Properties of Electron Ion Acoustic Solitary Waves in a Four Component Degenerate Quantum Plasma

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(1)z. Available online at http://www.journalcra.com. INTERNATIONAL JOURNAL OF CURRENT RESEARCH International Journal of Current Research Vol. 10, Issue, 06, pp.70044-70058, June, 2018. ISSN: 0975-833X. RESEARCH ARTICLE PROPERTIES OF ELECTRON--ION ION ACOUSTIC SOLITARY WAVES IN A FOUR COMPONENT DEGENERATE QUANTUM PLASMA *1Promi Halder, 2Mukta, K. N. and 2Mamun, A. A. 1Department 2Department. of Physics, Jagannath University, Dhaka-1100, 1100, Bangladesh of Physics, Jahangirnagar University, Savar, Dhaka-1342, Dhaka 1342, Bangladesh. ARTICLE INFO. ABSTRACT. Article History:. The research focuses on the nonlinear propagation of electron-ion electron ion (El) acoustic solitary waves in a degenerate quantum plasma (containing Maxwellian electrons, vortex-like vortex like negative ions, cold mobile positive ions, and arbitrarily charged static dust grains) have have been theoretically investigated. The Korteweg Korteweg-de Vries (K-dV) and modified K-dV K (mK-dV) dV) equations are derived by adopting the reductive perturbation method. Their stationary solutions are derived and analyzed analytically as well as numerically to study some new basic fea¬tures of the El acoustic solitary structures that are commonly found to exist in degenerate quantum plasma. It is found that the basic properties (viz. amplitude, width, and phase speed, etc.) of the El acoustic waves are significantly modified by the effects of relativistic ally degenerate electrons and light ions, quantum pressure, number densities of plasma particles, and external magnetic field, etc. The results of this theoretical investigation may be useful for understanding the formation fo and fea-tures tures of the solitary structures in astrophysical compact objects like white dwarfs, neutron stars, active galactic nuclei, pulsars etc. are pinpointed.. Received 14th March, 2018 Received in revised form 27th April, 2018 Accepted 20th May, 2018 Published online 28th June, 2018. Key words: Electron-Ion Acoustic Waves, Solitary waves, Relativistic Effect, Degenerate Pressure, Astrophysical Compact Objects.. Copyright © 2018, Promi Halder et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Citation: Promi Halder, Mukta, K. N. and Mamun, A. A. 2018. “Properties of electron-ion ion acoustic solitary waves in a four component degenerate quantum plasma”, International Journal of Current Research, Research 10, (06), 70044-70058.. INTRODUCTION Progress in the understanding of dense quantum plasmas has received an immense interest not only because of their omnipres omnipresence in many astrophysical compact objects (like white dwarfs, neutron stars, active galactic nuclei, etc. (Woolsey et al., 2004; Shapiro, 1983; Garcia-Berro, Berro, 2010; Mamun, 2010)), but also for their available application in the laboratory laboratory i.e., intense laser laser-solid matter interaction experiment periment (Berezhiani, 1992; Murkhmd, 2006). The degenerate compact objects e.g. white dwarfs, neu neutron stars, etc. are usually consist of extremely dense iron/oxygen/carbon and helium nuclei (Shiikla, 2011; Massey, 1976) are ideal examples of degenerate plasma systems where the quantum mechanical effects (i.e., when the de Broglie thermal wavelength of the charged particles is equal or larger than the average inter-particle inter particle distance) play an important role in the plasma dynamics (Abdikian, 2016). The plasma in the interior of white dwarfs and in the crust of neutron stars are extremely dense and highly degenerate (Mamun, 2010; Chabrier, 2006; Lai, 2001; Harding, 2006) (plasma particles number densities densities can be of the order of 1030cm~3, or even more and the magnetic field strength can also be very large, i.e., B >> 109G). For such plasmas, the Fermi tempe temperature TFis high and the quantum-mechanical mechanical effects associated associated with the quantum statistical pressure and the quantum force involving the Bohm potential (causing the plasma particles tunneling) are expected to play an important important role (Manfredi, 2005). The Therefore, by adding the quantum statistical tistical pressure term (the Fermi-Dirac Fermi Dirac distribution) and the quantum diffraction term (the Bohm potential) to the fluid model generalized QHD model (Manfredi, 2001; Manfredi, 2005) is obtained. A number of works (Rouhani, 2014; Hossain, 2011; Haas, 2003; Bhowmik, 2007; Saeed-ur-Rehman, Saeed Rehman, 2012; Hossen, 2015) have been done by con considering the quantum effects of plasma particles in different ferent plasma medium. The behaviour of high densities and low temperatures quantum plasmas was first studied ied by Pines (Pines, 1961). Haas et al. (Haas et al., 2003) found in their investigation vestigation that the Bohm potentials associated with the plasma particles significantly modify the basic features of the nonlinear IA waves. Bhowmik Bhowmiket al. (2007) investigated the effects of quantum diffraction parameter H, and the equilibrium density ratio of the plasma species in modifying the electron-acoustic (EA) waves in a quantum tum EI plasma.. *Corresponding author: Promi Halder, Department of Physics, Jagannath University, Dhaka-1100, Dhaka Bangladesh. DOI: https://doi.org/10.24941/ijcr.30766.06.2018 .2018.

