Dust Acoustic Solitary Waves in Saturn F-ring s Region

Transcription

1 Commun. Theor. Phys. 55 ( Vol. 55, No. 1, January 15, 011 Dust Acoustic Solitary Waves in Saturn F-ring s Region E.K. El-Shewy, 1, M.I. Abo el Maaty, H.G. Abdelwahed, 1 and M.A. Elmessary 1 Theoretical Physics Group, Faculty of Science, Mansoura University, Mansoura, Egypt Mathematics and Physics Department Faculty of Engineering, Mansoura University, Mansoura, Egypt (Received April 7, 010; revised manuscript received June 8, 010 Abstract Effect of hot and cold dust charge on the propagation of dust-acoustic waves (DAWs in unmagnetized plasma having electrons, singly charged ions, hot and cold dust grains has been investigated. The reductive perturbation method is employed to reduce the basic set of fluid equations to the Kortewege-de Vries (KdV equation. At the critical hot dusty plasma density N h0, the KdV equation is not appropriate for describing the system. Hence, a set of stretched coordinates is considered to derive the modified KdV equation. It is found that the presence of hot and cold dust charge grains not only significantly modifies the basic properties of solitary structure, but also changes the polarity of the solitary profiles. In the vicinity of the critical hot dusty plasma density N h0, neither KdV nor mkdv equation is appropriate for describing the DAWs. Therefore, a further modified KdV (fmkdv equation is derived, which admits both soliton and double layer solutions. PACS numbers: 5.35.Sb, Fg, 5.35.Fp, 5.35.Mw Key words: dust-acoustic waves, hot and cold dust charge, double layer solutions 1 Introduction Dusty plasmas or complex plasmas are plasmas containing normal electron-ion plasma with an additional highly charged component of small micron or submicron sized extremely massive charged particulates dust grains. This state of plasma is ubiquitous in the universe, e.g., in interstellar clouds, in interplanetary space, in cometary tails, in ring systems of gaint planets (like Saturn F- ring s, in mesopheric noctilucent clouds, as well as in many Earth bound plasma, see for instance. [1 ] Applications of dusty plasma range from astrophysics to strongly coupled dusty plasmas and dusty plasma crystals to technology plasma etching and deposition. [3 6] In a dusty plasma, the dust grains may be charged negatively by plasma electron and ion currents or positively by secondary electron emission, UV radiation, or thermionic emission etc. [7] Due to the higher thermal velocity of electron than ion, the dust grains usually acquire a negative charge in low temperature laboratory dusty plasma. [8] However, in a laboratory Q machine, positively charged dust grains may be produced by replacing the plasma electrons with negative ions whose thermal velocity is smaller than that of positive ions. [9] From theoretical perspective, Chow et al. [10] have shown that due to the size effect on secondary emission insulating dust grains with different sizes in space plasmas can have opposite polarity (smaller ones being positive and larger ones being negative. This is mainly due to the fact that the excited secondary electrons have shorter (longer distances to travel to reach the surface of the smaller (larger dust grains. So it is easy to say that dust grains of different sizes in many circumstances can acquire different polarities, large grains becoming negatively charged, and small grains positively charged. Such differences in sign of the charge on the dust grains can greatly modify the proporties of the plasma. Therefore, it is so important and interesting to deal with the dust plasma systems if in addition to the electrons and ions, there are two types of dust particles with different masses, charges, and temperatures. Consquently, there