1 Introduction. Commun. Theor. Phys. 63 (2015) Vol. 63, No. 6, June 1, B.C. Kalita 1, and R. Kalita 2,

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1 Commun. Theor. Phys Vol. 63, No. 6, June, 05 A New Approach to Energy Integral for Investigation of Dust Ion Acoustic DIA Waves in Multi-Component Plasmas with Quantum Effects in Inertia Less Electrons B.C. Kalita, and R. Kalita, Department of Mathematics, Gauhati University, Guwahati-7804, Assam, India Research Scholar, Department of Mathematics, Gauhati University, Guwahati-7804, Assam, India Received November 6, 04; revised manuscript received February 9, 05 Abstract Dust-ion acoustic waves are investigated in this model of plasma consisting of negatively charged dusts, cold ions and inertia less quantum effected electrons with the help of a typical energy integral. In this case, a new technique is applied formulating a differential equation to establish the energy integral in case of multi-component plasmas which is not possible in general. Dust-ion acoustic DIA compressive and rarefactive, supersonic and subsonic solitons of various amplitudes are established. The consideration of smaller order nonlinearity in support of the newly established quantum plasma model is observed to generate small amplitude solitons at the decrease of Mach number. The growths of soliton amplitudes and potential depths are found more sensitive to the density of quantum electrons. The small density ratio r= f with a little quantized electrons supplemented by the dust charges Z d and the in-deterministic new quantum parameter C are found responsible to finally support the generation of small amplitude solitons admissible for the model. PACS numbers: 5.7.Lw, 5.35.Mw Key words: dust-acoustic solitary waves, quantum effects Introduction In the last few years, there has been an extensive study in the new domain of plasma physics relating to dusty plasmas. Besides the usual components of ions and electrons, heavier charged dust grains are seen to play characteristic role in the formation of solitary waves in space plasma. This is indeed a fascinating structure in space laboratory that can also be verified experimentally in terrestrial atmosphere. Dust grains are abundantly found in planetary rings, cometery tails likewise in Halley s comet, Saturn s E and F rings, neighborhood of stars, [ ] in inter stellar medium, in asteroid zones and Earth s lower ionosphere magnetosphere. The presence of dust particles acting as negatively charged in general dust grains of micrometer or submicrometer size because of field emission, plasma current could change the properties of plasma waves in space. It has been found that the presence of static charged dust grains in plasma, [3] can modify the existing wave spectra and can generate extremely low frequency dust acoustic wave, [] in absence of magnetic field. The fluctuations of dust charge are found to act as damping source to dust acoustic waves. Many results with minor corrections of negative ion plasmas can be adapted to dusty plasma for its low frequency behavior when the wave length and the inter-particle distance are much larger than the grain size. By means of distribution of immobile dust particles in plasmas, creation of spatial in homogeneity can be seen in interpreting the low frequency noise enhancement observed by the Vega and Goitto space probes in the dusty regions of Halley s Comet. Dust acoustic waves DA, [] which is propagating linearly as a normal mode and non-linearly as supersonic solitons of either positive or negative electrostatic potentials in a dusty plasma with inertial charged dusts and Boltzmann distributed electrons and ions have been studied. Many authors [,4 7] have investigated dust ion acoustic waves DIAWS under various plasma compositions. In the study of nonlinear dust acoustic waves, [8] in an unmagnetized dusty plasma with the effects of vortex-like and non thermal ion distributions, large amplitude rarefactive as well as compressive solitons are reported to exist. In a similar approach with sufficient non thermality in the ions following Cairns distribution, [9] and sufficient negative charges on the dust along with electrons, [0] investigated the existence of both positive and negative solitary structures in some parameter space. Besides, taking the non thermal parameter in the non thermal distribution of ions as variable, DA solitary waves in dusty plasma, [] studied with the help of pseudo-potential method. Dust acoustic solitary waves, [] are also studied through the Zakharov Kuznetsov ZK equation in obliquely propagating magnetized dusty plasma with mobile negatively charged dusts, two temperature Maxwellian ions and non bckalita3@gmail.com rekhakalita973@gmail.com c 05 Chinese Physical Society and IOP Publishing Ltd

