Damped dust-ion-acoustic solitons in collisional magnetized nonthermal plasmas
aa r X i v : . [ phy s i c s . p l a s m - ph ] J a n Damped dust-ion-acoustic solitons in collisional magnetized nonthermal plasmas: therole of dissipation and ions temperature
M. R. Hassan and S. Sultana Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh
A multi-species magnetized collisional nonthermal plasma system, containing inertial ion species,non-inertial electron species following nonthermal κ − distribution, and immobile dust particles, isconsidered to model the dissipative dust-ion-acoustic (DIA) soliton modes, both theoretically andnumerically. The electrostatic solitary modes are found to be associated with the low frequencydissipative dust-ion-acoustic solitary waves (DIASWs).The ion-neutral collision is taken into ac-count and the influence of ion-neutral collisional effects on the dynamics of dissipative DIASWs areinvestigated. It is reported that most of the plasma medium in space and laboratory are far fromthermal equilibrium, and the particles in such plasma system are well fitted via the κ − nonthermaldistribution than via the thermal Maxwellian distribution. The reductive perturbation approachis adopted to derive the damped KdV equation, and the solitary wave solution of damped KdVequation is derived via the tangent hyperbolic method to analyze the basic features (amplitude,width, speed, time evolution, etc.) of dissipative DIASWs. The propagation nature and also thebasic features of dissipative DIASWs are seen to influence significantly due to the variation of theplasma configuration parameters and also due to the variation of the supethermality index κ inthe considered plasma system. The implication of results of this study could be useful of betterunderstanding the electrostatic localized disturbances, in the ion length and time scale, in spaceand experimental dusty plasmas, where the presence of excess energetic electrons and ion-neutralcollisional damping are accountable. I. INTRODUCTION
The origin and propagation of nonlinear electrostaticwaves in dusty plasma have drawn the attention of re-searchers in the last few decades after the first predic-tion of the existence of these waves in Saturn’s ring byBliokh and Yarashenko in 1985 [1], after which scientistsdiscovered several new modes of nonlinear excitations,e.g., dust ion acoustic waves (DIAWs) [2], dust acousticwaves (DAWs) [3], dust lattice waves (DLWs) [4], etc.Dust ion acoustic solitary structures are a type of dust-ion-acoustic wave (DIAW) are formed in a plasma systemdue to the compromise between the dispersion and thenonlinearity and [43] form stable hump or dip shapedstructures. In a DIAW, the mass density of ions is con-sidered to provide the inertia, while the electrons thermalpressure is assumed to give the restoring force to gener-ate the wave, and the negatively charged dust species areapproximated to remain static. Viking Satellite [5]andTHEMIS mission [6] have already identified this typeof wave in both spaces (viz., solar atmosphere [7], Sat-urn’s magnetosphere [7], pulsar magnetosphere [8], activegalactic nuclei [9], neutron star’s polar region [10], etc.)and laboratory environments (viz., semiconductor plas-mas [9], cathode discharge [8], tokamaks [9], etc.).From the rudimentary stage of plasma physics, it wasestablished that only a fractional part of electrons ac-quire higher kinetic energy than that of the most re-mainings and thus was modeled to follow the Maxwelldistribution [20]. After observing the dominating partof electrons possessing excess energy in space plasmas[21–24] and laboratory environments [40–42], they wereformulated to follow the superthermal κ -distribution [25–27]. The researchers have carried out much research work in the last few decades to study the origin, propaga-tion, and characteristics of the rogue structures [28–30],shock structures [31–33], envelope solitons [34–36], soli-tary structures [15, 37–39, 54], etc. plasmas containingkappa-distributed electrons/ positrons/ ions.There is always a chance for the solitary pulses to getdamped due to dissipation in the plasma medium gradu-ally. Apart from the existence of the collision betweendifferent plasma components [44, 45], the ion-acousticsolitary structures can also get dissipated if the ions nolonger remain cold instead become hot enough comparedto that of the electron species, i.e., due to the ion tem-perature effect [46–48]. As most of the practical systemsaround us are not in equilibrium, almost every wave hasto suffer a certain amount (more or less) of dissipation.This influenced the researchers to study solitary wave’sattenuation in various plasma mediums [12–15] and non-linear optics [16–19]. Mamun [46] in 1997 considered aplasma medium containing no dust and studied the ef-fect of ion temperature on electrostatic solitary struc-tures in unmagnetized collisionless non-thermal plasmas.Later, Chatterjee et al. [47] carried out a similar investi-gation to determine ion temperature’s effect on the soli-tary structures in a quantum electron-ion plasma. Royet al. [48] also conducted a research work analogous toMamun in which he assumed a plasma consisting of aq-nonextensive electron species. Sing [53] extended thework of Chatterjee by considering the influence of themagnetic field. Sultana [54] took a huge step to considerthe magnetic field effect and included dust in her system,but she did not take the effect of hot ion fluid into ac-count in her investigation. This motivated us to observethe origin, evaluation, streaming, and the fundamentalproperties of nonlinear solitary excitations in a plasmamedium comprised of negatively charged dust, hot ionfluid colliding with neutral particles, and super-thermalelectrons in the presence of an external magnetic field.The investigation of such a plasma system will enable usto analyze and understand the nonlinear structure’s be-havior in a magnetized complex dusty plasma in whichthe ion temperature effect and the energy dissipation ef-fect are considered at the same time.This paper is organized in the following manner: Thebasic governing equations for describing our plasma sys-tem are shown in section II. Section III contains thederivation of the damped KdV equation using the reduc-tive perturbation model. The analytical solutions rep-resenting dissipated solitary pulses are given in sectionIII A. The numerical simulations are shown in sectionIII B. Finally, the summary of our research work is dis-cussed briefly in section IV. II. THE PLASMA MODEL AND BASICFORMALISM
To study the nature of the low frequency obliquelypropagating solitary waves in ion time (length) scale, weconsider a three-component magnetized collisional com-plex/dusty plasma system consisting of • inertial ion fluid of mass m i and charge z i e with e being the electronic charge and z i being the ioncharge state; • inertialess electrons of mass m e and charge − e fol-lowing nonthermal κ − distribution; and • micron/submicron sized massive negativelycharged static dust of mass m d and charge − z d e with z d being the dust charge state.Thus, at equilibrium the plasma quasi-neutrality condi-tion reads: n e + z d n d = z i n i , where n s represents thenumber density of plasma species s at equilibrium (here s = e, i, d correspond to electron, ion, and dust respec-tively). We also assume that the ambient constant mag-netic field is acting along the z -axis (i.e., B = B ˆ z ). Itis noted that the phase speed of DIAWs in such a plasmamedium is much greater than the thermal speed of ionand dust but much smaller than the electron’s thermalspeed (i.e., v th,e ≫ v ph ≫ v th,i,d ). The ion-neutral colli-sions and also the thermal effect of inertial ion populationon the dynamics of obliquely propagating DIASWs aretaken into account. One can, therefore, describe the dy-namics of DIASWs in such a dusty plasma medium bythe following set of fluid equations ∂n i ∂T + ∇ . ( n i u i ) = 0 , (1) ∂ u i ∂T + ( u i . ∇ ) u i = − z i em i ∇ Φ + z i eB m i ( u i × ˆ z ) − k B T i m i n