will be two basic dust-acoustic modes propagating with two different velocities. It means that the linear and nonlinear waves will become interesting and may have many application in space and astrophysical plasmas. For example, Rao et al. [11] and Mamun et al. [1] studied a hot non-thermal dusty plasma consisting of a hot dust fluid and a non-thermally distributed ion, and studied the effects of the dust fluid temperature and the non-thermal distribution of ions on the arbitrary amplitude compressive (density hump and rarefactive (density dip electrostatic solitary structures that have been found to coexist in such a non-thermal dusty plasma model. Sakanaka and Shukla [13] derived the Sagdeevs pseudopotential for dust acoustic waves (DAWs in an unmagnetized four component dusty plasma and studied the coexistence of both soliton and DA double layers (DLs. Also, the DASW and DIASW have also been investigated in the presence of hot dust in unmagnetized plasmas. [14 15] In most of the previous investigations of DAW, the dust has been assumed to be cold or hot. Recently, Akhtar et al., [16] derived the Sagdeevs pseudopotential for dust acoustic waves (DAWs in an unmagnetized two types of dust fluids (one cold and the other is hot in the presence of Bolltzmannian ions and electrons, and studied the existence of rarefactive solitons. The major topic of this work is to study the effect of e k el shewy@mans.edu.eg, emadshewy@yahoo.com c 011 Chinese Physical Society and IOP Publishing Ltd

2 144 Communications in Theoretical Physics Vol. 55 hot and cold dust charge grains on the formations of both rarefactive and compressive solitons and shocks for a small amplitude dust acoustic waves (DAWs. This paper is organized as follows. Section contains the basic set of governing equations as well as the derivation of Korteweg de Vries equation. In Sec. 3, at the critical hot dust density, we derive the modified mkdv equation. In Sec. 4, in the vicinity of the critical hot dust density, neither the KdV nor mkdv equation is appropriate for describing the system. Therefore, the further modified KdV (fmkdv equation is derived. The soliton as well as double layer solutions are discussed in Sec. 5. Finally, some remarks, discussion, and conclusions are given in Secs. 6 and 7. Basic Equations We consider a homogeneous unmagnetized plasma having electrons, singly charged ions, hot and cold dust species. The electrons and ions are assumed to follow the Boltzmann distributions with T e, T dh where T e, are the temperatures of the electrons and ions, respectively and T dh is the temperature of hot dust used as T h. The hot dust is assumed to be heated adiabatically. The one-dimensional continuity and momentum equations for cold dust, respectively, are, n c t + x (n cu c = 0, (1a u ( c t + u c u c e Z c φ x m c x = 0. (1b Corresponding equations for hot dust particles are, n h t + x (n hu h = 0, (1c u ( h t + u h u h e Z h φ x m h x + 1 P h m h n h x = 0, (1d For adiabatic hot dust, we have ( nh γ P h = P h0, (1e N h0 where γ = (N +/N, and N is the number of degrees of freedom. The present work N = 1, and hence γ = 3 and P h0 = N h0 T h. The Boltzmann distributed electrons and ions follow the density equations, respectively, as, ( eφ n e = n e0 exp, (a T e ( eφ n i = n i0 exp. (b The Poisson equation can be written as, φ x = 4πe(n e n i + Z c n c + Z h n h. In equilibrium we have, (c n i0 = n e0 + Z c N c0 + Z h N h0, (3 where φ is the electrostatic potential, Z c (Z h are the charge numbers for negatively charged cold and hot dust, respectively. u c (u h is the cold (hot dusty plasma velocity, respectively. n h, n c, n e, n i, N h0, N c0, n e0, n i0 are the perturbed and equilibrium number densities of the