(2) 70045. Promi Halder et al. Properties of electron-ion acoustic solitary waves in a four component degenerate quantum plasma. On the other hand, Ali et al. (2007) analyzed the IA waves in an EPI plasma, and found that the nonlinear properties of the IA waves are significantly affected by the inclusion of the quantum terms in the momentum equations of electrons and positrons. But none of them considered the effects of external magnetic field and the presence of the heavy ions. It is well known that (Miller, 1987; Plastino, 1993; Gervino, 2012) the presence of external magnetic field (which causes the obliqueness of the wave propagation) plays a vital role in modifying the basic features of the linear and nonlinear waves in space and astrophysical plasmas (Mahmood, 2008; Sultana, 2010; El-Tantawy, 2012; Shahmansouri, 2013; Alinejad, 2013; Ashraf et al., 2014), and that the most of astrophysical degenerate quantum plasma systems like white dwarfs and neutron stars usually contain degenerate electrons and light ions along with heavy ions (Koester, 1990). This means that the effects of heavy ions and magnetic field must be considered, specially for the study of the nonlinear phenomena in the degenerate astrophysical objects (Woolsey, 2004; Shapiro, 1983; Torres et al., 2010; Mamun, 2010). Obliquely propagating electron-acoustic (EA) solitary waves (SWs) in a two electron population quantum rnag-netoplasma was theoretically investigated by Masood and Mushtaq (2008). They found that propagation characteristics of the EA SWs are significantly affected by the presence of quantum corrections and the ratio of hot to cold electron concentration. Recently, Hossen and Mamun have theoretically investigated the nonlinear positron-acoustic (PA) waves propagating in the fully relativistic electron-positron-ion plasma and found that the effects of relativistic degeneracy of electrons and positrons, static heavy ions, plasma particles velocity, and enthalpy, etc. have significantly modified the basic properties of the PA SWs. Using a fully relativistic set of two fluids plasma equations, Lee and Choi (Lee, 2007), Tribeche and Boukhalfa (2011), Saberianet al, (2011), and Akbari-Moghanjoughi (2011) have studied the characteristics of the nonlinear IA waves in different fully relativistic plasmas. Therefore, in our present work, we have examined the basic properties of the El acoustic waves propagating in a degenerate quantum plasma composed of relativistically magnetized quantum electrons and light ions in the presence of stationary heavy ions. We have considered the quantum mechanical (such as tunneling associated with the Bohm potential) effects for both electrons and light ions which abundantly occurs in different astrophysical situations (viz. white dwarfs, neutron stars, active galactic nuclei, etc. (Woolsey, 2004; Shapiro, 1983; Garcia-Berro, 2010; Mamun, 2010), and laboratory plasmas like intense laser-solid matter interaction experiment (5. 