i ∇ n i − υ in u i , (2) ∇ Φ = 4 πe ( n e + z d n d − z i n i ) , (3)where n i , n e , and n d are number densities of ions,electrons, and dust grains, respectively. u i is the velocityof inertial ion fluid, Φ is the electrostatic potential, T i is the characteristic temperature of ion fluid, and υ in isthe ion-neutral collision frequency.The number density expression for the non-thermal κ − distributed electron has the form [32, 54, 55] n e = n e (cid:20) − e Φ k B T e ( κ − / (cid:21) − κ + , (4)here T e is the characteristic temperature of the electron, k B is the Boltzmann constant, and κ is the superthermal-ity parameter which determines strength of superther-mality or non-thermality of the plasma medium. It isclear in (4) and also in Refs. [11], [27], and [7] that fora physically meaningful particle distribution, one shouldconsider κ > /
2. It should also be noted from existingresearch [28, 32] that the values of κ can lie in the range ∞ ≥ κ ≥ / κ defines strongernon-thermality and larger values of κ means weaker non-thermality, and the plasma medium is considered to beMaxwellian for the limit κ → ∞ .To normalize our plasma model equations (1) - (3), weconsider the following scaling factors ∇ → ∇ λ D , t → Tω − pi , u → u i C i , n → n i n i ,φ → e Φ k B T e , υ in → ω pi υ i . The Debye length λ D and the characteristic plasma fre-quency ω pi are given by λ D = q k B T e πn i z i e ,ω pi = q πn i z i e m i , (5)so that the condition of ion thermal speed C i = ω pi λ D =( z i k B T e /m i ) / can be fulfilled. Now if we take µ = z d n d /z i n nio , Ω ci = ω ci /ω pi (in which ω ci = z i eB /m i ),and σ = T i / ( z i T e ), the normalized form of equations (1)-(3) can be written as ∂n∂t + ∇ . ( n u ) = 0 , (6) n ∂ u ∂t + n ( u . ∇ ) u = − n ∇ φ + Ω ci n ( u × ˆ z ) − σ ∇ n − nυ i u , (7) ∇ φ = 1 − n + pφ + qφ . (8)The coefficients p and q [appear in equation (8)], whichcontain the information of plasma non(super)thermalityvia the superthermality index κ , are expressed as p = (1 − µ )( κ − . κ − . q = (1 − µ )( κ − . κ +0 . κ − . . (9)As mentioned above, for the limit κ → ∞ , the dis-tribution will no longer be nonthermal and the particledistribution approaches to the thermal Maxwellian. Wewill consider the numerical value of superthermality in-dex κ ≥ . < κ ≤ III. DAMPED DIASWs: PERTURBATIVEAPPROACH
To examine the nonlinear dynamics of small but finiteamplitude solitary waves in magnetized collisional plas-mas, we introduce the stretched coordinates as ξ = ǫ / ( l x x + l y y + l z z − V t )) ,τ = ǫ / t, ) (10)where ǫ is a small expansion parameter which charac-terizes the weakness of the dispersion (0 < ǫ <
1) and V is the phase speed of DIAWs (normalized by C i ) andto be determined later. Here l x , 1 y , and l z symbolizethe directional cosines of the wave vector k along x , y ,and z -axes, respectively to the magnetic field B so that l x + l y + l z = 1. As we considered weak damping inour plasma medium due to ion-neutral collision, we cantherefore assume the following scaling for the ion-neutralcollision frequency υ i as υ i = ǫ / υ ′ . (11)Now, we expand the physical variables n , u , and φ nearequilibrium in the power series of ǫ as nu x,y u z φ = + ǫ n ǫ / u x,y u z φ + ǫ n u x,y u z φ + · · · (12)Now, we substitute our assumptions in (10) - (12)into the equations (6)-(8) and separating the lowest or-der terms of ǫ from the resultant equations, we obtainthe phase speed of obliquely propagating dust- acousticwaves (DAWs) in the form V = s l z p + σl z . (13) It is seen from Eq. (13) that the phase speed of DIAWsdepends on the obliquity angle δ (= cos − l z ), the numberdensities of dust and ions (via the dust-to-ion numberdensity µ ), the ion and electron temperature (via σ = T i /T e )a ratio of ion, and the superthermality index κ . ∆ = ∆ =
20 ° ∆ =
10 ° ∆ = Κ V (a) σ = . σ = . σ = . σ = . μ V (b) FIG. 1: (a) Phase speed V versus the superthermalityindex κ for different values of obliquity angle δ , whereion-to-electron temperature σ = 0 . µ = 0 .
