Multi-dimensional instability of dust-ion-acoustic solitary structure with opposite polarity ions and non-thermal electrons

An attempt has been made to study the multidimensional instability of dust-ion-acoustic (DIA) solitary waves (SWs) in magnetized multi-ion plasmas containing opposite polarity ions, opposite polarity dusts and nonthermal electrons. First of all, we have derived ZakharovKuznetsov (ZK) equation to study the DIA SWs in this case using reductive perturbation method as well as its solution. Small-k perturbation technique was employed to find out the instability criterion and growth rate of such a wave which can give a guideline in understanding the space and laboratory plasmas, situated in the D-region of the Earth’s ionosphere, mesosphere, and solar photosphere, as well as the microelectronics plasma processing reactors.


Introduction
Very recently many authors have studied the nonlinear propagation of ion-acoustic (IA) solitary waves (SWs) [1][2][3][4] dust-ion-acoustic (DIA) SWs [5][6][7][8][9], dust-acoustic (DA) SWs [10][11][12][13][14] in a multi-ion plasmas, as negative ions can be found in space [15,16] and laboratory [17][18][19] as well as positive ions. In multi-ion plasmas, there exist fast and slow mode. In fast mode, positive and negative ions oscillate out of phase (180 ) where as positive and negative ions oscillate in phase. In the case of low temperature plasmas, both in space (the Earth's ionosphere, planetary rings, interstellar clouds, planetary atmospheres, interstellar media, protostellar disks, interstellar and circumstellar clouds, asteroid zones, cometary tails and nebula) and laboratory (plasma processing and plasma crystal) some micron or sub-micron sized dust particles might be present. Due to the size effect on secondary emission [20], the dust particles can have the opposite polarity. Larger particles are found to be negatively charged and smaller ones are positively charged [20]. These opposite polarity ions and opposite polarity dusts [10][11][12][13][14] can significantly modify the basic properties of linear and nonlinear DIA SWs and DA SWs. The anomalous dissipation in complex plasmas, which originates from the dust charging process, makes possible the existence of a new kind of shock wave [21][22][23][24][25]. When the dissipation is weak or absent at the characteristic dynamical time scales of the system, the balance between nonlinearity and dissipation is weak at the characteristic dynamical time scales of the system, the balance between nonlinearity and dissipation effects can result in the formation of a symmetrical SWs. The non-thermal particle distributions turn out to be a characteristic feature in space plasmas, as in auroral zone [26]. The observation of non-thermal electrons made by the Swedish Viking satellite [27] and Freja satellite [28] have shown electrostatic solitary structures in the magnetosphere with density depression. This non-thermal electron populations might be distributed isotropically in velocities or possess a net streaming motion. Cairns et al. [29] considered the influence of non-thermal electrons on the existence conditions of ion-acoustic (IA) solitary structures. The presence of non-thermal electrons allows for the existence of both positive and negative density perturbations. The non-thermal electron distribution of Cairns et al. [29] is a more general class of the electron distribution including a population of fast or energetic electrons. The non-thermal electron n e can be written as where a is a parameter determining the fast particles present in this plasma model and w is the wave potential. Sayed et al. [9] have studied dust-ion-acoustic (DIA) solitary structures in unmagnetized plasmas containing positive and negative ions with positive and negative dust and Maxwellian distributed electrons. This mode is only valid if a complete depletion of the background electrons and ions is possible, and both positive and negative dust fluids are cold. However, in real dusty plasma, the effect of finite ion temperature cannot be neglected and the electron behavior can be strongly modified by non-linear potential of the localized DIA structures by generating a population of fast energetic electrons [30]. Gill et al. [1] have studied double layered IA SWs containing positive and negative ions of two different temperatures and non-thermal electrons by deriving K-dV and mK-dV equations. Mannan et al. [3] have studied nonplanar (cylindrical and spherical) Gardner solitons and double layers (DLs) in an unmagnetized plasma composed of positive and negative ions, and non-thermal electrons. Mamun and Shukla [4] have studied the effects of the non-thermal distribution of electrons as well as the polarity of the net dust-charge number density on nonplanar DIA SWs. It is found from the different observations that most of the plasmas in reality are magnetized, and it can change its characteristics according to the wave direction.
Haider [5] studied the DIA SWs by deriving K-dV equation, associated with a plasma system containing opposite polarity ions, opposite polarity dusts and nonthermal electrons where restoring forces are provided by the plasma thermal pressure of electrons and the inertia is due to the ion mass. But the instability criterion and growth rate is steel unknown. To do this, Zakharov-Kuznetsov (ZK) equation has derived employing reductive perturbation method [31] and using its constant instability criterion and growth rate has been studied by small-k perturbation technique in a magnetized dusty plasma consisting of ions with opposite polarity (negatively andy positively charged), dusts with opposite polarity (negatively and positively charged) and non-thermal electrons following Cairn distribution [29]. The manuscript is organized as follows. The basic equations are given in Sect. 2. The Zakharov-Kuznetsov (ZK) equation in Sect. 3. The solitary-wave solution of the ZK equation is obtained in Sect. 4. Instability criterion with growth rate is in Sect. 5 and a brief discussion is finally given in Sect. 6.

