Arbitrary amplitude nucleus-acoustic solitary waves in thermally degenerate plasma systems
AArbitrary amplitude nucleus-acoustic solitary waves in thermally degenerate plasmasystems
A. Mannan,
1, 2, ∗ S. Sultana, and A. A. Mamun † Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh Institut für Mathematik, Martin Luther Universität Halle-Wittenberg, D-06099 Halle (Saale), Germany (Dated: August 20, 2020)A rigorous theoretical investigation is made of arbitrary amplitude nucleus acoustic solitary wavesin a fully ionized multi-nucleus plasma system (consisting of thermally degenerate electron speciesand non-degenerate warm light as well as heavy nucleus species). The pseudo-potential approach,which is valid for the arbitrary amplitude solitary waves, is employed. The subsonic and supersonicnucleus-acoustic solitary waves (which are found to be compressive) along with their basic featuresare identified. The basic properties of these subsonic and supersonic nucleus-acoustic solitary wavesare found to be significantly modified by the effects of non and ultra-relativistically degenerate elec-tron species, dynamics of heavy nucleus species, number densities as well as adiabatic temperaturesof light and heavy nucleus species, etc. It shown that the presence of heavy nucleus species with non-degenerate (isothermal) electron species supports the existence of subsonic nucleus-acoustic solitarywaves, and that the effects of electron degeneracies and light and heavy nucleus temperatures reducethe possibility for the formation of these subsonic nucleus-acoustic solitary waves. The amplitudeof the supersonic nucleus-acoustic solitary waves in the situation of non-relativistically degenerateelectron species is much smaller than that of ultra-relativistically degenerate electron species, butis much larger than that of isothermal electron species. The rise of adiabatic temperature of lightor heavy nucleus species causes to decrease (increase) the amplitude (width) of the subsonic andsupersonic nucleus acoustic solitary waves. On the other hand, the increase in the number densityof light or heavy nucleus species causes to increase (decrease) the amplitude (width) of the subsonicand supersonic nucleus acoustic solitary waves. The results of this investigation are found to beapplicable in laboratory, space, and astrophysical plasma systems.
PACS numbers: 52.35.Sb; 47.35.Fg; 94.05.Fg; 43.25.Rq
I. INTRODUCTION
Mamun [1] has first introduced the electron degener-ate energy along with corresponding wave speed ( C l ) andwave scale length ( L q ) associated with the degenerateelectron pressure [2–8], and has also identified the de-generate pressure driven (DPD) nucleus-acoustic (NA)waves, and has pinpointed their new basic features in de-generate plasma systems [2–11], which are composed ofcold degenerate electron species (DES) [2–4], cold non-degenerate light nucleus species (viz. H [2], or He [3] or C [5] or O [5]), and stationary heavy nucleusspecies (viz. Fe [12] or Rb [13] or Mo [13]). Thelinear dispersion relation for such DPD NA waves in sucha cold degenerate plasma is given by [1] ω = (cid:114) γ e µ kC l (cid:113) γ e µ k L q , (1)where ω = 2 πf and k = 2 π/λ with f ( λ ) being the DPDNA wave frequency (wavelength); µ = Z h N h /Z l N l with Z l e ( Z h e ) being the charge of the light (heavy) nu-cleus species, and N l ( N h ) being the equilibrium num- ∗ Electronic address: [email protected] † Also at Wazed Miah Science Research Centre, Jahangirnagar Uni-versity, Savar, Dhaka-1342, Bangladesh. ber density of the light (heavy) nucleus species; C l =( Z l E e /m l ) / is the DPD NA wave speed with m l beingthe mass of a light nucleus; E e = KN γ e − e is the colddegenerate electron energy [1] associated with degener-ate electron pressure [2–4] P e = KN e γ e at equilibrium; K (cid:39) π ¯ h / m e [1–7] for γ e = 5 / (non-relativisticallyDES [2–4]); K (cid:39) hc/ [1–7] for γ e = 4 / (ultra-relativistically DES [2–4]); L q = C l /ω pl is the DPD NAwave length scale with ω pl = (4 πN l Z l e /m l ) / beingthe nucleus plasma frequency and m l mass of a light nu-cleus; m e ( ¯ h ) is the electron rest mass (reduced Planck’sconstant), c is the speed of light in vacuum, and e isthe charge of a proton or the magnitude of the chargeof an electron. We note that N e = Z l N l + Z h N h atequilibrium. It is important to mention that in any colddegenerate plasma K is unknown for γ e = 1 , which, thus,cannot be considered in (1) since the latter is not validfor any cold degenerate plasma system. The dispersionrelation for the long wavelength DPD NA waves (viz. kL q (cid:28) , which is the appropriate limit for these waves)becomes ω (cid:39) (cid:114) γ e µ kC l , (2)which indicates that the degenerate electron pressure(nucleus mass density) provides the restoring force (iner-tia) in these DPD NA waves, and that the phase speed ofthese DPD NA waves decreases (increases) with the rise a r X i v : . [ phy s i c s . p l a s m - ph ] A ug of µ ( γ e ). This dispersion relation