4, and (b) V versus µ for differentvalues of σ , where κ = 3 and δ = 15 ° .The variation of phase speed V versus the superther-mality index κ for different values of obliquity angle δ isdepicted in Figure 1a. On the other hand, Figure 1bshows the variation of V versus the dust-to-ion num-ber density µ for different values ion-to-electron tem-perature σ . We see in Figure 1a that the phase speedof obliquely propagating DAWs is higher in Maxwellianplasmas in comparison to that in superthermal plas-mas, i.e., the phase speed in seen to lower for smallervalues of κ (stronger superthermality) and it becomeshigher for larger values of κ (moderately nonthermal orMaxwellian). Figure 1a also suggests that when the DI-AWs propagate through the plasma medium parallel (i.e., δ = 0 ° ) to the magnetic field, they achieve the higherphase speed. However, when the waves advance obliquely(i.e., δ = 0), the phase speed is predicted to decreasewith the increase in obliquity angle. Figure 1b displaysinfluence of the dust-to-ion number density µ and theion-to-electron temperature σ on the linear properties(i.e., the phase speed) of obliquely propagating DAWs.we found that the higher values of µ leads to the forma-tion/propagation of DAWs with higher phase speed inthe considered plasma medium, while the increase (de-crease) in ion (superthermal electron) temperature maylead to propagate the DAWs with higher phase speed, asdepicted in Figure 1b.Now we derive the x and y -components in terms ofelectric potential φ from the momentum equation bytaking the same coefficients as z -component, and theyare given as u x = − (cid:18) l y Ω ci + σl y l z Ω ci ( V − σl z ) (cid:19) ∂φ ∂ξ ,u y = (cid:18) l x Ω ci + σl x l z Ω ci ( V − σl z ) (cid:19) ∂φ ∂ξ . (14)We now separate the next order of ǫ from our modelequations (6)-(8), combine the resultant equations andeliminate n , u x,y,z and φ from the resultant equations(algebraic details are omitted here), and finally we con-sider φ = ψ and get the time evolution equation in theform of damped KdV (dKdV) equation as ∂ψ∂τ + αψ ∂ψ∂ξ + β ∂ ψ∂ξ + υψ = 0 , (15)where the nonlinear term α , dispersion term β , and dis-sipation (or damped) term υ are given, respectively, asfollows α = V (cid:18) p + p pσ ) − qp (1 + pσ ) (cid:19) , (16) β = V p (cid:18) pσ ) + (1 − l z )(1 + pσ )Ω ci (cid:19) , (17) υ = υ ′ . (18)It is well known that the mutual balance between thenonlinear term α and the dispersion term β form the soli-tary waves, where α determines the steepness/sharpnessof the solitary excitations and β measures the broaden-ing of the solitary waves. On the other hand, the dissi-pation/damping term υ measures the decay of the soli-tary wave over time while propagating. This is why itis essential to thoroughly study the variations of thesecoefficients with the parameters on which they depend.From Eqs. (16) and (17), it is seen that α and β bothdepend on some common plasma parameters (such asthe superthermality index κ , obliquity angle δ , dust-to-ion number density µ , and ion-to-electron temperature σ ,etc.). We see in equation (17) that the external magneticfield B has influence only on the dispersion term β viaΩ ci and the influence of B diminishes as l z = cos θ = 0 ◦ ,i.e., for parallel propagation.We now see how different plasma configuration param-eters influence the nonlinear term α and the dispersionterm β . We now plot α against κ for different values δ = °δ (cid:0) °δ (cid:1) °δ (cid:2) ° (cid:3) α (a) δ = ° δ = ° δ = ° δ = ° κβ (b) FIG. 2: The variation of (a) the nonlinearity coefficient α with µ for different values of σ for δ =? and κ =?,and (b) the dispersion coefficient β with κ for different δ , where µ = 0 . σ = 0 .
10. For both panel, we chooseΩ ci = 0 . δ in Figure 2a to trace the influence of nonthermalityand obliqueness while Figure 2b shows the variation of β versus κ for different values of obliquity angle via δ .From these figures, it is clearly seen that both α and β attain comparatively higher value in a thermallydistributed (Maxwellian) plasma than in a superther-mally distributed plasma, which suggests that the solitonformed in a Maxwellian plasma will be taller and broaderthan that in a superthermal plasma medium. Neverthe-less, the coefficients show opposite characteristics for thevariation of obliquity angle, δ i.e., nonlinearity decreaseswith obliquity angle while the dispersion increases as thevalue of the obliquity angle are increased. In Figure 3a,we depict the variation of α against µ for different valuesof ion and/or electron temperature via σ , and β versus µ for different values of external magnetic field B (viaΩ ci ) in Figure 3b. We see in Figure 3a that the valueof α gets smaller for the higher value of µ , i.e., the non-linearity decreases with increasing dust number densitycompared to that of ion, and after a particular criticalvalue, it downfalls. So our plasma model is valid for bothdust ion-acoustic soliton of positive and negative poten-tial. This figure also indicates that α increases with theincrease of σ , which means that if the ion temperatureincreases with keeping the electron temperature fixed,the nonlinearity also rises for our considered medium. Sothe solitary structure will become less tall for the moresignificant value of ion to electron temperature ratio σ .Though the expression of β contains σ , in reality, β doesnot vary significantly with σ while α does. So we haveplotted β against µ for various values of Ω ci instead of σ in Fig. 3b, which shows β increases with µ , in con-trast, decreases with Ω ci , i.e., the dispersion coefficient isseen to be increased with the ratio of dust to ion numberdensity unlike the nonlinearity coefficient α and with theincrease of the magnetic field B the dispersive term getsdecreased while the nonlinearity is not affected at all. σ = σ = σ = σ = - (cid:4) α (a) Ω (cid:5) Ω (cid:6) Ω (cid:7) Ω (cid:8) μβ (b) FIG. 3: The variation of (a) the nonlinearity coefficient α versus κ for different values of δ with σ = 0 .