Basic equations
The propagation of DIA SWs have been studied, in the present work, in a collisionless magnetized dusty plasma consist of (i) opposite polarity (negatively and positively charged) mobile ions, (ii) opposite polarity stationary dust, and (iii) non-thermal electrons.
It is also considered that there is an external static magnetic field B 0 acting along the z-direction (B 0 ¼kB 0 ), wherek is the unit vector along the z-direction which is very strong that the electrons and dusts are moving along the magnetic field direction very fast, i.e. the response of electrons and dusts look like as that in the unmagnetized plasma. The nonlinear dynamics of the DIA SWs propagating in such a multi-component dusty plasma is governed by ou n ot þ ðu n Á rÞu n ¼ r/ À x cn u n Âk ð3Þ where n s (n n /n p ) is the ion number density (negative/positive) normalized by its equilibrium value n s0 , u n (u p ) is the negative (positive) ion fluid speed normalized by C n ¼ ðk B T e =m n Þ 1=2 , with k B is the Boltzmann constant, T e is the temperature of electrons and m n being the rest mass of negative ions. / is the DIA wave potential normalized by k B T e =e, with e being the magnitude of the charge of an electron. x cn is the negative ion cyclotron frequency ðeB 0 =m nc Þ normalized by plasma frequency x pn ¼ ð4pn n0 e 2 =m n Þ 1=2 with c being the speed of light. The time variable (t) is normalized by x pn À1 , the space variables are normalized by Debye radius k D ¼ ðk B T e =4pn n0 e 2 Þ 1=2 . At equilibrium we have where, jn d0 ¼ n dþ À n dÀ with n dþ being the positive dust number density and n dÀ being the number density of negative dust. j ¼ 1 for the condition n dþ [ n dÀ and j ¼ À1 for the condition n dþ \n dÀ , i.e. the value of j dependents on net charge of dust grain and b is the mass ratio of negative ion to positive ion (m n =m p ). We can also write where, l 0 ¼ n e0 =n n0 , l p ¼ n p0 =n n0 and l d ¼ n d0 =n n0 .

Derivation of ZK equations
To derive a dynamical equation for the nonlinear propagation of the electrostatic waves in a dusty plasma, under consideration, have to employed in Eqs. (1)(2)(3)(4)(5). To do so, the following stretched coordinates [31][32][33][34][35][36][37] are introduced where is a smallness parameter (0\\1) measuring the weakness of the dispersion and V 0 is the Mach number (the phase speed normalized by C n ). n s , u s and / can be expanded about their equilibrium values in a power series of , viz., where s represents the species (n for negative ions and p for positive ions).
Using the stretched coordinates and Eq. (8) into Eqs. (2)(3)(4)(5) and (1), one can develop equations in various powers of . To the lowest order of is Equating the coefficients of from Poissons equation, we get Using the value of n ð1Þ n and n ð1Þ p from Eq. (9) into Eq. (10), we can get the linear dispersion relation To the next higher order of , i.e., equating the coefficients of 2 , we can express xÀ and yÀcomponents of the momentum equations for both negative and positive ions, and Poisson's equation as Now, using (9-17), we can readily obtain