2 76 Communications in Theoretical Physics Vol. 63 inertial electrons. Rarefactive solitons in magnetized non thermal dusty electronegative plasma in which width is shown to decrease at the increase of B o. [3] But it is also reported that the amplitude remains unaffected. The nonlinear DIA waves, [4] which are leading to shock waves are also observed experimentally. The nonlinear DIA waves and dust ion-acoustic shock waves, [5] are also studied which are implicitly related to KdV-Burger equation due to non adiabatic charge variation of dust particles. Further in IA waves in dusty plasma, [6] the troughs of the first sinusoidal pulse is found to become shallow and mach velocity of the peak in the oscillation experimentally decrease when negatively charged dusts are mixed in the plasma. The dust ion acoustic DIA solitons, [7] found in a dusty plasma having high electron energy distribution. The consideration of quantum effects in the formation of solitary waves in plasma is a recent trend of research. The de-broglie wavelength of the charge carriers i.e., λ B = /πmv becomes comparable to the spatial scale of quantum system and therefore the quantum effects are expected to play an important role in plasma dynamics. Hence in such a case, at an extremely low temperature, the ultra-cold dusty plasma behaves like a Fermi gas and the quantum mechanical effects affectively change the behavior of charged particles. The quantum hydro-dynamical model QHM enshrined in Schrodinger Poisson SP equation generalizes the fluid model for plasmas taking into consideration macroscopic variables like density, velocity field, stress tensor and electrostatic potentials. Quantum drift waves and quantum surface waves are studied in Refs. [8] [9]. Quantum effects and applications are remarkably perceived in electronic device, [0] expansion of electron quantum electron gas into vacuum, [] quantum plasma echos, [] laser-produced plasma, [3] dense astrophysical environments, [4 5] quantum Penrose diagram, [6] Fermi gases. [7] The range of validity of quantum plasmas, [6] and regimes for several mathematical models, [8 9] in quantum plasmas are described along with examples. The linear, weakly nonlinear and fully nonlinear characteristic behavior of quantum ion-acoustic waves QIAW, [8] and for ultra-cold plasma, [30] pure quantum effects and applications are also perceived. Also, dust acoustic solitary waves were studied through the KdV equation by using quantum hydrodynamic model. [3] Further the amplitude modulation of dust-acoustic waves in three-species plasmas is studied by, [3] using non-linear Schrodinger equation NLSE with quantum effects. They have reported that the quantum effects can suppress the instability. On the other hand, the authors, [33] have found the existence of both compressive and rarefactive double layers only for positively charged dust particles. The exact soliton solutions for complex nonlinear Schrodinger equation, [34] were established using the semi-inverse variational principle SVP. The multiple wave solutions of nonlinear partial differential equations NLPDEs, [35] were found by using the multiple exp function method. Dust ion-acoustic DIA solitary waves, [36] in magnetized quantum dusty plasma employing Zakharov Kuznetsov ZK equation for small amplitude perturbations have been studied. In their analysis, the effects of quantum corrections, concentration of dust grains and angles of propagation are investigated on the formation of DIA solitons. The stationary dusts at the background plasma and inertia less electrons with quantum effects have been included in the magnetized two-dimensional ion-fluid plasma. The low frequency electrostatic drift like waves in a non-uniform collisional quantum magnetoplasma is also investigated. [37] In this paper, we have investigated dust ion-acoustic DIA solitary waves in plasma with negatively charged dusts, ions, and inertia less electrons with quantum effects. Basic Equations Governing the Dynamics of Motion The basic equations governing this model of plasma are as follows: For negatively charged dusts, n d t + n dv d x = 0, For ions, n i t + n iv i v d t + v d v d x = φ x. x = 0, 3 v i t + v v i i x = φ Q x, with Q = m i. m d 4 For inertia less electrons with quantum effects, 0 = φ x + H e x with B = v Fe α me Cs r He ωpd =, Z d kt i Q d and Poisson equation n e n e ne x Bn e x, = v Fe Q d m i Cd, r Q d = m e m d, 5 d φ dx = n e + z d n d n i, 6 where we consider the system of Eqs. 6 in nondimensional form by normalizing densities by the equilibrium ion density n io, velocities by the dust-acoustic speed C d = Z d kt i /m d /, distances by the Debye length λ Di = Z d kt i /4πn io e /, electrostatic potential φ by Z d kt i /e and time by the λ Di /C d according to the procedure of Ref. [0].