10 and µ = 0 .
4, and (b) the dispersion coefficient β versus µ fordifferent values of Ω ci with κ = 3, δ = 15 ° , and σ = 0 . β is always positive for any given values of plasma su-perthermality index κ , as expected; and the polarity ofthe DIASWs depends only on the sign of the nonlinearterm α . The dust-to-ion number density threshold (i.e.,the critical number density of dust-to-ion), we define as σ = σ = σ = σ = κ (cid:9) c FIG. 4: Showing the variation of critical values ofdust-to-ion number density µ c with κ for different σ andfor δ = 15 ° .TABLE I: Critical values of dust-to-ion number density µ c with and without ion temperature effect, and fordifferent values of κ κ µ c ( σ → µ c ( σ → . .
533 0 . .
606 0 . .
632 0 . .
660 0 . .
663 0 . µ c , can be obtained by solving α [ l z , σ, µ, κ ] = 0 for µ , hasthe form µ c = 116 κ σ − κσ + 4 σ × (cid:20) (16 σ + 12) κ − (16 σ + 24) κ + (4 σ + 9) ± (2 κ − × q (2 κ − (cid:8) (16 σ + 18) κ + (8 σ − (cid:9)(cid:21) . (19)It is worth to note that the nonlinear term α > µ > µ c and α < µ < µ c . We see in equation(19) that the critical number threshold µ c , depending onwhich the considered dusty plasma medium may form thepositive or negative potential solitary excitations, is anexplicit function of the superthermality parameter κ andthe ion-to-electron temperature σ . We depict the vari-ation of critical values of dust-to-ion number density µ c with the superthermality parameter κ for different val-ues of ion or electron temperature via σ (= T i /z i T e ). Wefound that the critical number density is seen to increasewith the increase (decrease) in ion (electron) temperaturefor the fixed values of z i , while µ c is smaller (in magni-tude) in superthermal (lower κ values) plasmas in com-parison to that in moderately non-thermal or Maxwellian(higher κ values) plasmas, as shown in Figure 4. A. Analytical solution
If we consider υ = 0 (i.e., there is no collision betweenions and neutral particles), equation (15) reduces to thestandard form of KdV equation as ∂ψ∂τ + αψ ∂ψ∂ξ + β ∂ ψ∂ξ = 0 , (20)upon which the integration can be performed completelyand it follows an infinite set of conservation laws. If E isassumed as the soliton’s energy, the energy conservationis then expressed as ∂E∂τ = 0 , (21)where E = Z ∞−∞ ψ ( ξ, τ ) dξ and the solution of equation (19) is ψ ( ξ, τ ) = Ψsech (cid:20)s α Ψ12 β (cid:0) ξ − α Ψ3 τ (cid:1)(cid:21) . (22)The conditions for the formation of localized structures ψ → , ∂ψ∂ξ → , ∂ ψ∂ξ → (23)are fulfilled as ξ → ±∞ . In Eq. (21), Ψ(= U α ) isthe soliton amplitude, L (= q αU = q βα Ψ ) is the solitonwidth, and U (= α Ψ3 ) is the soliton speed. That is, theKdV equation (20) is fully integrable and one can find theexact solution of KdV equation when there is no collisionor damping is present in the plasma medium (i.e., υ = 0).In the presence of collision or damping/dissipation (i.e., υ = 0), it is not possible to solve the dKdV equation(15) analytically for an exact solution and one has toconsider an approximate solution for the weak dissipa-tion due to ion-neutral collisional effect (i.e., for the as-sumption υ ≪ E will definitely not be conserved, and ∂E∂τ = − υτ, ⇒ E ( τ ) = E (0) e − υτ . The time dependent form of the soliton amplitude, speed,and width [51, 54]) are, respectively, expressed