6). The manuscript is organized as in Sec. II, the basic equations governing our plasma model are presented; in Sec. Ill and IV, the K-dV and mK-dV equations along with their solutions are derived; in Sec. V, a brief discussion is given. Model Equations: The propagation of the EIA waves in a degenerate dense quantum plasma system containing degenerate electron and positron fluids (both non-relativistic and ultrarelativistic), and inertial nonrelativistic light ion fluid has been considered. We consider a collision less plasma system with an ambient magnetic field directed along the z axis, i.e., (Bo =. zˆBo ).. where zˆ is a unit vector in the z direction. At equilibrium, the quasi-neutrality condition can be expressed as. Z hho  io  eo  0,. where.  ho ,io , and  eo. are the equilibrium number densities of immobile heavy ions, light ions, and. Z. electrons, respectively, and h is the immobile heavy ions charge state respectively. The dynamics of the electrostatic EIA waves propagating in such plasma system is governed by the following set of normalized equations.. N s  * . N sU s   0, T. (1). . qs  * 1  U s  U s . * U s     c U s  B0  m  e  T. . .  * Ps*  2 *  1 *2     N s ,  me N s 2me  ns . (2).  *2   4eN i  N e  Z h N h , (3) is. * s. P and the full expression for it is (Mon, 1935). Ps* . me4 c 5 3h. 3. R R s. 2 s.  2 R. 1. 1/ 2. in which the relativity parameter. N  N  Ns1, Ns1. s s0 Taking order, Eq. (4) turns to. 2 s. . .  3  3 sinh 1 Rs . Rs  pFs / mec. and. (4). pFs  (3h 3ns / 8 )1 / 3. is the perturbation of the s'th species with. is the Fermi relativistic momentum of s'th species.. Ns1. Nsoand the Taylor expansion up to second.

(3) 70046. International Journal of Current Research, Vol. 10, Issue, 06, pp.70044-70058, June, 2018.  P *  1   2 Ps*  Ps*  Ps 0   s  N s1   N s31 2   N 2  N s  Ns  Ns 0 s  Ns  Ns 0  .  Ps 0 . . . 2  Fs 3R02  4  Fs 2 N s1  N s1 ' 3 s 0 9 s30 N s 0.  s 0  1  Rs20'. R  P / m c, P  (3h3n / 8 )1 / 3 ,. s0 e s0 s0 where , s0 basic Eqs. (l)-(3), we use the following normalized parameters. T  t pe ,* .  pe uFe. n N s  s ,  ps  ns 0. uFs  Ps* . ,  . (5). and.  Fs   2 (3 2 ns ) 2 / 3 / 2me . In order to normalize the. e u , Us  s uFe 2  Fe.  pe 4e 2 ns 0 , Hs  me 2  Fe. 2  Fs , c  eB0 , c   c me  pe me c. Ps n ,    Fe ,   h 0 2 Fe ns0 ne 0 .  Fi. (6). The normalized equations for the ion quantum fluid can be written as. ni  .niui   0, t. (7). ui P  ui .ui    c (ui  zˆ)  i t ni .  1 2  H2   ni , 2/3  n  2(1  Z h )  i . (8). and the normalized equations for the electron quantum fluid can be written as. ne  .neue   0, t. (9). ue Pe  ue .ue    c (ue  zˆ)  t ne .  H 2  1   2 n p ,  2  ne . (10). and the normalized form of the Poisson equation is. 2  ne  ni (1  Z h )  Z h . K-dV EQUATION. (11).