3 No. 6 Communications in Theoretical Physics 763 The linear dispersion relation of our dust-ion acoustic mode is established as ω = [{ V Qr C dk { Q r} Fe me Cd + m d 4 H eλ Dik } { V Fe me + m d 4 H eλ Dik } λ ] Di k. r C d For negligible value of V Fe /C d m e/m d, it can be put in the form ω = Q C dk { Q r} 4r H eλ Dik 4r H eλ 4 Dik 4. For small value of the Fermi velocity V Fe than usual, though VFe /C d >, the quantum effect due to H e is explicitly accountable for linear waves as observed from the sound form in presence of heavy dust mass m d in the ion background plasma. But for the usual Fermi velocity, the quantum effect in this dusty plasma as seen from the first expression is not dominantly accountable for the growth of waves. Due to the presence of heavy dust mass with dominant ion background in the plasma, small amplitude waves in the thermal electron limit are expected. Introducing the new coordinate η defined by η = x Mt, we use the boundary conditions n i = n io, n d = n do, n e = n eo, v i = 0, v d = 0 and φ = 0 at η, integrating results of Eqs. 5 and from Eq. 6 we get, v d = M n do, 7 n d n d = n do + φ /, 8 M v i = M n io, 9 n i n i = n io φ /, 0 QM B n eo = φ B n e + n e H e ne η, d φ dη = n e + z d n d n i. By observing the nature of the terms of Eq., it is expected that n e must be a function of φ. We consider ne = fφe φ with unknown function fφ and get the following expression, n e dφ η = e φ [fφ + f φ + f φ] dη + e φ [fφ + f φ] d φ dη. 3 To transform n e / η in terms of d φ/dη for the pursuit of obtaining the energy integral, we equate the coefficient of dφ/dη to 0 in search of maximum φ always for some fφ, so that fφ+f φ+f φ = 0, which gives the solution fφ = C + C φe φ. 4 Using the above solution of Eq. 4, Eq. 3 can be reduced to, n e η = C d φ dη, 5 with n e = C + C φ n e = C + C C φ + C φ and C = n eo. 6 It is seen from Eq. 5 that the quantum related particle density term involves C along with the wave potential φ. Otherwise, if C = 0, then the quantum related density term will not survive even if H e 0. 3 Validity of the Model for Exclusive Support of Smaller Order Nonlinearity in Our Assumption of Electron Density in Relation to an Equivalent Compatible Known Model for Determination of the Range of Quantum Related Parameter C Observations by Viking satellite, [38] counter-streaming electron distribution functions indicate that they are a common feature of the auroral zone. For the presence of distribution of non-thermal electrons with an excess of energetic particles, the nature of ion sound solitary structure changes and solutions with depletions are observed as predicted by Viking satellite [39] and Freja satellite. [40] In this context, an electron distribution with fast particles in one-dimensional structure can be modeled Eq. by Ref. [9] as n o f e v = 3α + πve [ + α v v e ] exp v ve where v is velocity of the structure, v e is thermal velocity of the electrons. In presence of a non-zero potential, the electron distribution in a steady state being function of electron energy can be found replacing v /ve by v /ve φ in the above expression. Since v /ve is very small therefore the replacement of it by v /ve φ is justified if φ is also small. Finally, the electron density is expressed Ref. [9] for v /ve as n e = n o βφ + βφ exp φ, where β = Equivalently, we may consider exp v [ exp v ] φ so that v e v e = exp = expφ λ, where λ = v ve, 4α 3α +. 7 φ <, v ve