also indicate that theDPD NA waves completely disappear in absence of theelectron degenerate pressure, which is independent of thetemperature of any plasma species. Thus, the DPD NAwaves [1] defined by (1) is completely different from thewell known ion-acoustic (IA) waves [14–16] due to thefact that the IA (DPD NA) waves are driven by the elec-tron thermal (degenerate) pressure, and that the speed,length scale, and time scale of the IA waves are far differ-ent from those of the DPD NA waves. However, cold de-generate plasma systems under different conditions havebeen considered by many authors to study the nonlinearpropagation of the IA waves during the last ten years[17–39].Recently, there has been a great deal of interest inunderstanding the physics of linear and nonlinear prop-agation of DPD NA waves [1] in degenerate plasma sys-tems under different situations [2–13] not only becauseof their basic difference from the IA waves [14–16], butalso because of the existence of the degenerate plasmasystems [2–13] in enormous number of astrophysical com-pact objects [2–8] and laboratory devices [9–11], wherethe degenerate pressure is comparable to or greater thanall other pressures like thermal, electrostatic, and self-gravitational pressures [2–13].We are now interested in deriving a more general andrealistic dispersion relation by considering thermally DES(TDES) [instead of the cold degenerate DES consideredin (1)], and cold mobile heavy nucleus species [insteadof stationary heavy nucleus species considered in (1)].The dynamics of the light nucleus species is as before.The dispersion relation for the DPD NA waves in suchthermally degenerate plasma system (TDPS) is given by ω = (cid:115) γ e (1 + µS h )1 + µ + γ e k λ q kC q , (3)where S h = Z h m l /Z l m h ; C q = ( Z l E e /m l ) / in which E e = E ed + E et with E ed ( E et ) being the electron degener-ate (thermal) energy associated with electron degenerate(thermal) pressure; and λ q = C q /ω pl . The dispersion re-lation for the long-wavelength DPD NA waves ( kλ q (cid:28) ,which is the appropriate limit for these waves) in a TDPSbecomes ω (cid:39) (cid:115) γ e (1 + µS h )1 + µ kC q , (4)which indicates that the dispersion relation (4) for theDPD NA waves in such a TDPS can be interpreted asfollows • The dispersion relations (2) and (4) are identicalfor S h = 0 (indicating stationary heavy nucleusspecies) and E et = 0 (indicating cold DES). • The phase speed ( ω/k ) of the DPD NA waves in-creases with rise of the value of S h . The rate ofincrease of ω/k with µ in the case of S h (cid:54) = 0 isslower than that in the case of S h = 0 . • It is obvious that C q > C l and λ q > L q . Thismeans that the phase speed (wavelength) for E et (cid:54) =0 is higher (lower) than that for E et = 0 . Thisis due to the rise of the volume of the degeneratemedium caused by the outward thermal pressure ofthe TDES.There are also a number of investigations [40–48] on non-linear NA waves in degenerate plasma systems during thelast five years. The limitations of these works are as fol-lows. • The works [40–48] are valid only for cold degen-erate electron and nucleus species. So the worksare not valid for warm degenerate plasma systems,particularly for hot white dwarfs [49–53]. • The works [40–48] are based on the reductive per-turbation method [54], which is valid for small am-plitude nonlinear waves. Thus, the works are notvalid for large amplitude nonlinear waves. • The wave speed and length scale, which are inde-pendent of the degenerate electron pressure, are notproperly defined in the works [40–48] from whichone cannot get the linear dispersion relation for theDPD NA waves defined by (2) or (4). Therefore,the linear and nonlinear features of the DPD NAwaves were not properly identified by these works,which are correct for other kind of nonlinear NAwaves, but not for the DPD NA waves defined by(2) or (4).To overcome the limitations of the works [40–48], weconsider a thermally degenerate plasma system [con-taining thermally degenerate electron species, and non-degenerate warm light and heavy nucleus species, andinvestigate the arbitrary amplitude DPD NA solitarywaves (SWs) by the pseudo-potential approach [55, 56].The thermally degenerate plasma system under ourpresent consideration is so general that it is valid forhot white dwarfs [49–53] as well as in many space [57–60] and laboratory [61–64] plasma environments, wherenon-degenerate electron-ion plasma with heavy positivelycharged particles (as impurity or dust) occur.The structure of the manuscript is as follows. Thethermally degenerate plasma model is illustrated in Sec.II. The criteria for the existence of subsonic and super-sonic DP NA SWs and their basic features for differentsituations of thermally degenerate plasmas are investi-gated by the pseudo-potential approach in Sec. III. Thethermally degenerate plasma model under consideration,results obtained from this investigation, and some impor-tant applications are pinpointed as a brief discussion inSec. IV.