as ψ m ( τ ) = ψ (0) e − υ τ ,U ( τ ) = U (0) e − υ τ ,L ( τ ) = s βα ψ (0) e υ τ , and the approximate analytical process gives the solutionof equation (15) as [49–51] ψ ( ξ, τ ) = ψ m ( τ )sech (cid:20)s α ψ m ( τ )12 β (cid:0) ξ − α Z τ ψ m ( τ ) dτ (cid:1)(cid:21) , ⇒ ψ ( ξ, τ ) = ψ (0)e − υ τ sech (cid:20)s U ( τ )4 β (cid:26) ξ − α ψ (0)2 υ (cid:0) − e − υ τ (cid:1)(cid:27)(cid:21) , (24)where ψ (0) is the soliton amplitude at time τ = 0, whichwould remain the same with time unless the collisionalparameter is present in the system. As the time eval-uates, the soliton gets damped due to the term υ andeventually diminishes over time while propagating. B. Parametric and numerical analysis
We are now interested to examine the characteristicsof obliquely propagating damped DIASWs in a mag-netized collisional κ − nonthermal dusty plasma mediumparametrically and numerically with the help of MATH-EMATICA, in order to trace the effect of different plasmacompositional parameters (especially, the effect of iontemperature via σ , the damping effect via υ , the plasmanonthermality or superthermality via κ , the effect of ionand dust density µ , and the influence of obliqueness via δ ,etc.) on the dynamics of damped DIASWs. An approx-imate solution of damped KdV equation (15) for weakdamping ( υ ≪
1) [given in equation (24)] is used toanalyse the nature of obliquely propagating damped DI-ASWs parametrically. It is already mentioned in Sec.III A that in the presence of ion-neutral collision (i.e., υ = 0), it is no longer a completely integrable Hamilto-nian system. We, therefore, consider the solitary wavesolution of equation (15) in the absence of collision is ψ ( ξ,
0) = Ψsech (cid:20)q α Ψ12 β ξ (cid:21) . But in the presence of col-lision, to solve equation (15) numerically for examiningthe dynamics of obliquely propagating damped DIASWs,we use the approximate solution in Eq. (24).In Figure 5, we depict the evolution of DIASWs inthe considered plasma medium by solving the dampedKdV equation (15) numerically. In the first numericalexperiment in Figure 5a, we choose equation (22) for theplasma parameters υ = 0 , κ = 3 , µ = 0 . , δ = 10 ° , Ω ci =0 . , U = 0 .
05 as an initial condition, and the propaga-tion of DIASWs is also considered for the same plasmaparametric values. We found that the DIASW is seen tomaintain it’s stability over time while propagating, i.e.,the amplitude and the width of the pulse remain con-stant with time while advancing through a collisionlessmagnetized dusty plasma. On the other hand, solitarywave solution (22) for collisionless plasma (i.e., υ = 0)with plasma parameters U = 0 . µ = 0 . δ = 10 ° , -
20 0 20 Ξ Τ Ψ (a)(b)(c) FIG. 5: Evolution of dust-ion-acoustic solitons in aplasma with (a) υ = 0 and σ = 0, (b) υ = 0 .
01 and σ = 0, and (c) υ = 0 .
01 and σ = 0 .
20 . Other plasmaparameters are fixed at U = 0 . µ = 0 . δ = 10 ° ,Ω ci = 0 . σ = 0 .
10, and κ = 3.Ω ci = 0 . σ = 0 .
10, and κ = 3, is considered to propa-gate in a collisional (collisional parameter υ = 0 .
01 isassumed) magnetized superthermal plasma in absence δ = ° δ = ° δ = ° δ = ° - -
10 0 10 200.000.020.040.060.080.100.12 ξψ FIG. 6: Dust-ion-acoustic solitary potential ψ versus ξ for different δ at τ = 30 in a collisional magnetizednonthermal dusty plasma for µ = 0 .