4 764 Communications in Theoretical Physics Vol φ λ φ λ φ λ expφ λ = + φ λ = λ + λ + λ + λ φ + λ + 4 λ φ + higher orders of φ = A + Aφ + Aφ, neglecting higher powers of small φ where A = λ + λ. Substituting the above expression in Eq. 7, we get n e = βφ + βφ expφ λ = βφ + βφ A + Aφ + Aφ, normalized n e = A + A βφ + Aφ. Comparing this subsequent result with Eq. 6 i.e., equating the coefficients, we obtain n eo = A = λ + λ, C = A = λ + λ. The Cairn distribution of electrons with characteristic parameter β 0 is widely accepted which includes the Boltzmann relation n e = e φ for inertia less electrons when β = 0. Contextually, the contribution from the inertia less electrons in quantum sense to the electron fluid equation is given by H e [ ] ne x ne x. Consideration of inertia less electrons in the background assumption of both the cases prompts us to take n e = fφe φ and needs to search for suitable fφ through the above quantum affective particle density term. It is known to us that the transition from particle behavior physically to wave behavior mathematically is crucial in quantum sense. With this assumption, the relation 5 viz. n e / η = C d φ/dη in wave frame η= x Mt space is obtained. It is satisfied by Eq. 6 which is restricted to smaller order nonlinearity upto oφ. In this context, C is said as the indeterministic parameter related to quantum affective term in the electron fluid motion. For physical justification to support smaller order nonlinearity and to explore the basic range of the in-deterministic parameter C connected with quantum affective term, this section is incorporated. The Cairn distribution 7, reduces to Boltzmann distribution n e = n o exp φ for β = 0 and hence β 0 can be said to be deviation parameter from inertia less electrons imbibing therein smaller order nonlinearity as explained above. Noticeably, the quantum parameter H e occurs in the inertia less fluid equation of electrons is also dependent on electron inertia m e, since H e = ωpd Z d kt i md m e. In strict sense, electrons fluid equation with quantum effect H e is not fully independent of electron inertia. Hence H e 0 can also be said as deviation parameter from inertia less electrons and in this sense, β 0 and H e 0 can be put into the same footing. But under the small amplitude limit, the Cairns distribution 7 of electrons β 0 becomes equivalent to 6 if C 5/6 for the maximum value of λ = /. In conformity with small amplitude limit or small φ in quantum sense, the Cairns distribution will also admit this condition for β 0 within the already established range of β. Further, the indeterministic parameter C corresponding to the quantum affective term appears in the restricted smaller order nonlinear expression of n e subject to λ = /v /ve < and it is correlated with the quantum affected parameter C. The deviation parameter β 0 imbibes the notion of inertia less electrons and Eq. 6 with the appearance of C. Inertia less electrons being the genus of both the cases which also admit second order non-linearity of φ in the expansion of n e. Thus, i Inertia less electrons is the initial common basis for both Cairns distribution and the electrons fluid equation with quantum effects in this case, ii The deviation parameters β 0 and H e 0 can be put at the same footing, and iii The small amplitude wave potential φ inherent in quantum sense is also represented by Cairns distribution for some β 0 which correlates with C of ne = C + C φ, the derived relation from the quantum affective term of Eq.. For its established range and admittance of conditional imposition to represent small φ, β plays the role of bridging factor between H e and C. Hence so far as small amplitude plasma wave is concerned, C can be said to be an ad-hoc quantum parameter within the range of β. By means of our observation, the interval < C < of C is determined on the basis of the range of β. 4 Derivation of Energy Integral Using the corresponding result of n e in Eq., we get