II. MODEL EQUATIONS
We consider a general and realistic TDPS containingthe TDES and warm adiabatic degenerate heavy andlight nuclei species. We also consider the propagationof thermally degenerate pressure driven (TDPD) nucleusacoustic (NA) waves in such a TDPS. The dynamics ofthe TDPD NA waves in such a TDPS is described by ∂N j ∂T + ∂∂X ( N j U j ) = 0 , (5) ∂ P jq ∂T + U j ∂ P jq ∂X + γ j P jq ∂U j ∂X = 0 , (6) ∂∂X ( P ed + P et ) − N e e ∂ Φ ∂X = 0 , (7) ∂U l ∂T + U l ∂U l ∂X = − Z l em l ∂ Φ ∂X − N l m l ∂∂X ( P ld + P lt ) , (8) ∂U h ∂T + U h ∂U h ∂X = − Z h em h ∂ Φ ∂X − N h m h ∂∂X ( P hd + P ht ) , (9) ∂ Φ ∂X = 4 πe ( N e − Z l N l − Z h N h ) , (10)where Φ is the electrostatic NA wave potential; N j ( U j )is number density (fluid speed) of the plasma species j (with j = e for TDES, j = l for degenerate adiabaticallywarm light nucleus species, and j = h for degenerateadiabatically warm heavy nucleus species; P jq in (6) − (9)is the outward pressure for the species j of the type q (with q = d for the degenerate pressure and q = t for thethermal pressure); γ j is adiabatic index for the plasmaspecies j ; X ( T ) is the space (time) variable.To derive the expression for P jq from (5) and (6), wefirst make all the dependent variables to depend only ona single variable ζ = X − M T , where M is the nonlinearwave speed. Now, expressing (5) and (6) in terms of ζ and using the steady state condition ∂/∂T → , weobtain − M dN j dζ + ddζ ( N j U j ) = 0 , (11) − M d P jq dζ + U j d P jq dζ + γ j P jq dU j dζ = 0 . (12)Now, integrating (11) with respect to ζ with the ap-propriate equilibrium conditions (viz. N j → N j and U j → ), one can write U j = M (cid:18) − N j N j (cid:19) . (13)Inserting (13) into (12) and dividing the resulting equa-tion by N γ j − j , we obtain ddζ (cid:32) P jq N γ j j (cid:33) = 0 . (14)By integrating (14) once with respect to ζ , one can ex-press P jq as P jq = K jq N γ j j , (15)where K jq = E jq N (1 − γ j ) j is the proportional-ity/integration constant [in which E jq is equilibrium en-ergy associated with the outward pressure for the species j of type q ]. We also write the expression for n e (= N e /N e ) in termsof φ (= e Φ / E e ) , where E e = E ed + E et ), as n e = (cid:18) γ e − γ e φ (cid:19) γe − , (16)which derived by using (5) − (6). We note that (16)is valid for the arbitrary value of γ e , and is, thus,valid for non-relativistically ( γ e = 5 / ) as well as ultra-relativistically ( γ e = 4 / ) TDES. We also note that fora cold DES, E et = 0 and E e = E ed = K ed N ( γ e − e ,which mean that φ = e Φ / E ed . On the other hand, fora non-degenerate thermal electron species, E ed = 0 and E e = E et = k B T e , which indicate that φ = e Φ /k B T e .It is worth noting that we cannot directly use γ e = 1 in (16). To use γ e = 1 in (16), we expand the latter as n e = (cid:18) γ e (cid:19) φ + (cid:18) γ γ e (cid:19) φ + (cid:18) γ γ γ e (cid:19) φ + · · · , (17)where γ = 2 − γ e and γ = 3 − γ e , and by substituting γ e = 1 into (18), one obtains n e as n e = 1 + φ + φ
2! + φ
3! + · · · = exp( φ ) . (18)Thus, after expressing (16) in the form of (18), it is validfor γ e = 1 which yields n e = exp( φ ) with φ = e Φ /k B T e .It is convenient to introduce dimensionless quantitiesinto (5) − (10). Thus, substituting P ld and P lt as obtainedfrom (15) into (8) and (9), our basic equations (5), (8)and (9) for nucleus species, and the Poisson’s equation(10) can be rewritten in dimensionless form as ∂n l ∂t + ∂∂x ( n l u l ) = 0 , (19) ∂n h ∂t + ∂∂x ( n h u h ) = 0 , (20) ∂u l ∂t + u l ∂u l ∂x = − ∂φ∂x − σ l n l ∂n γ l l ∂x , (21) ∂u h ∂t + u h ∂u h ∂x = − S h ∂φ∂x − σ h n h ∂n γ h h ∂x , (22) ∂ φ∂x = (1 + µ ) n e − n l − µn h , (23)where we have normalized the variables as x = X/λ q , t = T ω pl , n l = N l /N l , n h = N h /N h , u l = U l /C q , u h = U h /C q , φ = e Φ / E e , σ l = E l /Z l E e (with E l = E ld + E lt ), and σ h = E h S h /Z h E e . We note that we haveredefined E e , and that the newly defined E e must beused in defining C q and λ q . However, as before C q = λ q ω pl . III. NA SOLITARY WAVES