1, Ω ci = 0 . σ = 0 . U = 0 . κ = 3, υ = 0 . μ = μ = μ = μ = - -
10 0 10 20 - - ξψ FIG. 7: Positive potential DIASWs for µ < µ c andnegative potential DISWs for µ > µ c at τ = 0 in theconsidered plasma for δ = 15 ° , Ω ci = 0 . σ = 0 . U = 0 . κ = 3, υ = 0 . σ = 0)to see the influence of dissipation due to collision only,as depicted in Figure 5b. It is expected and also clearfrom our numerical result in Figure 5b that the ampli-tude (width) of the obliquely propagating DIASWs de-creases (increases) with time and thus, the soliton prop-erty ( amplitude × width = constant ) remains conserved.In Figure 5c, we paint the dissipation due to both the iontemperature effect and collisional effect combinedly withkeeping the other parametric values the same as that weconsidered in Figure 5b. The combined effect of thesetwo parameters consequences in a more prominent dis-sipation of the soliton than that with considering thecollision only.We now see the effect of obliqueness δ if Figure 6 anddust-to-ion number density µ in Figure 7. Figure 6 showsthe geometrical characteristics of the solitary structuresin space for different obliquity angle in such a plasmamedium which has the following parameters µ = 0 . ci = 0 . σ = 0 . U = 0 . κ = 3, υ = 0 . τ = 30. The plot suggests that both the ampli-tude and width of the pulse increase with the obliquityangle i.e., when the pulse propagates along the externalmagnetic field B ( δ = 0 ° ), amplitude and width havethe smallest values and as the value of obliquity angleincreases, both the amplitude and width also increasewhich means the solitary structure becomes taller andwider with the increase of obliquity angle. On the otherhand, the influence of dust-to-ion number density onthe propagation nature/characteristics of obliquely prop-agating damped DIASWs is studied in the consideredplasma for fixed plasma parameters δ = 15 ° , Ω ci = 0 . σ = 0 . U = 0 . κ = 3, υ = 0 .
01, and τ = 0. Wehave seen that our considered plasma system allows thepropagation of positive potential DIASWs for µ < µ c and negative potential DIASWs for µ > µ c , and boththe amplitude and the width of the observed DIASWsare seen to increase with the increase (decrease) of µ forthe positive (negative) potential solitary excitation re-gion, as depicted in Figure 7.The influence of ion (nonthermal electron) tempera-ture via σ (= T i /z i T e ) on the dynamical properties ofobliquely propagating dust-ion-acoustic solitary struc-tures is analysed numerically in Figure 8 in a collisionalmagnetized plasma medium in which δ = 15 ° , Ω ci = 0 . µ = 0 . U = 0 . κ = 3, υ = 0 .
01, and τ = 0. It isexamined that as the ion temperature effect increases onthe plasma medium, it causes the DIAW solitary wavesto become less steep but thicker. From Figures 8b and8c, it can be implied that the higher the ion (electron)temperature, the greater (lesser) the amplitude of thesolitary structure will be. This interpretation goes op-posite for the case of the width of the solitary structure,i.e., with the increase (decrease) of the value of ion (elec-tron) temperature in the plasma medium, the solitonsbecome more (less) comprehensive. This was anticipatedas a higher pulse is narrower, as predicated by the esti-mated solution above. We now see the effects of externalmagnetic field B (via Ω ci ) in Figure 9 in a nonthermalcollisional magnetized plasma with compositinal param-eters δ = 15 ° , σ = 0 . µ = 0 . U = 0 . κ = 3, υ = 0 .
01, and τ = 0. It is found that the external mag-netic field does not have any effect on the amplitude ofDIASWs, but B has a significant effect on the widthof DIASWs, which is clearly depicted in Figure 9 andagrees with previous research in collisional and collision-less plasma contexts [56].Figure 10 displays the effects of σ (ion and elec-tron temperature effect) in a collisionless (see upper twocurves in Figure 10) as well as in a collisional plasma (seelower two curves in Figure 10). The considered plasmamedium suggests the formation of smaller (in amplitude)and wider (in width) solitons in the presence of colli-sion in comparison to those are formed in a collisionlessplasma. It is also seen that for negligible temperatureand collisional effects (i.e., σ = 0 , υ = 0), the solitarystructure does not get damped hence the amplitude ismaximum. On the other hand, in the presence of con-siderable amount of temperature and no collision, the σ = σ = σ = σ = - -
10 0 10 200.000.020.040.060.080.100.120.14 ξψ (a) σ = σ = σ = σ = τΨ ( τ ) (b) σ = σ = σ = σ = τ L (cid:10) τ (cid:11) (c) FIG. 8: Showing the variation of (a) ψ versus ξ at time τ = 30, (b) damped DIASWs amplitude ψ m ( τ ) versus τ , and (c) damped DIASWs width L ( τ ) versus τ , fordifferent σ ; where other plasma compositionalparameters are fixed at δ = 15 ° , Ω ci = 0 . µ = 0 . U = 0 . κ = 3, υ = 0 . σ = 0 , υ = 0), the amplitudeof the structure becomes minimum.In Figure 11, we depict the DIASWs profile versus ξ for different values of electron’s superthermality index κ Ω = Ω = Ω = Ω = - -
10 0 10 200.00 ξψ FIG. 9: Effect of external magnetic field B (via Ω ci ) ondamped DIASWs at τ = 0 in a magnetized collisionalnonthermal plasma for δ = 15 ° , σ = 0 . µ = 0 . U = 0 . κ = 3, υ = 0 . σ = ν = σ = ν = σ = ν = σ = ν = - -
10 0 10 200.00 ξψ FIG. 10: Showing the variation of solitary profile ψ versus ξ for different σ and υ , where µ = 0 .