5 No. 6 Communications in Theoretical Physics 765 HeC d φ C + C φ dη = B C + C φ 4 B n eo φ He d φ dη = B C C + C φ 5 B C n eoc + C φ φ C C + C φ. 8 Adding Eqs. and 8 and putting n d, n i from Eqs. 8 and 0, we obtain + H e d φ dη = C + C φ + Z d n do + φ M / nio φ QM / + B C C + C φ 5 B C n eo C + C φ φ C C + C φ. Multiplying both sides of the above equation by dφ/dη gives us, d dφ ] [ = dη[ dη + He C + C φ + Z d n do + φ / nio M φ / B + QM C + C φ C B n C eoc + C φ φ ] dφ C + C φ C dη. We get the energy integral after integration as dφ + V φ = 0, where 9 dη [ { V φ = + He {C + C φ 3 C 3 3C } + M Z d n do + φ / } { M + QM n io φ / } QM + B 6C {C + C φ 6 C 6 } Bn eo C {C + C φ C } φ C C + C Equation 9 is the energy integral in terms of the Sagdeev potential V φ given by 0. φ 3 ] Conditions for the Existence of Solitary Waves Appropriate and necessary conditions for the existence of localized solitary wave solutions can be obtained by studying the behavior of the potential V φ near φ = 0, and φ = φ m, where φ m is the maximum value of φ i.e., the amplitude of the solitary wave pulse. For solitary wave solutions, the conditions required are V φ = V φ = 0, at φ = 0, and V φ < 0 between φ = 0 and φ = φ m. Setting V φ m = 0, we can derive the non-linear dispersion relation for amplitude φ m as { {C + C φ m 3 C 3 3C } + M Z d n do + φ / } { m M + QM n io φ / } m QM + B 6C {C + C φ m 6 C 6 } Bn eo C {C + C φ m C } φ m C C + C φ 3 m = This gives the soliton amplitude φ m. Expanding V φ by Taylor s expansion theorem near φ = 0 and φ = φ m, we get [ V φ 0 = φ m + He C C + 4Bn Zd n do eo M + n io QM + C ], 4 C V φ φ m = φ [ m + He C + C φ m + Z d n do + φ / m nio M φ / m B + QM C + C φ m 5 C Bn eo C + C φ m ] C φ m + C φ C C m. 5 From the relation 4, we observe that solitons will exist subject to if C C + 4Bn eo Zd n do M + n io QM + C > 0. C Using the relation n eo + Z d n do = n io, so that n eo /n io = f with f = Z d n do /n io otherwise n eo /n io =r = f, above condition can be put in the form C + 4Bn eo neo C QM + f M r > 0. 6 For φ m > 0, the expression of the parenthesis of Eq. 5

6 766 Communications in Theoretical Physics Vol. 63 must also be positive and for φ m < 0, it must be negative. 6 Discussion In this model of multi-component un-magnetized dusty plasmas with three distinct components, we have skillfully derived the Energy integral 9 to characterize solitary waves in such a situation which is usually not possible in general. From the nature of nonlinear dispersion equation 3, it is seen that for real solution of solitons amplitudes φ m, φ m /QM <, otherwise φ m < QM / and φ m < M /. This is quite a new investigation in three-component quantum plasmas in presence of dust particles to yield solitary waves through the energy integral. To make the Sagdeev potential V φ < 0 in between φ = 0 and φ m maximum = 0, the suitable positive and negative values of C characteristically related to the quantum affective term that involve in the solution of the generic differential equation and are assigned from the range already established in Sec. 3. For the characteristic new parameter C positive or negative of our quantum plasma model of investigation, both subsonic compressive and rarefactive as well as supersonic compressive and rarefactive DIA solitons are found to exist. The expected small amplitude for the model rarefactive DIA solitons for positive values of C and compressive solitons for negative values of C are shown to exist. The consideration of smaller order nonlinearity in this quantum plasma model is responsible for the generation of small amplitude solitons based on the characteristic parameter C. of plasma particles. The small density ratio r= f with a little quantized electrons supplemented by the dust charge Z d and the deterministic parameter C are responsible to finally support the generation of small amplitude solitons admissible for the new model. The small amplitude rarefactive supersonic solitons in this quantum plasma model are found also to increase with the decrease of the new parameter C = 0.38, 0.36, 0.34 Fig. for the fixed Mach number M =.6 > and the other fixed parameters as shown in the figure. Smaller values of C compared to those of Fig. are seen to produce Fig. relatively higher amplitude supersonic instead of subsonic solitons for the same set of parameters subject to the condition 4 within the specific restricted domain. Fig. Sagdeev Potential V φ versus amplitude φ of rarefactive solitons in quantum plasma for fixed values of M =.6, Q = 0.5, Q d = , H e = 