SW solution of the ZK equation
To study the properties of the solitary waves propagating in a direction making an angle d with the Z-axis, i.e., with the external magnetic field and lying in the (Z-X) plane, we first rotate the coordinate axes (X, Z) through an angle d, keeping the Y-axis fixed. Thus, we transform our independent variables to J Theor Appl Phys (2016) 10:297-305 299 This transformation of these independent variables allows us to write the ZK equation in the form where We now look for a steady state solution of this ZK equation in the form where Z ¼ n À u 0 t in which u 0 is a constant speed normalized by the ionacoustic speed ðC i Þ. Using this transformation we can write this ZK equation in steady state form as Now, using the appropriate boundary conditions, viz., / ð1Þ ! 0, ðd/ ð1Þ =dZÞ ! 0, ðd 2 / ð1Þ =dZ 2 Þ ! 0 as Z ! AE1, the SWs solution of this equation is given by where / m ¼ 3u 0 =d 1 is the amplitude and j ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u 0 =4d 2 p is the inverse of the width of the solitary waves. As u 0 [ 0, it is clear from (11), (19), and (22) that depending on the sign of B, the solitary waves will be associated with either positive potential ð/ m [ 0Þ or negative potential ð/ m \0Þ. Therefore, there exists solitary waves associated with positive (negative) potential when B [ 0 ðB\0Þ. Figure 1 represents the variation of the phase speed V 0 of DIA SWs with l p and l d for positive as well as negatively charged stationary dust particles. This figure indicates that the phase speed is larger for negatively charged dust (upper surface) than for positively charged dust (lower surface). It is also found from this figure that the phase speed decreases with increasing the number density of both positive ions and dust particles in our present dusty plasma system. Figures 2,3 represent the variations of the potential / m of DIA SWs with various parameters. From Fig. 2, it has been found that the SWs associated with negative potential only for propagation angle (d ! 0 -90 ) and it associated with positive potential for the propagation angle (d ! 90 -180 ). That means below the propagation angle 90 the solitary wave structure associates with dip shape and above this value the solitary wave structure associates with hump shape. Figure 3 indicates that the potential of the SW decreases with increasing the number density of both positive ions and dust particles but the potential is larger for negatively charged dust (upper surface) than for positively charged dust particles. Figures 4 and 5 show the variation of the width of SWs with propagation angle d, l p and l d respectively. From Fig. 4, it has been found that for lower limit of the angle ð0 -50 Þ the width increases with it and decreases for higher limits of the angle ð50 -90 Þ for both positively and negatively charged dust particles. It is also clear from this figure that the external magnetic field leads to a decrease in the potential width, i.e., a stronger magnetic field leads to steeper and thus narrower soliton profiles. Similar observations are found in the work of Haider [5] and the work of Sabetkar and Dorranian [11,12,14] while studying the DIA and DA SWs with the presence of magnetic field. The width of the SWs is lower for positive dust grains than for negative dust grains, thus the positive dust grains makes the soliton profile more steeper. Figure 5 shows how the width changes with l p and l d and this figure indicate that the width decreases with increasing the value of both the number density of positive ions and dust particles as found in the work of Haider [5]. From the above analysis we have seen that the magnetic field has an effect only on the width of the SW but it has no effect on the amplitude of the waves.