To study arbitrary amplitude TDPD NA SWs, we firstassume that all dependent variables in (19) – (23) dependon a single independent variable ξ = x − M t , where M is the the Mach number. This transformation along withthe steady state condition ( ∂/∂t → ) leads our basic setof equations to M dn l dξ − ddξ ( n l u l ) = 0 , (24) M dn h dξ − ddξ ( n h u h ) = 0 , (25) M du l dξ − u l du l dξ = dφdξ + σ l n l dn γ l l dξ , (26) M du h dξ − u h du h dξ = S h dφdξ + σ h n h dn γ h h dξ , (27) d φdξ = (1 + µ ) n e − n l − µn h . (28)Now, by imposing the appropriate boundary conditions(namely, n l = 1 , n h = 1 , u l = 0 , u h = 0 , and φ = 0 ), theintegration of (24)-(27) gives rise to u l = M (cid:18) − n l (cid:19) , (29) u h = M (cid:18) − n h (cid:19) , (30) M u l − u l − φ − γ σl [ n ( γ l − l −
1] = 0 , (31) M u h − u h − S h φ − γ σh [ n ( γ h − h −
1] = 0 , (32)where γ σl = 2 σ l γ l / ( γ l − , γ σh = 2 σ h γ h / ( γ h − .Again, substituting u l and u h [given by (29) and (30)],respectively, into (31) and (32), one can obtain equationsfor n l and n h as γ σl n ( γ l +1) l − ( M + γ σl − φ ) n l + M = 0 , (33) γ σh n ( γ h +1) h − ( M + γ σh − S h φ ) n h + M = 0 . (34)It is important to note that (33) and (34) are valid for thearbitrary value of γ e , and ( γ l , γ h ) > . Thus, they canbe used for cold ( σ l = σ h = 0 ) as well as adiabatic ( γ l = γ h = 3 ) non-degenerate light and heavy nucleus species.We also note that we have ignored the the effect of thenucleus degeneracy in our present investigation, becausethe degeneracy in both light and heavy nuclei species isinsignificant compared to that in electron species [1, 7,40, 41].For the cold light and heavy nucleus species limit ( σ l = σ h = 0 ), we can solve (33) and (34) for n l as n h as n l = 1 (cid:113) − φ M , (35) n h = 1 (cid:113) − S h φ M . (36)On the other hand, for both non-degenerate adiabaticlight and heavy nucleus species ( σ l = σ lt (cid:54) = 0 , σ h = σ ht (cid:54) = 0 , and γ l = γ h = 3 ), (33) and (34) can be expressed,respectively, as γ l n l − ( M + 3 σ lt − φ ) n l + M = 0 , (37) γ h n h − ( M + 3 σ ht − S h φ ) n h + M = 0 , (38)where σ lt = T l /Z l T e and σ ht = S h T h /Z h T e . It is obviousthat (37) and (38) are quadratic equations for n l and n h ,respectively. Therefore, the solution of (37) and (38) for n l and n h are given by n l = (cid:20) σ lt (cid:18) Φ l − (cid:113) Φ l − σ lt M (cid:19)(cid:21) , (39) n h = (cid:20) σ ht (cid:18) Φ h − (cid:113) Φ h − σ ht M (cid:19)(cid:21) , (40)where Φ l = M +3 σ lt − φ and Φ h = M +3 σ ht − S h φ .The multiplication of (28) first by dφ/dξ , and then theintegration of the resulting equation with respect to ξ [under appropriate boundary conditions, ( dφ/dξ ) → at ξ → ±∞ ] give rise to an energy integral in the form (cid:18) dφdξ (cid:19) + V ( φ ) = 0 , (41)where V ( φ ) = − (cid:90) [(1 + µ ) n e − n l − µn h ] dφ , (42)in which n e is given by (16). The latter is valid for γ e = 5 / (non-relativistically TDES) and γ e = 4 / (ultra-relativistically TDES), and (18) is valid for γ e = 1 (Boltzmann distributed electron species). The energy in-tegral (41) [with the pseudo-potential V ( φ ) defined by(42)] gives rise to the TDPD NA SWs if [ d V /dφ ] φ =0 < so that the fixed point at the origin is unstable [56] and ifat the same time [ d V /dφ ] φ =0 > < for the TDPDNA SWs with φ > ( φ < ). We note that V (0) = 0 and [ dV /dφ ] φ =0 = 0 are automatically satisfied becauseof the integration constant chosen and the equilibriumcharge neutrality condition, respectively. We now studythe basic features of the TDPD NA SWs for two specialsituations of TDPS in following two subsections. A. Cold non-degenerate nucleus species