(4) 70047. Promi Halder et al. Properties of electron-ion acoustic solitary waves in a four component degenerate quantum plasma. In order to examine the characteristics of the nonlinear propagation of the electrostatic perturbation modes in the relativisticallymagne-tized quantum plasmas (under consideration), we will derive the K-dV equation employing the reductive perturbation method (47). So, we first introduce the stretched coordinates (Washimi, 1966) as.  1/ 2 (lz  l y  lz  Vpt),. (12).  3 / 2 t ,. (13). where. Vp [  /  (. is angular frequency and k is the wave vector)] is the wave phase speed,  ( 0  1) is a measure of the. solitary wave amplitude i.e., a measure of the weakness of the dispersion or of the nonlinear effect, 2. cosines of the wave vector k along the x, y, and z axes, respectively, so that. ni , ne , usx , y ,usz '. lx , l y ,. and. lz are the direction. 2. lx  lx  1.. We then expand the perturbed quantities. and  about their equilibrium values in power series of  as. ni  1  ni(1)  2 ni( 2 )  3 n4(3)  ......,. (14). ne  1  ne(i )  2 ne( 2)  3 ne(3)  ......,. (15). usx, y 3/ 2 usx(1). y 2 usx(2.)y 5/ 2 usx(3.)y ......,. (16). u sz  u sz(1)  2 u sz( 2 )  3 u sz( 3)  ......,. (17).   (1)  2  (2)  3  (3)  ........ (18). Now, substituting Eqs. (12)-(18) in Eqs. (7)-(ll), and equating the coefficients for the lowest order of  , we obtain the first order continuity equations, the z -component of the momentum equations, and Poisson's equation, which after simplification, we can write as. ni. (1). . 3 i 0lz2  (1) , 3 i 0V p2  lz2. 3 i 0lzVp. (1). uiz . ne. (1). (1). . 2 i0 p. 3 V  l.  (1) , (20). 3 e 0l z2  (1) , l z2  32e 0V p2. uez . vp . 2 z. 3 e0lz2Vp 2 z. 2 e0 p. l  3 V. (19). (21).  (1) ,.  e 0 l z2   i 0 l z2 ( l  Z h ) , 3  e 0  i 0 ( 2  Z h ). (22). (23). We define lz = cos  ,where  is the angle between the directions of the wave propagation vector k and the external magnetic field B0- The Eq. (23) represents the dispersion relation for the acoustic type electrostatic waves in the degenerate quantum plasma under consideration. We can write the first order x— and y—components of the momentum equations as.

(5) 70048. International Journal of Current Research, Vol. 10, Issue, 06, pp.70044-70058, June, 2018. uiy . e (3 i 0Vp2  lz2 ) . uix  . (24). c (3 i 0Vp2  lz2 )   (1). 3ly e0Vp2. (1). uey  . ,.  (1). 3l y i 0Vp2. (1). u(ez1) .  (1). 3lx i 0Vp2. (1). c (lz2  3 e0Vp2 )  2 y e0 p 2 2 z e0 p. 3l  V. . , (25). , (26). (1). ,. c (l  3 V ) . (27). Eqs. (24)-(27), represent the x— and y—components of the ion-electron electric field drift, respectively. Again, substituting Eqs. (12)-(18) in Eq. (8) and Eq. (10) and using Eqs. (19)-(27), we obtain the next higher order x-and y—components of the momentum equations as. u iy . c2 (3 i 0V p2  l z2 )  2. c2 (3 i 0V p2  l z2 )  2. u(ey2)  . ( 2).  2 (1). 3Vp3 i 0l x. ( 2). u iy .  2 (1). 3Vp3 i 0l y. ( 2). u ex  . 3 p e0 y. 3V  l 2 c. 2 z. , (28). ,. . 2 e0 p.  (l  3 V )  2 3Vp3 e0lx. (29). 2 (1). 2 (1). c2 (lz2  3 e0Vp2 )  2. , (30). , (31). Eqs. (28)-(31), denote the y— and x—components of the ion-electron polarization drift, respectively. Further, substituting Eqs. (12)-(18) in Eqs. (7)-(ll), we obtain the next higher order continuity equations, the z- component of the momentum equations, and Poisson's equation, which can be given as ( 2). u ni(1) n( 2) u ( 2 ) u ( 2)  Vp i  lz ix  1y iy  lz iz       (l ) (l ) ni uiz   0,  lz . (32). (1)  ( 2) uiz(1) uiz( 2 ) l ni( 2 ) (1) uiz  Vp  lz uiz  1z  z     3 i 0 . . . l z 3Ri20  4 (l ) ni( l ) l z (l ) ni(l )  ni  ni 9 i30  3 i 0  H 2 l z  3 ni(1)   0, 4  3. (33) ( 2). u ne(1) n( 2) u ( 2) u ( 2)  Vp e  lx ex  l y ey  lz ez     .