Instability analysis
We now study the instability of the obliquely propagating SWs, discussed in the previous section, by the method of small-k perturbation expansion technique [33][34][35][36][37][38]. We first assume that / ð1Þ ðZ; f; g; sÞ ¼ / 0 ðZÞ þ wðZ; f; g; sÞ ð26Þ where / 0 ðZÞ is defined by (25), and for a long-wavelength plane wave perturbation in a direction with direction cosines ðl f ; l g ; l n Þ, wðZ; f; g; sÞ is given by wðZ; f; g; sÞ ¼ uðZÞe Àixsþikðl f fþl g gþl n ZÞ ð27Þ in which l 2 f þ l 2 g þ l 2 n ¼ 1, and for small k, uðZÞ and x can be expanded as uðZÞ ¼ u 0 ðZÞ þ ku 1 ðZÞ þ k 2 u 2 ðZÞ þ Á Á Á ð28Þ Now, substituting (26) into (21), and linearizing with respect to w, we can express the linearized ZK equation in the form  Fig. 4 Variation of the width D of DIA SW with d for u 0 ¼ 0:1, The solid curve is for the value of j ¼ 1 and dotted curve is for the value of j ¼ À1 with x cn ¼ 0:1 (red curve), x cn ¼ 0:3 (green curve), x cn ¼ 0:5 (blue curve) Our main objective is to find x 1 by solving the zeroth-, first-, and second-order equations obtained from (27)(28)(29)(30). zeroth-order equation can be written, after integration, as where C is an integration constant. It is clear from (24) that the homogeneous part of this equation has two linearly independent solutions, namely Therefore, the general solution of this zeroth-order equation can be written as where C 1 and C 2 are two integration constants, and d 2 is defined in (22). Now, evaluating all integrals, the general solution of this zeroth-order equation, for u 0 not tending to AE1 as Z ! AE1, can finally be simplified to The first-order equation, i.e.,, the equation with terms linear in k, obtained from (27)(28)(29)(30) and (35), can be expressed, after integration, as We have graphically obtained the parametric regimes (values of d, x cn , l p , l f , and l g ) for which the SWs become stable and unstable. This is shown in Fig. 6, which indicates that for the parameters above (below) the surface the SWs become unstable (stable). This figure represents S i ¼ 0 surface plot showing the variation of x cn with l g and l f for l p ¼ 1, b ¼ 1 and d ¼ 10 . This figure shows that the values of x cn increases with increasing the values of l f and it decreases with increasing the values of l g . If this instability criterion S i [ 0 is satisfied, the growth rate C ¼ ð! À X 2 Þ 1=2 of the unstable perturbation of these SWs is given by The Eq. (49) represents that the growth rate ðCÞ of the unstable perturbation is a linear function of IA wave speed u 0 , but a nonlinear function of propagating angle ðdÞ, negative ion-cyclotron frequency ðx cn Þ, the ratio of equilibrium positive ion density to negative number density l p , the mass ratio of negative ion to positive ion b and direction cosines (l f , and l g ). To study instability/ stability and growth rate of SWs after ZK equation, some perturbation techniques (ordinary, multi-scale about k ¼ 0 and k 2 ¼ 5, small-k for k ( cos a and k [ cos a) discussed in the work of Allen and Rowlands [39,40] based on different cases. Using these techniques Williams and Kourakis [41] studied stability analysis with growth rate of solitary structures. The instability analysis using small-k perturbation technique and process of obtaining growth rate in present form is very suitable to understand, because it express instability criterion and growth rate in terms of basic plasma parameters like ratio of number density, cyclotron frequency, direction cosine etc. which are used to describe the concern plasma situation. Many authentic authors have studied instability criterion as well as growth rate using this perturbation technique [36][37][38]42] as it is a well established technique. The nonlinear variations of C with d, x cn , l g , and l f are shown in Figs. 7 and 8. Figure 7 shows how C changes with d and x cn for u 0 ¼ 0:1, l g ¼ 0:5, l f ¼ 0:5 and l p ¼ 2, and

Results and discussions
Here a magnetized dusty plasma system consisting of nonthermal electrons, cold mobile positive as well as negative ions, and opposite polarity stationary dust has been considered. The ZK equation by using the reductive perturbation method has been derived. Then their multidimensional instability by the small-k perturbation expansion technique have analyzed. The results, which have been found in this investigation may be pointed out as follows: But the magnitude of phase velocity is larger for negative dust grains than for positive dust grains. (4) The magnitude of the external magnetic field B 0 has no effect on the solitary wave amplitude. However, it has a direct effect on the width of the SWs, and it is found that the width of the waves decreases as the magnitude of magnetic field B 0 increases, i.e. the magnetic field makes the solitary structures more spiky. (5) The width of the SWs increase for the lower range of d, i.e., from 0 to about 50 , but decrease for its higher range, i.e., from about 50 to 90 . As d ! 90 , the width goes to 0 and the amplitude goes to 1. It is likely that for large angles the assumption that the waves are electrostatic is no longer valid, and one should look for fully electromagnetic structures. (6) It has found that the values of a has no other effect whether the SWs are stable or unstable. However, stability strongly depend on the external magnetic field and the propagation directions of both the nonlinear wave and its perturbation mode. (7) It has been found that the value of x cn for which the SWs become unstable increases with increasing the values of a and l p . On the other hand it decreases with both of l f and l g . (8) The growth rate C of the unstable perturbation decreases with increasing the value of d and x cn and it increases with both of l f and l g which is shown in Figs. 7 and 8, respectively.
However, our theory is valid for small but finite amplitude solitary waves long wavelength perturbations, but not for large amplitude solitary waves and short wavelength perturbation modes. We, therefore, propose a more exact theory for stability analysis of arbitrary amplitude SWs and arbitrary wavelength perturbation modes, through a generalization of this work. The results which we obtained may be useful for understanding the localized electrostatic disturbances in some space (viz. auroral plasma, Saturns Eand F-rings, etc.) [42][43][44][45] and laboratory [45][46][47] dusty plasmas.