1, Ω ci = 0 . δ = 10 ° , U = 0 . κ = 3, and τ = 30 . κ = κ = κ = κ = - -
10 0 10 200.00 ξψ FIG. 11: Variation of ψ with ξ for different κ , where δ = 15 ° , σ = 0 . µ = 0 . U = 0 .
05, Ω ci = 0 . υ = 0 . τ = 0 . to trace the influence of nonthermality on the dynami-cal properties of DIASWs in the considered plasma. Wehave observed another notable result from the numeri-cal simulation, which shows that as the distribution of the plasma medium tends to go toward the thermal dis-tribution from the superthermal one, the value of bothamplitude and width of solitary waves increase. IV. CONCLUSION
In this manuscript, we have investigated the proper-ties of DIASWs in a three component nonthermal (or su-perthermal) collisional magnetized dusty plasma medium(consisting of static dust particles, inertial ions, and in-ertialess electrons following nonthermal κ − distribution)analytically and numerically. The ion temperature andthe ion-neutral collisional effects are taken into account.The reductive perturbation approach is adopted to derivethe damped KdV equation and the solitary wave solutionof damped KdV equation id obtained to analyse the for-mation nature and the characteristics of DIASWs in sucha plasma system. The influences of different fundamentalplasma parameters (e.g., obliquity angle, superthermalityindex, unperturbed dust to ion ratio, cyclotron frequency,etc.) in the presence of collision and ion temperature ef-fects, are studied, and we have succeeded to extract someexciting results from our investigation which are brieflystated as follows:1. Ion temperature has a meaningful impact on thefundamental properties of DIASWs (e.g., stabil-ity, speed, amplitude, width, etc.) while advancingthrough a collisional magnetized dusty plasma.2. The Phase speed, V , is seen to be lesser whilepropagating diagonally (i.e., 0 ° < δ < ° ) thanwhile propagating along (i.e., δ = 0 ° ) the magneticfield. The DIASWs are seen to have more incredi-ble phase speed in plasma containing thermally dis-tributed electrons than that containing superther-mally distributed electrons.3. If we consider the effect of ion temperature, thephase speed acquires a higher value than the caseof not considering the ion temperature effect, andas the ion temperature increases, the phase speedalso gets increased. Similarly, the value of V alsogets increased when the unperturbed dust densityis higher than that of dust.4. The nonlinear coefficient α acquires both positiveand negative value (depends on the concentrationof dust and ion); thus, the propagation of ion-acoustic solitary structures of both positive andnegative potential is possible in our consideredplasma model. At a particular value of µ , the non-linearity coefficient has a value of zero, which indi-cates that the amplitude of the solitary structurefor this condition will be infinite.5. Though the nonlinearity increases with ion tem-perature, the dispersion coefficient B does not suf-fer any mentionable impact caused by ion temper-ature.06. The dust ion-acoustic solitary waves suffer dissi-pation not only by the collision but also by iontemperature. However, the amount of dissipationcaused by ion temperature drops down prominentlyin the presence of the ion-neutral collision.7. And the increasing intensity of the magnetic fielddoes not increase the amplitude of the solitarystructures but contributes to the width to getbroader.8. The ion temperature effect decreases the amplitudewhile increases the width of solitary waves, i.e., itplays the same role as the collision in this specificperspective, which is the same as we have found insome earlier works [46–48].9. If we consider both the ion temperature and thecollision effect, the solitary wave width possessesa higher value in superthermal plasma compar- atively than Maxwellian thermal plasma as ex-pected. 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