(6) 70049. Promi Halder et al. Properties of electron-ion acoustic solitary waves in a four component degenerate quantum plasma.  (1) (1) ne u ez  0, . .  lz. . (34). (1)  ( 2) lz ne( 2 ) uez(1) uez( 2 ) (1) uez  Vp  lxuez  lz      3 e 0 . lz (3R2e0  4) (1) ne(1) l z (1) ne(1)  ne  ne 9 e30  3 e30  H 2l z  3ne(1)   0, 4(1  Z h ) 2 / 3  3  2 (1) . 2. (35).  ne( 2)  (1  Z h )ni( 2) (36). Now, simplifying Eqs. (32)-(36) by using Eqs. (19)-(31), and combining each other, we finally obtain our desired equation in the form.  2 (1) .  A. (1).  (1) . B. 3 (1)  3.  0, (37). Eq. (37) is the well known K-dV equation describing the dynamics of the SWs propagating in the degenerate quantum plasma. The nonlinear coefficient, A and the dispersion coefficient, B are given as. A. B. L1  L2  L3  L4 , 6 V p a 3b  6 2p 0V p ab 3 (1  Z h ). (38). K1  K 2  K 3  K 4  K 5 , 72 V p l z2 a 2 c2  72 i20V p l z2b 2  c2 (1  Z h ). (39). 2 e0. 2 e0. Where. a  (3 i 0Vp2  lz2 ), b  (lz2  3 e0Vp2 ), L1  27 e30Vp2a3lz2 ), L2  a3lz4 (3Re20  3 e20  4), L3  27 i30Vp2b3lz2 (1  Zh ), L4  b 3lz4 (3Ri20  3 i20  4)(1  Z h ), 2 4 2 2 K1  4a 2b2c2 , K2  36 e0Vp a (1  lz ),. K 3  9 e20l z4 H 2 a 2 c2 ,. K4  36 i20Vp4b2 (1  l z2 ), (1  Z h ), K 5  9 i20lz4 H 2b 2c2 ), (1  Z h )1/ 3 ,. (40). Now, to investigate the properties of the El acoustic SWs, we are interested in the SWs solution of the K-dV equation. To do so, we introduce another stretched coordinates. solution of the SWs can be written as.   ,  1 .     u0 .. After the coordinate transformation, the steady state ( /   0).  (1)  m sech 2 . (41).

(7) 70050. International Journal of Current Research, Vol. 10, Issue, 06, pp.70044-70058, June, 2018. where the amplitude whereas. m  3U 0 / A. and the width,. 1  4B / U 0 . It is seen that the quantum parameter H is present in 1,. m is totally independent of these parameter.. mK-DV EQUATION We now follow the K-dV equation (Eq. (37)) is the result of the second order calculation in the smallness parameter  , where the. A (1) (1) /  ,. quadratic nature has been revealed by the nonlinear term . For plasmas with more than two species as like our system, however, there can arise cases where A vanishes at a particular value.  , and Eq. (37) fails to describe nonlinear evolution of perturbation. So, higher order calculation is needed. of a certain parameter at this critical value (Washimi, 1966).    c . For. this reason, to derive the mK-dV equation, we apply the following stretched coordinates.  1/ 2 (l x  l y  l z  Vp t ),. (42).  3 / 2 t ,. (43). We then expand the variables. ni , ne , u sx , y , u sx ,. , and  , in power series of  as. ni  1 1 / 2 ni(1)   ni( 2 )  3 / 2 ni( 3)  ...,. (44). ne  1 1 / 2 ne(1)   ne( 2)  3 / 2 ne(3)  ...,. (45). uix. y usz(1.)y  3 / 2 usx(2.)y  2 usx(3.)y  ...,. (46). usz 1 / 2 usz(1)   usz( 2)  3 / 2 usz(3)  ...,. (47).  1/ 2 (1)   ( 2)  3/ 2 (3)  ...,. (48). By using Eq. (42)-(48) in Eqs. (7)-(ll), we found the values of. n i(1) , n e(1) , u iz(1) , u ez(1) , u ix(1,)y , u ex(1). y. and Vpas like as that of the K-dV. n (1 ) , n e(1 ) , u iz(1 ) , u ez(1) , u ix(1,)y , u ex(1). y equation. To the next higher order of  , we obtain a set of equations, which, after using the values of i and Vpcan be simplified as. ni( 2 ) . . . 2 p. 2 3 z. 2(3 i 0V  l ).  . (1) 2. 3l z2 i 0  ( 2) (3 i 0V p2  l z2 ). ( 2) e. n. 81V p2l z4 i30  3l z6 (3 Ri20  3 i30  4). . (49). 81Vp2lz4 e30  3lz6 (3Re20  3 e20  4) 2 z. 2 3 e0 p. 2(l  3 V ).  . (1) 2. 3lz2 i 0  (2) (lz2  3 e 0Vp2 ). uiy( 2 ) . ( 2) i. (50) 4 x z. 2 i0. 2 i0. (1). lx n l l (3R  3  4) (1)    3 c i 0   i 0 c (3 i 0V p2  lz2 )2 .