0, α = 0., r = 0.000, B = with C = 0.34 dashed line, C = 0.36 solid line and C = 0.38 dotted line. Fig. Sagdeev Potential V φ versus amplitude φ of rarefactive solitons in quantum plasma for fixed values of M = 0.6, Q = 0.5, Q d = , H e = 0, α = 0., r = 0.000, B = with C = 0.46 solid line, C = 0.5 dashed line and C = 0.54 dotted line. The small amplitude in conformity with the model of rarefactive subsonic solitons of the quantum plasma increases with the decrease of the new parameter C = 0.54, 0.5, 0.46 Fig.. The potential depths of small amplitude rarefactive solitons are found smaller for higher values of C which accommodate relatively less number Fig. 3 Sagdeev Potential V φ versus amplitude φ of compressive solitons in quantum plasma for fixed values of M =., Q = 0.3, Q d = , H e = 0, α = 0.5, C = 0.4 and B = , , corresponding to r = dotted line, solid line, dashed line. The slightest increase in r causes to increase the amplitude as well as the potential depths Fig. 3 of the supersonic M =. > compressive solitons for C = 0.4 and other parameters as in the figure. The faster plasma particles M =. are trapped in a deeper potential to produce higher amplitude compressive solitons Fig. 3 for small C = 0.4. To the contrary, relatively slower plasma particles M = 0.94 are

7 No. 6 Communications in Theoretical Physics 767 contained in a much shallow potential to yield smaller amplitude compressive solitons Fig. 4 for the higher value of C = The other parametric values are same for both the figures. Fig. 4 Sagdeev Potential V φ versus amplitude φ of compressive solitons in quantum plasma for fixed values of M = 0.94, Q = 0.3, Q d = , H e = 0, α = 0.5, C = 0.54 and B = , , corresponding to r = dotted line, solid line, dashed line. increases to its upper limit. Of course, amplitude increases at the marginal increase in r and decrease of parameter B. The growth of the rarefactive supersonic solitons is dependent on the new parameter C characterizing this quantum model of investigation. The small amplitude subsonic DIA rarefactive solitons are seen to increase with the Mach number M = 0.5, 0.56, 0.6 Fig. 6 for the three positive values of C within the defined range and for the other parameters r = 0.000, Q = 0.5, Q d = , α = 0., H e = 0, and B = The small wave speed M justifiably supports generation of small amplitude solitons valid for the model. It is observed, smaller the values of M, smaller is the soliton amplitude for the other fixed parameters as in Fig. 6. Thus, the trend of generation of gradually smaller amplitude rarefactive solitons at the decrease of the Mach number definitely justifies the feasibility of the model. Fig. 5 Amplitude φ of rarefactive solitons versus C in quantum plasma for fixed values of M =.6, Q = 0.5, Q d = , α = 0., H e = 0 and B = , , corresponding to r = dotted line, solid line, dashed line. Fig. 7 Sagdeev Potential V φ versus amplitude φ of compressive solitons in quantum plasma for fixed values of Q = 0.3, Q d = , H e = 0, α = 0.5, r = 0.000, B = and C = 0.3, 0.36, 0.4 corresponding to the Mach numbers M =.6,.,.04. Fig. 6 Sagdeev Potential V φ versus amplitude φ of rarefactive solitons in quantum plasma for fixed values of Q = 0.5, Q d = , H e = 0, α = 0., r = 0.000, B = and C = 0.45, 0.5, 0.55 corresponding to the Mach numbers M = 0.6, 0.56, 0.5. In this model of quantum plasma, the small amplitude supersonic M =.6 rarefactive soliton tends to be smaller and smaller for all r Fig. 5 at the fixed parameters as shown in the figure when the new parameter C Fig. 8 Graph between B versus r for fixed values of M =.6, α = 0.5, 0., 0.5, Q = 0.5, Q d = and H e = 0. The small amplitude compressive solitons are found to increase for the three negative values of C = 0.3, 0.36, 0.4 corresponding to the supersonic speeds M =.04,.,.6 Fig. 7 and for the other fixed parameters as in the figure. Otherwise, smaller the value of M, smaller is the amplitude of the compressive solitons also in conformity with the model.