species, respectively..1 Theory for Small-Amplitude Dust Acoustic- Waves According to the general method of reductive perturbation theory, we introduce the slow stretched coordinates: [17] τ = ǫ 3/ t, ζ = ǫ 1/ (x λt, (4 where ǫ is a small dimensionless expansion parameter measuring the strength of nonlinearity and λ is the wave speed. All physical quantities appearing in (1 are expanded as power series in ǫ about their equilibrium values as: n c = N c0 + ǫn c1 + ǫ n c + ǫ 3 n c3 +, u c = ǫu c1 + ǫ u c + ǫ 3 u c3 +, n h = N h0 + ǫn h1 + ǫ n h + ǫ 3 n h3 +, u h = ǫu h1 + ǫ u h + ǫ 3 u h3 +, φ = ǫφ 1 + ǫ φ + ǫ 3 φ 3 +. (5 We impose the boundary conditions that as ζ, n c = N c0, n h = N h0, u c = u h = 0, φ = 0. Substituting (4 and (5 into (1 and (, and equating coefficients of like powers of ǫ. Then, from the lowest-order equations in ǫ, the following results are obtained: n c1 = e N c0 Z c λ φ 1, u c1 = e Z c φ 1, m c λ m c e Nh0 n h1 = Z h λ φ 1, m h N h0 3P h0 eλn h0 Z h u h1 = λ φ 1. (6 m h N h0 3P h0 Poisson s equation gives the linear dispersion relation N c0 Zc Nh0 λ + Z h m c λ n e0 n i0 = 0. (7 m h N h0 3P h0 T e Proceeding to order of ǫ, and by eliminating the second order perturbed quantities n c, n h, u c, u h, and φ we obtain the desired KdV equation for the first-order perturbed potential: where φ 1 τ + AB φ φ 1 1 ζ + A 3 φ 1 ζ 3 = 0, (8 A = ( 8 e πλm h Z h N3 h0 (λ m h N h0 3P h0 + 8e πn c0 Z c λ 3 m c 1, B = 1πN c0z 3 ce 3 λ 4 m c πn3 h0 (m hn h0 λ + P h0 Z 3 h e3 (λ m h N h0 3P h0 3 4π( n e0 Te n i0 e 3 T e. A stationary solitary wave solution of the KdV equation can be obtained by transforming the space variable to: η = (ζ vτ, (9 where v is an arbitrary parameter similar to Mach number, which allows the possibility of solitons moving with velocities different from the phase velocity of the wave, by imposing the boundary conditions for localized perturbations, viz. φ 1 = 0, dφ 1 /dη = 0, and d φ 1 /dη = 0 for

3 No. 1 Communications in Theoretical Physics 145 η ±. Thus, the steady state solution of (8 can be expressed as φ 1 = φ 0 sech [ η ], (10 where φ 0 = 3v A AB, =. v Our system can support two kinds of potential structure namely, compressive and rarefactive pulses. Depending on the sign of the coefficient of the nonlinear term a 1 = 1πN c0zc 3e3 λ 4 m c a 4 = 3 (AB, compressive soliton exists if N h0 > N h0c while rarefactive soliton exists if N h0 < N h0c. However, one can find that the KdV equation becomes a dispersive equation without nonlinear term for N h0 = N h0c, dispersion can destroy a solitary wave and the KdV equation is not appropriate for describing the system. This situation will discuss in the next section, where with N h0c = a P h0 a 1 a 3 3 4a a 3 ± 4πe 3( n e0 T n i0, a e T = 1e 3 πzh 3, a 3 = λ m h, i a 1 a P 5 h0 + 4a 1 a a 3 3 P 4 h0 a 1 a4 P 10 h0 49a3 1 a3 a3 3 P 9 h0 + 16a4 1 a a6 3 P 8 h0, a 5 = 15 3 a 1 P 3 h0 a 4 a 6 = 15 3 a 1 P 3 h0 a 4 ( (a P h0 a 1 a a 1a 3 P h0 + (a P h0 a 1 a 3 3 a 4a + 3a 4 3, a 3 a a 3 1a 1a 3 P h0 + (a P h0 a 1 a 3 3 3a 4 a 3 a a 3 a 3 a3 3 a a a 1P h0 (a P h0 a 1 a 3 3 a a5 + 16a 1Ph0 (4 a 5 1. a ± a6, (11 3 At Critical Hot Dust Density In order to describe the solitary waves at the critical hot dust density region, higher order nonlinearity must be considered and the corresponding stretched coordinates are in the form, The lowest-order in ǫ gives the same equations (6. The next-order in ǫ yields n c = 3e N c0 Z c φ 1 λ 4 m c τ = ǫ 3 t, ζ = ǫ(x λt. (1 en c0z c φ λ, u c = e Zc φ 1 m c λ 3 m ez cφ, c λm c n h = en h0 Z h((3/en h0 (m h N h0 λ + P h0 Z h φ 1 (λ m h N h0 3P h0 φ (λ m h N h0 3P h0 3, u h = eλn h0z h ((1/eN h0 (m h N h0 λ + 9P h0 Z h φ 1 (λ m h N h0 3P h0 φ (λ m h N h0 3P h0 3. (13 Proceeding to order of ǫ, and by eliminating