We consider here cold non-degenerate nucleus species( σ l = σ h = 0 ) which is valid for ( ω/k ) (cid:29) ( k B T l /m l ) / .Inserting (16), (35), and (36) into (42), we obtain thepseudo-potential as V ( φ ) = C − (1+ µ ) (cid:18) γ e − γ e φ (cid:19) γe − γe −M (cid:114) − φ M − M µS h (cid:114) − S h φ M , (43)where C = 1 + µ + M + M µ/S h is the integrationconstant which has been chosen in such a manner that V ( φ ) = 0 at φ = 0 .To analyze V ( φ ) defined by (43) analytically, for φ < ,we can expand V ( φ ) as V ( φ ) ≈ C φ + C φ + · · · , (44)where C = 12! (cid:20) S h µ M − γ e (1 + µ ) (cid:21) , (45) C = 13! (cid:20) S h µ ) M − γ e (2 − γ e )(1 + µ ) (cid:21) . (46)It is obvious from (43) and (44) that V ( φ ) = dV ( φ ) /dφ =0 at φ = 0 . Therefore, NA solitary wave solution of(41) exist if (i) d V ( φ ) /dφ < at φ = 0 so thatthe fixed point at the origin is unstable [56] and (ii) [ d V /dφ ] φ =0 > ( < )0 for the NA SWs with φ > ( φ < )[56]. Under the above assumption the NA SWs existif C < , i.e. if M > M c , where M c is the criticalMach number, which corresponds to the vanishing of thequadratic term in (44), and is given by M c = (cid:115) γ e (1 + S h µ )1 + µ . (47)At this critical value of M , the NA SWs with φ > ( φ < ) will exist if C > < , where C ( M = M c ) is given by C ( M = M c ) = (cid:18) µ γ e (cid:19) (cid:20) µ )(1 + S h µ )(1 + S h µ ) − γ e (cid:21) . (48)It is observed that C ( M = M c ) > for µ ≥ , S h > and γ e ≥ . Therefore, our plasma system under con-sideration only supports the NA SWs with φ > forany possible values of µ , S h , and γ e . Figure 1 showshow the critical Mach number M c varies with µ for theisothermal electron species γ e = 1 (red solid curve), ultra-relativistically DES γ e = 4 / (green dotted curve), andnon-relativistically DES γ e = 5 / (blue dashed curve).It is seen that as the non-degenerate heavy nucleus num-ber density increases, the critical Mach number ( M c )decreases. It also indicates that the isothermal electronspecies supports the formation of both subsonic and su-personic NA SWs. The existence of subsonic NA SWsregion is represented by the shadow area, as shown inFig. 1. This region becomes broader with the increase in µ . The supersonic NA SWs region is found above the pur-ple dot-dashed line ( M c = 1 ). On the other hand, theultra-relativistic and non-relativistic DES support onlythe supersonic NA SWs for < µ < .We first investigate the properties of small amplitudeNA SWs by considering the approximation [given by(44)]. Inserting (44) into (41) and upon integrating alongwith the condition V ( φ ) = 0 at φ → φ m , we obtain, in the small amplitude limit, the NA solitary wave solution[63] φ = (cid:18) − C C (cid:19) sech (cid:32)(cid:114) − C ξ (cid:33) . (49) μ ℳ c FIG. 1: The variation of the threshold Mach number M c with µ for S h = 0 . , γ e = 1 (red solid curve), γ e = 4 / (greendotted curve), and γ e = 5 / (blue dashed curve). The purpledot-dashed line corresponds to M c = 1 . The profiles (indicating the amplitude and width) ofthe small amplitude subsonic ( M c < M < ) and super-sonic ( M > and M > M c ) NA SWs associated withthe positive potential are graphically displayed in Figs. 2- 4. We also investigate the properties of arbitrary ampli-tude NA SWs by numerical analyses of (43). Our directnumerical analysis of (43) also show the existence of pos-itive NA SWs potential. Figures 5 - 7 displays the forma-tion of the potential wells in the positive φ -axis for thesame set of plasma parameters as that in small amplitudelimit. It is found for the small amplitude limit that thesubsonic NA SWs with φ > exist for the non-degenerateisothermal electron, but both the ultra-relativistic andnon-relativistic degenerate electron supports the super-sonic NA SWs with φ > . It is observed that the am-plitude (width) of the NA SWs increases (decreases) asthe number density of heavy nucleus species increases.Thus, the effect of the ultra-relativistic degenerate elec-tron significantly modifies the basic features of NA SWs.It is found that the amplitude of NA SWs in the non-relativistically DES is much smaller than that in ultra-relativistically DES, but is larger than that in Boltzmanndistributed electron species (BDES). Note that the widthof supersonic NA SWs in ultra-relativistically degenerateelectron species is much wider than that in both otherelectron species. On the other hand, for arbitrary ampli-tude limit Figs. 5 - 7 provide a visualization of the ampli-tude ( φ m ), which is the intercept on the positive φ -axis,and the width ( φ m / (cid:112) | V m | , where | V m | is the maximumvalue of V ( φ ) in the potential wells formed in the positive φ -axis. - -