8 768 Communications in Theoretical Physics Vol. 63 Further, the implicit occurrence of plasma temperature α = T e /T i in B through C d plays an important role as B in this dusty quantum plasma. For small α, B attains high value primarily for high Fermi-ion speed ratio V Fe /C and becomes small as r increases Fig. 8 to Otherwise, for small r and α giving rise to higher value of B as evident in Fig. 8, it becomes more effective in the formation of DIA solitons in this quantum plasma. The important consequences in the behavior of high density astrophysical plasmas where H e is nearer to unity, and diagnostics of charged dust impurities in microelectronic devices, quantum effects can be used. Such studies may be helpful to understand the interior of super-dense white dwarfs, neutron stars, super-dense giant planets. [4] Acknowledgments The authors acknowledge the referee with gratitude for constructive suggestions to improve the standard of the paper. References [] N.N. Rao, P.K. Shukla, and M.Y. Yu, Planet Space Sci [] F. Verheest, Planet Space Sci [3] P.V. Bliokh and V.V. Yaroshenko, Sov. Astron [4] A. Barkain, R.L. Merlino, and N. D Angelo, Phys. Plasmas [5] P.K. Shukla and A.A. Mamun, Introduction to Dusty Plasma Phys, IOP, London 00. [6] W.S. Duan, X.R. Hong, Y.R. Shi, and J.A. Sun, Chaos, Solitons and Fractals [7] Y.Y. Wang and J.F. Jhang, Phys. Lett. A [8] A.A. Mamun, R.A. Cairns, and N. D Angelo, Phys. Plasmas [9] R.A. Cairns, A.A. Mamun, R. Bingham, R. Dendy, R. Bostrm, C.M.C. Nairns, P.K. Shukla, Geophys. Res. Lett [0] F. Verheest and S.R. Pillay, Phys. Plasmas [] H.R. Pakzad, Astrophys. Space Sci [] M.M. Masud, M. Asaduzzaman, and A.A. Mamun, Phys. Plasma [3] N.R. Kundu, M.M. Masud, K.S. Ashraf, and A.A. Mamun, Astrophys. Space Sci [4] Y. Nakamura, H. Bailung, and P.K. Shukla, Phys. Rev. Lett [5] S. Ghosh, S. Sarkar, H. Khan, and M.R. Gupta, Phys. Lett. A [6] Y. Nakamura and A. Sarma, Phys. Plasmas [7] M. Shahmansouri and M. Tribeche, Commun. Theor. Phys [8] B. Shokri and A.A. Rukhadze, Phys. Plasmas [9] B. Shokri and A.A. Rukhadze, Phys. Plasmas [0] P.A. Markowich, C.A. Ringhofer, and C. Schmeiser, Semiconductor Equations, Springer, Vienna 990. [] S. Mola and G. Manfredi, J. Plasma Phys [] G. Manfredi and M. Feix, Phys. Rev. E [3] D. Kremp, Th. Bomath, M. Bonitz, and M. Schlanges, Phys. Rev. E [4] Y.D. Jung, Phys. Plasmas [5] M. Opher, L.O. Silva, D.E. Dauger, V.K. Decyk, and J.M. Dawson, Phys. Plasmas [6] F. Hass, G. Manfredi, and M. Feix, Phys. Rev. E [7] G. Manfredi and F. Hass, Phys. Rev. B [8] F. Hass, L.G. Garcia, J. Goedert, and G. Manfredi, Phys. Plasmas [9] G. Manfredi, Fields Inst. Commun [30] T.C. Killian, Nature London [3] S. Ali, P.K. Shukla, Phys. Plasmas [3] A. Misra, A. Roy Chowdhury, Phys. Plasmas [33] W.M. Moslem, P.K. Shukla, S. Ali, and R. Schlickeiser, Phys. Plasmas [34] M. Najafi and S. Arbabi, Commun. Theor. Phys [35] M.T. Darvishi, M. Najafi, L. Kavitha, and M. Venkates, Commun. Theor. Phys [36] S.A. Khan, A. Mushtaq, and W. Masood, Phys. Plasmas [37] S. Ali, N. Shukla, and P.K. Shukla, Europhys. Lett [38] R. Lundin, L. Eliasson, B. Hultqvist, and K. Stasiewicz, Goephys. Res. Lett [39] R. Bostrom, IEEE Trans. Plasma Sci [40] P.O. Dovner, A.I. Eriksson, R. Bostrom, and B. Holback, Geophys. Res. Lett [4] G. Chabier, F. Douchin, and A.Y. Potekhin, J. Phys. Condens. Matter [4] F. Verheest, Waves in Dusty Space Plasmas, Kluwer Academic, Dordrecht 000.