the second order perturbed quantities n c, n h, u c, u h, and φ we obtain the the desired mkdv equation for the first-order perturbed potential: φ 1 τ + AC φ 1 φ 1 ζ + A 3 φ 1 ζ 3 = 0, (14 where e πλ 3 m c (λ m h N h0 3P h0 C = 4(πm c m h Nh0 3 Z h λ4 + πn c0 (λ m h N h0 3P h0 Zc ( 15Nc0 Zc 4 λ 6 m 3 + 3N4 h0 (5m h N h0 λ4 + 30m h N h0 P h0 λ + 9P h0 Zh 4 c (λ m h N h0 3P h0 5 n e0 T 3 n i0 e T 3 i The solution of equation (14 can be obtained using the travelling-wave equation (9 to yield ( 6v 1/ [( v φ 1 = ± sech AC A. 1/η ]. (15 4 In Vicinity of Critical Hot Dust Density In the vicinity of the critical hot dust plasma density, neither the KdV nor mkdv equation is suitable for describing the DAWs. In the previous section, Poisson s equation reads: πez c n c + πez h n h = πn e0φ 1 e3 T e + πn i0φ 1 e3 πn e0φ e T e πn i0φ e, (16

4 146 Communications in Theoretical Physics Vol. 55 we now assume that the deviation of the system from the state of critical density is O(ǫ, the coefficient of φ 1 is O(ǫ thus the Poisson s equation up to O(ǫ 3. So, in Poisson s equation up to O(ǫ 3, we have to take this quantity into account. [18] Then Poisson s equation up to O(ǫ 3 becomes πz c (n c3 + n c e + πz h (n h3 + n h e = πn e0φ 3 1 e4 3T e 3 πn i0φ 3 1 e4 πn e0φ 1 φ e 3 3T 3 i T e + πn i0φ 1 φ e 3 πn e0φ 3 e πn i0φ 3 e + 1 T e φ 1 ζ. (17 Substituting for the second- and third-order perturbed quantities into equation (17, after some algebraic manipulations, one obtains the further modified KdV (fmkdv ζ φ 1 + (AB φ 1 + AC φ 1 ζ φ 1 + A 3 ζ 3 φ 1 = 0, (18 which is the evolution equation in the vicinity of the critical density. It consists of the nonlinear terms of the KdV equation (8 and of the modified KdV equation (14. Thus the evolution equation (18 can be studied for the particular cases of the KdV or the mkdv equation, and serves especially as the transitive link between the various KdV equations. 5 Solutions of fmkdv Equation It is well known that, Sagdeev [19] investigated the fully nonlinear ion-acoustic waves in an unmagnetized plasma system, which consists of cold and hot ions as well as isothermal electrons. In his original work, [19] the basic set of governing equations was reduced to the form of an energy integral of a classical particle in a potential well. The quasi-potential (which is also called the Sagdeev potential can be analyzed, in many cases, to predict the existence of localized solution. [0] Using the travelling-wave transformation (9, and integrating, by using vanishing boundary conditions at infinity, we obtain 1 ( φ1 η + V (φ1 = 0, (19 where V (φ 1 is the Sagdeev potential, which has the following explicit form v V (φ 1 = ( A φ 1 B 3 φ3 1 C 6 φ4 1. (0 5.1 Solitary Wave Solutions Now, it is obvious from (0 that V (φ 1 = 0 at φ 1 = 0, hence the solitary wave solution of (18 exists only when (i V (φ 1 φ 1 φ1=0 = 0, and (ii V (φ 1 < 0. φ1=0 φ 1 Using last condition (ii into Eq. (0, it is found that the soliton exists under the following condition A > 0. The solitary wave solution(s for Eq. (19 can be recast to ( φ1 ( v = η A φ 1 B 3 φ3 1 C 6 1 φ4 = C 3 φ 1 (φ 1 Ψ 1 (φ 1 Ψ, (1 where Ψ 1 = 1 AC (AB R, Ψ = 1 ( AB + R, AC and R = A B + 6vAC. ( Integrating Eq. (1, one can obtain the compressive and rarefactive soliton solutions as [1] respectively, as φ 1+ = φ 1 = 6v/AC Ψ 1 sinh [ v/aη] Ψ cosh [ v/aη], (3 6v/AC Ψ sinh [ v/aη] Ψ 1 cosh [ v/aη]. (4 Due to Eq. (, it is clear that to have soliton solutions R must take positive sign. 5. Double Layer Solution For the double layer solution, Sagdeev potential V (φ 1 should be negative between φ 1 = 0 and φ 1m, noting that φ 1m is some double roots of the potential. Hence, for the formation of double layers, V (φ 1 must satisfy the following conditions