10 0 10 200.000.050.100.15 ξ ϕ FIG. 2: The variation of the small amplitude subsonic NASWs for different values of µ = 0 . (red solid curve), µ = 0 . (green dotted curve), and µ = 0 . (blue dashed curve) at γ e = 1 , M = 0 . , and S h = 0 . . - -
10 0 10 200.000.050.100.150.20 ξ ϕ FIG. 3: The variation of the small amplitude supersonic NASWs for different values of µ = 0 . (red solid curve), µ = 0 . (green dotted curve), and µ = 0 . (blue dashed curve) at γ e = 4 / , M = 1 . , and S h = 0 . . - -
10 0 10 200.000.050.100.15 ξ ϕ FIG. 4: The variation of the small amplitude supersonic NASWs for different values of µ = 0 . (red solid curve), µ = 0 . (green dotted curve), and µ = 0 . (blue dashed curve) at γ e = 5 / , M = 1 . , and S h = 0 . . - ϕ - - V ( ϕ ) FIG. 5: The formation of potential wells in positive φ -axisfor µ = 0 . (red solid curve), µ = 0 . (green dotted curve),and µ = 0 . (blue dashed curve) at γ e = 1 , M = 0 . , and S h = 0 . . - ϕ - - - V ( ϕ ) FIG. 6: The formation of potential wells in positive φ -axis for µ = 0 . (red solid curve), µ = 0 . (green dotted curve), and µ = 0 . (blue dashed curve) at γ e = 4 / , M = 1 . , and S h = 0 . . - ϕ - - V ( ϕ ) FIG. 7: The formation of potential wells in positive φ -axis for µ = 0 . (red solid curve), µ = 0 . (green dotted curve), and µ = 0 . (blue dashed curve) at γ e = 5 / , M = 1 . , and S h = 0 . . The increase in µ causes to increase (decrease) the am-plitude (width) of both subsonic and supersonic NA SWs.The depth of potential wells for the ultra-relativisticallyDES is much larger than that in isothermal and non-relativistically electron species. The effects of γ e showsthe similar results as that in the case of small amplitudelimit. It is concluded from this visualization that thevariation of the amplitude and the width with µ in thecase of arbitrary amplitude NA SWs is almost the sameas that in the case of small amplitude NA SWs. B. Adiabatically warm non-degenerate nucleusspecies
We finally consider non-degenerate warm adiabatic nu-cleus species where the light [heavy] number density de-fined by (39) [(40)]. The nucleus number densities [givenby (39) and (40)] are valid when P jd (cid:28) P jt which isvalid not only for hot white dwarfs [49–53], but also formany space [57–60] and laboratory [61, 62] plasma envi-ronments. Now, inserting (16), (39), and (40) into (42),and following the same procedure as mentioned before,we can obtain the pseudo-potential V ( φ ) as V ( φ ) = C σ − (1 + µ ) (cid:20) (cid:18) γ e − γ e (cid:19) φ (cid:21) γeγe − − √ √ σ lt (cid:16)(cid:112) Φ l − Φ l (cid:17) (cid:18) Φ l + 12 Φ l (cid:19) − µ √ S h √ σ ht (cid:16)(cid:112) Φ h − Φ h (cid:17) (cid:18) Φ h + 12 Φ h (cid:19) , (50)where C σ = 1 + µ + σ lt + M + µ ( M + σ ht ) /S h is theintegration constant chosen in such a way that V ( φ ) = 0 at φ = 0 , Φ l = M + 3 σ lt − φ , Φ l = (cid:112) Φ l − σ lt M , Φ h = M + 3 σ ht − S h φ , and Φ h = (cid:112) Φ h − σ ht M .To find the solitary wave solution of (41), the pseudo-potential V ( φ ) must satisfy the necessary conditions asmentioned before. Therefore, to find the conditions forthe existence of the NA SWs, we expand V ( φ ) as V ( φ ) ≈ C σ φ + C σ φ + · · · , (51)where C σ = 12! (cid:20) S h µ M − σ ht + 1 M − σ lt − γ e (1 + µ ) (cid:21) , (52) C σ = 13! (cid:20) S h µ ( M + σ ht )( M − σ ht ) + 3( M + σ lt )( M − σ lt ) − γ e (2 − γ e )(1 + µ ) (cid:21) . (53)The coefficient of φ (viz. C σ ) indicates from [ d V /dφ ] φ =0 < ) that the solitary wave solution of (41)with (50) exists if and only if C σ < . Thus, the NA SWsexist if M > M σc , where M σc is given by M σc = (cid:32) b + √ b − ac a (cid:33) / , (54) a = 1 + µ , b = 3 a ( σ lt + σ ht ) + γ e ( S h µ + 1) , and c =3 γ e ( S h µσ lt + σ ht ) + 9 aσ lt σ ht . We get M σc = M c if weneglect the temperature of light and heavy ions species(i.e. σ lt = σ ht = 0 ). On the other hand, the NA SWsexist with φ > ( φ < ) if C ( M = M σc ) > < .It has been checked that C ( M = M σc ) > for µ ≥ , σ lt ≥ , σ ht ≥ , and γ e ≥ . Therefore, the NA SWsonly with φ > exist for all possible