(8) 70051. Promi Halder et al. Properties of electron-ion acoustic solitary waves in a four component degenerate quantum plasma. 3Vp3l y i 0. 2(1)  2 , c (3 i 0Vp2  lz2 )2  2 uix( 2)  . (51). 4 2 2 ni( 2) 1y lz (3Ri 0  3 i 0  4) (1) (1)   3c i 0   i 0c (3 i 0Vp2  lz2 )2 . ly. 3Vp3ly i0. (1) , c2 (3 i 0Vp2  lz2 )2  2. uex( 2)  . (52). l y ni( 2) 1y lz4 (3Ri20  3 i20  4) (1)  (1)   3c i 0   i 0c (lz2  3li 0Vp2 )2 . 3Vp3ly e0. 2(1)  2 2 , c (lz  3 e0Vp2 )  2 uex(2)  . . 2(1) , c2 (lz2  3 e0Vp2 )  2. u. . . 9Vplz3 i30  lz5 (3Ri20  3 i20  4).  i 0 (3 i 0Vp2  lz2 )2. (54). (1)   (1). l z ( 2) lz  ni( 2) , Vp 3V p i 0. ( 2) ez. u. l y ne(2) 1x lz4 (3Re20  3 e20  4) (1) (1)   3c e0   e0c (lz2  3 e0Vp2 )2 . 3Vp3lx  e0. ( 2) iz. . (53). . (55). 9Vplz3 e30  lz5 (3Re20  3 e20  4).  e0 (lz2  3 e0Vp2 )2. (1)   (1). lz ( 2) l z ( 2 )   ne , Vp 3V p e 0. 1 2. (56).  ( 2)   A1(1)   0, 2. (57). where  T1  T 2 A1     l z2  3  e 0V p2. . . 3. . (1  Z h )( P1  P2 )  ' 3  3  i 0V p2  l z2. . . (58). Where. T1  81Vp2lz4 e30 ' T2  3l z6 (3Re20  3 e20  4)' P1  81Vp2lz4 i30 ' P2  3l z6 (3Ri2  3 i30  4)' To the next higher order of  , we obtain a set of equations which after simplification as follows. (59).