V (φ 1 = 0 at φ 1 = 0 and φ 1 = φ 1m, (5 V (φ 1 = 0 at φ 1 = 0 and φ 1 = φ 1m, (6 V (φ 1 < 0 at φ 1 = 0 and φ 1 = φ 1m. (7 Using the boundary conditions into (3, we can calculate φ 1m = B C, and v = AB 6C. (8 Inserting v and B from (8 into (0, we obtain V (φ 1 = C 6 φ 1(φ 1m φ 1. (9 Here, C < 0 has to be fulfilled, in order for V < 0, and thus reality to be ensured Ref. []. Notice that, from (8, the double root here has the same sign as B. Thus, double layers will only exist for B > 0. Inserting equation (9 into (19, the solution of Eq. (19 admits φ 1 = φ [ [ 1m C ]] 1 ± tanh φ 1m 1 η. (30 The existence of double layer requires C < 0. Beside that, equation (8 ensures that the nature of the double layer depends on the sign of B; i.e. for B > 0 a positive double layer exists (viz. φ 1m > 0; see Ref. [1], whereas for B < 0 we would have a negative double layer (φ 1m < 0. 6 Numerical Results and Discussion Nonlinear DAWs in a homogeneous unmagnetized plasma having electrons, singly charged ions, hot and cold dust grains have been investigated. we have assumed that the effect of gravity in the system is neglected (r d 1 as well as there are no neutrals. Saturn F-ring s are one of the space plasma observations that satisfy our conditions:

5 No. 1 Communications in Theoretical Physics 147 (i there are no neutrals, (ii the ratio between the intergrain distances between dust particles to plasma Debye radius is less than one, (iii coupling parameter Γ is less than one, and (iv r d is smaller than 1 µm. Hence, numerical studies have been made using plasma parameters close to those values corresponding to Saturn F-ring s i.e., the equilibrium electron and dust densities are n e0 = 10 cm 3, N h0 = 10 cm 3 and dust charge and mass are taken as Z h = , m h = m c = 10 1 m i, respectively, as given in Refs. [16, 3 5]. Generally speaking, the present system supports compressive and rarefactive soliton and shock waves. However, since one of our motivations was to study effects of the cold (hot dusty plasma density N c0 (N h0 on the formation of solitary waves, we have studied the effect of parameters like the charge numbers for negatively charged hot and cold dusty grains on the existance of solitary waves. Figure 1 shows that both compressive and rarefactive solitary pulse can propagate in this system. However, at critical hot dust plasma density N h0 the pulse cannot propagate. It is obvious from Figs. and 3 that compressive soliton amplitude increases with enhancing of N h0, Z h and decreases with N c0, Z c while its width increases with Z h and decreases with N h0, N c0, and Z c. On the other hand, the rarefactive soliton amplitude increases with N c0, Z c and decreases with N h0, Z h while its width increases with N c0, Z c and decreases with N h0 and Z h as shown in Figs. 4 and 5. To explain how the ratio of cold to hot dusty plasma density and charge N c0 /N h0 and Z c /Z h effects modify the properties of the DAWs, Figs. 6 and 7 are very interesting in the characteristic behaviours of nonlinear DAWs, the compressive soliton amplitude and width decreases with increases N c0 /N h0 and Z c /Z h while the rarefactive soliton amplitude and width increases with increases N c0 /N h0 and Z c /Z h. This means that, the structure of the solitary waves is modified in a way that depends upon N c0 /N h0 and Z c /Z h. always positive, therefore, the required condition for the existence of the DAWs in critical case is C > 0. Fig. The variation of the amplitude φ 0c and the width c of the compressive soliton with Z h for different values of N h0 for N c0 = 8, n e0 = 10, n i0 = , m h = m c = m i, Ti = Te = 1 ev, Z c = 1000, and Fig. 1 The variation of the amplitude φ 0 against N h0 for N c0 = 8, n e0 = 10, n i0 = , m h = m c = 10 1 m i, = T e = 1 ev, Z c = 1000, Z h = 4000, and At the critical case, we investigate the soliton profile for different values of Z h in Fig. 8. It is found that the soliton profile enlarges by reducing Z h. From Eq. (14 it is noted that the sign of AC must be positive, A is Fig. 3 The variation of the amplitude φ 0c and the width c of the compressive soliton with Z c for different values of N c0 for N h0 = 9, n e0 = 10, n i0 = , m h = m c = 10 1 m i, = T e = 1 ev, Z c = 1000, and