values of µ , σ lt , σ ht , and γ e . Figure 8 displays how the critical Machnumber ( M σc ) varies with σ lt for µ = 0 . . It is seenthat M σc increases with σ lt for all the possible values of γ e . The effects of the temperature of light and heavynucleus species reduce the region where the subsonic NASWs exist. In the presence of warm adiabatic light andheavy nuclei species, the region of subsonic SWs shrinksas the number density of light nucleus species increases.The similar effect of M σc with σ ht has been observed(which is not shown here). The non-relativistically andultra-relativistically degenerate electrons as well as thetemperature of light and heavy nuclei species are alsohere against the formation of subsonic NA SWs, but arein favor of the formation of supersonic NA SWs with φ > .For the small amplitude limit, the solitary wave so-lution of (41) with the approximation [given by (51)] aswell as the condition V ( φ ) = 0 at φ → φ m can be writtenas [63] φ = (cid:18) − C σ C σ (cid:19) sech (cid:32)(cid:114) − C σ ξ (cid:33) . (55)To study the role of nucleus temperature ( σ lt , σ ht ) onthe basic properties of both large and small amplitudessubsonic and supersonic NA solitary structures, we visu-alize the solution (55) and numerically solve the pseudo-potential V ( φ ) [given by (50)] for γ e = 1 (BDES), γ e = 4 / (ultra-relativistically DES), and γ e = 5 / (non-relativistically DES). σ lt ℳ c σ FIG. 8: The variation of the threshold Mach number M σc with σ lt for µ = 0 . , S h = 0 . , σ ht = 0 . , γ e = 1 (redsolid curve), γ e = 4 / (green dotted curve), and γ e = 5 / (blue dashed curve). The purple dot-dashed line correspondsto M σc = 1 . - - - ξ ϕ FIG. 9: The variation of the small amplitude subsonic NASWs for different values of σ lt = 0 . (red solid curve), σ lt =0 . (green dotted curve), and σ lt = 0 . (blue dashed curve)at γ e = 1 , M = 0 . , µ = 0 . , S h = 0 . , and σ ht = 0 . . - - - ξ ϕ FIG. 10: The variation of the small amplitude supersonicNA SWs for different values of σ lt = 0 . (red solid curve), σ lt = 0 . (green dotted curve), and σ lt = 0 . (blue dashedcurve) at γ e = 4 / , M = 1 . , µ = 0 . , S h = 0 . , and σ ht = 0 . . - - - ξ ϕ FIG. 11: The variation of the small amplitude supersonicNA SWs for different values of σ lt = 0 . (red solid curve), σ lt = 0 . (green dotted curve), and σ lt = 0 . (blue dashedcurve) at γ e = 5 / , M = 1 . , µ = 0 . , S h = 0 . , and σ ht = 0 . . - ϕ - - - - - V ( ϕ ) FIG. 12: The formation of potential wells in positive φ -axis for σ lt = 0 . (red solid curve), σ lt = 0 . (green dotted curve),and σ lt = 0 . (blue dashed curve) at γ e = 1 , M = 0 . , µ = 0 . , S h = 0 . , and σ ht = 0 . . - ϕ - - - V ( ϕ ) FIG. 13: The formation of potential wells in positive φ -axis for σ lt = 0 . (red solid curve), σ lt = 0 . (green dotted curve),and σ lt = 0 . (blue dashed curve) at γ e = 4 / , M = 1 . , µ = 0 . , S h = 0 . , and σ ht = 0 . . - ϕ - - - V ( ϕ ) FIG. 14: The formation of potential wells in positive φ -axisfor σ lt = 0 . (solid curve), σ lt = 0 . (dotted curve), and σ lt = 0 . (dashed curve) at γ e = 5 / , M = 1 . , µ = 0 . , S h = 0 . , and σ ht = 0 . . Note that we have used the same set of plasma pa-rameters for both large and small amplitude limit. Theresults are shown in Figs. 9 - 11 (Figs. 12 - 14) for small(large) amplitude NA SWs. It is observed that the effectof nucleus temperature reduces the possibility for the ex-istence of subsonic NA SWs. Therefore, more numberdensity or charge of heavy nucleus is required to have thesubsonic NA SWs as the nucleus temperature rises. Theamplitude (width) of both subsonic and supersonic NASWs decreases (increases) with the increases in nucleustemperature. We have observed the same behaviour, asfound in previous situation, that the ultra-relativisticallyDES ( γ e = 4 / ) and non-relativistically DES ( γ e = 5 / )do not support the formation of subsonic NA SWs, butthe BDES ( γ e = 1 ) does. The much wider NA solitarypulses in non-relativistically DES ( γ e = 5 / ) as comparedto the γ e = 1 and γ e = 4 / have also been observed here. IV. DISCUSSION
