Figure 3(e), it is observed that for higher values of temperature ratio σ i (> 1)these SWs transform to collapsible SWs. Figure 4 shows the solitonic solitons of the expression (22) which is obtained by solving the Energy Equation (15) considering higher order terms of φ upto 3 rd order. The graph shows collapsible wave. This type of wave was predicted by Labany and Taibany for higher expansion in φ and agrees with our graph.
order. This is a new type of spiky solitary wave of amplitude φ S .This is similar graphical representation with (Nejoh 1994) for electrostatic drift waves of inhomogeneous magnetized plasma. The spiky solitary wave is restricted to a region of positive potential. The existence of the spiky solitary wave in the astrophysical situation may be one reason to form spokes (Messenger 2000) in the Saturn's rings.  (Das et. al. 1998) which is shown in Figure6. The results we have obtained here is good agreement with the reports of many authors (Naranmandula andWang 2005, Deka et. al. 2018) for higher order expansion. Basically, these types of phenomenon depend on the coefficient of φ , which are functions of trapped electron coefficient, dust temperature, number of ions etc. The exclusive mode solutions are limited to an area where potential of the waves cross the upper limit of potential. It is observed that the spiky mode of the SWs exists within the area 0 0 < ( )< φ ξ φ while the explosive mode of the SWs grows in the region φ ξ ( )< 0 . So both mode of the solitary wave cannot be materialized simultaneously i.e. if explosive mode appears then the spiky solitary wave disappears.

Conclusion
In this work we have derived an Energy Equation, which contains Sagdeev Potential with the help of the nonperturbative approach in presence of trapped electrons, Boltzmann ions distribution and dust charge variation in a dusty plasma system. The SP has analysed imposing different condition which formed different types of SWs with rarefactive solitons. The dispersion relation is analytically derived in form of critical Mach number, and domain for the SWs is obtained in form of Mach number limit. The studies about the developments in nonlinearity of the Energy Equation for arbitrary amplitudes have been made for three different cases by expanding the SP in series . By further investigation we watch that the solution provides explosive solitary wave when φ < 0 and there is a possibility of double layer when the range is considered as 0 0 < < φ φ . In this report different kind of nonlinear mode of waves has been discussed independently. Many examples of each mode have been reported in space observations (Alfvén 1982, Temerin et. al.1982, Smith and Brecht 1985, so the results so obtained may helpful to better understanding the different astro-physical phenomena.

Acknowledgement
One of the authors (R.K.K.) is gratefully acknowledged the University Grant Commission (U.G.C.) of India for providing