6 148 Communications in Theoretical Physics Vol. 55 Z h change the profile (depth and amplitude of the potential as depicted in Fig. 10. It turns out that there is a potential well on the positive (negative φ-axis, resulting in the existence of compressive (rarefactive DAWs. Fig. 4 The variation of the amplitude φ 0r and the width r of the rarefactive soliton with Z h for different values of N h0 for N c0 = 8, n e0 = 10, n i0 = , m h = m c = 10 1 m i, = T e = 1 ev, Z c = 1000, and Fig. 6 The variation of the amplitude φ 0c and the width c of the compressive soliton with ratio N c0/n h0 for (Z c/z h = 1 bold line, Z c/z h = 1.1 thin line for n e0 = 10, n i0 = , m h = m c = 10 1 m i, Ti = Te = 1 ev, and Fig. 5 The variation of the amplitude φ 0r and the width r of the rarefactive soliton with Z c for different values of N c0 for N h0 = 0.98, n e0 = 10, n i0 = , m h = m c = 10 1 m i, = T e = 1 ev, Z h = 4000, and In the vicinity of critical hot dust plasma density N h0, increasing Z h the amplitude and width of compressive (rarefactive soliton decreases as shown in Fig. 9. To further explain this point we have also numerically analyzed the Sagdeev potential (0 and investigated how Fig. 7 The variation of the amplitude φ 0r and the width r of the rarefactive soliton with ratio N c0/n h0 for (Z c/z h = 1 bold line, Z c/z h = 1.1 thin line for n e0 = 10, n i0 = , m h = m c = 10 1 m i, Ti = Te = 1 ev, and

7 No. 1 Communications in Theoretical Physics 149 space and laboratory plasmas Fig. 8 Graph of φ Critical against η for different values of Z h for N h0 = , N c0 = 8, n c0 = 8, n e0 = 10, n i0 = , m h = m c = 10 1 m i, = T e = 1 ev, Z c = 1000, and Fig. 10 Graph of Sagdeev potentials V vs. potential φ for different values of Z h for N h0 = 0.97, N c0 = 8, n e0 = 10, n i0 = , m h = m c = 10 1 m i, = T e = 1 ev, Z c = 1000, and Fig. 9 Graph of the near critical compressive φ + and rarefactive soliton φ against η for different values of Z h for N h0 = 0.95, N c0 = 8, n e0 = 10, n i0 = , m h = m c = 10 1 m i, Ti = Te = 1 ev, Z c = 1000, and To examine the double layer formation, Sagdeev potential (0 under certain conditions (5 (7 gives the shock wave solution (30, double layer solution depends on the sign of B; i.e. for B > 0 a positive double layer exists (viz φ 1m > 0, whereas for B < 0 a negative double layer (φ 1m < 0 exists as shown in Fig. 11. In summary, it has been found that the presence of cold (hot dusty plasma density N c0 (N h0 and the charge numbers for negatively charged cold (hot dust Z c (Z h would modify the properties of the DAWs significantly and the results presented here should be useful in understanding salient features of localized electrostatic perturbations in Fig. 11 Graph of the shock waves φ + and φ against η for different values of Z h for N h0 = 0.9, N c0 = 8, n e0 = 10, n i0 = , m h = m c = 10 1 m i, = T e = 1 ev, Z c = 1000, and