The thermally degenerate pressure driven arbitraryamplitude nucleus acoustic solitary waves in a thermallydegenerate plasma system (containing thermally degen-erate electron species and non-degenerate light and heavynucleus species) have been investigated. The dynam-ics of light and heavy nucleus species has been studiedbased on equal footing. So, the solitary waves we in-vestigated can be either light nucleus-acoustic solitarywaves if n l m l (cid:29) n h m h or heavy nucleus-acoustic soli-tary waves if n l m l (cid:28) n h m h . The pseudo-potential ap-proach, which is valid for arbitrary amplitude solitarywaves, has been employed. The results, which have beenobtained from this theoretical investigation, can be pin-pointed as follows: • The phase speed of the thermally degeneratenucleus-acoustic waves decreases (increases) withrise of the value of µ ( S h ). The rate of decreaseof the phase speed with µ in the case of S h (cid:54) = 0 isslower than that in the case of S h = 0 . This is dueto same planarity of both dynamical species. How-ever, the result would be opposite if the planarityof two dynamical species would be opposite. • It is obvious that C q > C l and λ q > L q . This meansthat the phase speed (wavelength) for E et (cid:54) = 0 ishigher (lower) than that for E et = 0 . This is dueto the rise of the volume of the degenerate mediumcaused by the additional outward thermal pressureof the thermally degenerate electron species. • The consideration of Boltzmann distributed elec-tron species ( γ e = 1 ) makes the plasma systemnon-degenerate and gives rise to subsonic ther-mally degenerate nucleus-acoustic solitary waveswith φ > . However, the electron degeneracyand light and heavy nucleus temperature reducethe possibility for the formation of these supersonicsolitary waves. • The Mach number decreases as the charge densityof the heavy nucleus species increases which agrees with our linear analysis presented in introductionsection. We note that the Mach number definedhere is only valid if n l m l (cid:29) n h m h or if the wavesare formed due to the compression and rarefactionof light nucleus species. • The consideration of ultra-relativistically and non-relativistically degenerate electron species supportsonly the existence of supersonic solitary waves with φ > . • The amplitude (width) of both the subsonic andsupersonic solitary waves decreases (increases) withthe rise of values of γ e , σ lt , and σ ht . This is due tothe fact that the latter increases the random motionof both light and heavy nucleus species. • The height of the solitary structures in non-relativistically degenerate electron species ( γ e =5 / ) is much smaller than that in ultra-relativistically degenerate electron species ( γ e =4 / ), but is much larger than that in Boltzmanndistributed electron species ( γ e = 1 ). • The basic features obtained from analytical solitarywave solution of the energy integral with V ( φ ) = C φ + C φ , which is valid for small but finite am-plitude solitary waves, are found to be the same asthose obtained from the direct numerical analysisof the general form of V ( φ ) , which is valid for ar-bitrary amplitude solitary waves. This means thatthe basic features of the solitary waves identified inthis investigations are correct.The electron-helium-carbon thermal degenerate plasmasystem (for which Z l = 2 , Z h = 6 , m e = 9 . × − kg, and m l = 1 . × − kg, m h = 2 . × − kg) have been used in our numerical analyses. The widerange of values of other parameters, viz. σ lt = 0 - . , σ ht = 0 - . , and µ = 0 . - have been used. Thus,the results obtained from this investigation are appli-cable in understanding the salient features of localizedelectrostatic disturbances not only in astrophysical com-pact objects like hot white dwarfs [49–53], but also inspace environments [57–60] and laboratory devices [61–64] where the electrons species follow the Boltzmann re-lation, the ion species play the role as the light nucleusspecies does, and the positively charged particles (as posi-tively charged impurity or dust) play the role as the heavynucleus species does. Acknowledgments
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