8 150 Communications in Theoretical Physics Vol Conclusions In this work, we have investigated the properties of nonlinear DAWs in an unmagnetized plasma having electrons, singly charged ions, hot and cold dust grains. The reductive perturbation method has been used to reduce the basic set of fluid equations to the well-known KdV equation. We have shown that the presence of cold (hot dusty plasma density n c (n h and the charge numbers for negatively charged cold (hot dust Z c (Z h not only modifies the basic properties of the dust acoustic solitary potential structures, but also causes two different potential profiles, namely compressive and rarefactive pulses. In other word, the presence of additional dust components in our fluid model, two new parameters namely N c0 /N h0 and Z c /Z h can modify the basic properties of the dust acoustic solitary compressive and rarefactive structures. It is seen that there is a certain value that the hot dusty plasma density N h0 reaches to the so-called critical density value, where the KdV equation cannot describe the propagation of the solitary pulse. We have cleared that the compressive (rarefactive soliton amplitude increases (decreases as the wave propagates from low to high-density region. When the KdV equation is not appropriate to describe the system, we use a new stretching in order to derive mkdv equation for describing the system at the critical case. However, in the vicinity of critical hot dusty plasma density N h0, neither the KdV nor mkdv equations could describe the solitary wave propagating. Therefore, we have derived the fmkdv equation. Finally, we examined the double layer solution and existence conditions via Sagdeev potential analysis. It is found that this system supports both solitons and shocks for a plasma parameters satisfy certain conditions. To summarize, we have shown that the presence of additional dust components in our fluid model not only modifies the basic properties of the DAWs structures, but also causes the existence of two different potential profiles, namely, compressive and rarefactive pulses depends on the ratio of cold to hot dusty plasma density and charge N c0 /N h0 and Z c /Z h, which is a new feature in dusty plasma. We inject to that the analytical model demonstrated here can provide a useful basis for the interpretation of recent observations of solitary wave in dusty plasma environments. For example, the results presented may be applicable to dusty plasma existing in Saturn F-ring s region. References [1] F. Verheest, Waves in Dusty Plasma, Kluwer Academic, Dordrecht (000. [] P.K. Shukla and A.A. Mamun, Introduction to Dusty Plasma Physics, Institute of Physics, Bristol (00. [3] D.A. Mendis and M. Rosenberg, Annu. Rev. Astron. Astrophys. 3 ( [4] M. Horonyi, Annu. Rev. Astron. Astrophys. 34 ( [5] F. Verheest, Space Sci. Rev. 77 ( [6] P.K. Shukla and A.A. Mamun, New J. Phys. 5 ( [7] E.C. Whipple, Rep. Prog. Phys. 44 ( ; M. Hornayi, Annu. Rev. Astron. Astrophys. 34 ( [8] A. Barkan, N. D Angelo, and R.L. Merlino, Phys. Rev. Lett. 73 ( [9] N. D Angelo, J. Phys. D 37 ( [10] V.W. Chow, D.A. Mendis, and M.J. Rosenberg, Geophys. Res. [Space Phys.] 98 ( [11] N.N. Rao, P.K. Shukla, and M.Y. Yu, Planet. Space Sci. 38 ( [1] A.A. Mamun, R.A. Cairns, and P.K. Shukla, Phys. Plasmas 3 ( [13] P.H. Sakanaka and P.K. Shukla, Phys. Scr. 84 ( [14] R. Roychoudhury and S. Mukherjee, Phys. Plasmas 4 ( [15] S. Mahmood and H. Saleem, Phys. Plasmas 10 ( [16] N. Akhtar, S. Mahmood, and H. Saleem, Phys. Lett. A 361 ( [17] Y. Kodama and T. Taniuti, J. Phys. Soc. Jpn. 45 ( [18] S. Watanabe, J. Phys. Soc. Jpn. 35 ( [19] S.R. Sagdeev, Rev. Plasma Phys. 4 ( [0] S.G. Tagare, Phys. Plasmas 7 ( [1] M. Wadati, J. Phys. Soc. Jpn. 38 ( ; P.K. Shukla, and M.Y. Yu, J. Math. Phys. 19 ( [] A.E. Mowafy, E.K. El-Shewy, W.M. Moslem, and M.A. Zahran, Phys. Plasmas 15 ( [3] P.K. Shukla, D.A. Mendis, and T. Desai, (eds., Advances in Dusty Plasmas, World Scientific, Singapore (1997 p. 3. [4] S. Mahmood, Q. Haque, and H. Saleem, Chin. Phys. Lett. 18 ( [5] T. Farid, A.A. Mamun, P.K. Shukla, and A.M. Mirza, Phys. Plasmas 8 (