(9) 70052. International Journal of Current Research, Vol. 10, Issue, 06, pp.70044-70058, June, 2018. ( 3) 2 18Vplz2 i30 (1) ni 3lz  i 0 (3)    (3 i 0Vi 02  lz2 )2  (3 i 0Vp2  lz2 ) . . .  9Vp4 i20 1  lz2  9lz4 H 2 i20  2  2 2 2 2/3 2 2 2  c (3 i 0Vp  lz ) 4(1  Z h ) (3 i 0Vp  l z )   81 V p2 l z4  i30  3 l z6 ( 3 R i20  3  i30  4   3  (1 )    3  ( 3  i 0 V p2  l z2 ) 3  .  (1) 2  (1) F8  F9 [ (1) ( 2) ]  F1  F2  F3  F4 F5  F6  F7    '  2 2 5 2 2 4  2(3 i 0V p2  l z2 ) 5    2(3 i 0Vp  lz )  i 0 (3 i 0Vp  lz ).  . (60). (3) 18Vplz2 e20 (1) ne 3lz2 e0 (3)  2   (lz  3 e0Vp2 )2  (lz2  3 e0Vp2 )  2.  9Vp4 e20 (1  lz2 )  3 (1) 9l24 H 2 e20  2 2  2 2 2 2 2 3  c (lz  3 e 0Vp ) 4(lz  3 e 0Vp )    81V p2l z4 e30  3l z6 (3Re20  3 e30  4    (1)  ( 2)   (l z2  3 e 0V p2 ) 3     G  G  G3  G4 G  G6  G7  1 2 2  52 2 5 2 4  2(l z  3 e 0V p )  e 0 (l z  3 e 0V p )  (1) 2  (1) G8  G9  '  2(l z2  3 e 0V p2 )5    (1)  ne( 3)  (1  Z h )ni(3) . . .  . (61). (62). where 2 2 2 F1  486Vp2lz8 i50 , F2  18l 10 z  i 0 (3Ri 0  3 i 0  4) ,. F3  81Vp2lz8 i20 (3Ri20  3 i20  4), 2 2 2 F 4  3 l 10 z (3 R i 0  3 i 0  4 ) '. F5  81Vp2lz6 i40 , F6  9Vp lz8 i 0 (3Ri20  3 i20  4), 4 6 5 F7  3lz8 i 0 (3Ri20  3 i20  4), F8  729Vp lz  i 0. F9  27Vp2lz8 i20 (3Ri20  3 i20  4), 2 2 2 G1  486Vp2lz8 e50 , G2  18 2l10 z  e 0 (3 Re 0  3 e 0  4)'. G3  81Vp2lz8 e30 (3Re20  3 e20  4)' 2 2 2 G4  3 2l10 z (3 Re 0  3 e 0  4) '. G5  81Vp2lz6 e40 , G6  9Vplz8 e20 (3Re20  3 e20  4), 4 6 5. G7  3l z8 e 0 (3Re20  3 e20  4), G8  729Vp lz  e0 ,. G9  27Vp2lz8 e20 (3Re20  3 e20  4).. (63). Now, combining Eqs. (60)-(62) after simplifying by using Eqs. (49)-(57), we obtain the well-known mK-dV equation as follows.

(10) 70053. Promi Halder et al. Properties of electron-ion acoustic solitary waves in a four component degenerate quantum plasma. (1) 2   (1)  3 (1)  C  (1) B  0,    3.  . (64). Equation (64) is a modified konteweg-de Vries (K-dV) equation, exhibiting a stronger nonlinearity, smaller with and larger propagation velocity of the nonlinear wave. Where. a 2b 2 C 18V p a 2 e20l z2  18V pb 2 i20lz2 (1  Z h )  K1  R K G G   R F  F   5  42  8 5 9   1  Z h  5 1  42  8 5 9 , 2b   2a   2b  e0 b  2a  e0 a. (65). Where. K1  G1  G2  G3  G4 , K 2  G5  G6  G7 , R1  F1  F2  F3  F4 , R2  F5  F6  F7 .. (66) Now, taking the same stretching as like the solution of the K-dV SWs, the stationary SWs solution of Eq. (64) can be directly given as.    (1)   m sec h ,  2 . (67).   6U 0 / C. where the amplitude, m independent of these parameter.. and the width,.  2  B /U 0. , where. 2. depends H. On the other hand,. m. is totally. RESULTS AND DISCUSSION.  , (1) , 1, l ,. l ). We have graphically obtained the parametric regimes (values of and  for which the properties of the acoustic SWs in degenerate quantum plasmas are discussed in this section numerically. The effects of quantum diffraction are related not only in degenerate astrophysical plasmas but also may be meaningful in some dense laboratory plasma containing electrons and light ions in presence of stationary heavy ions. In this part, our intention is to explore numerically of the propagation angle, quantum pressure, relativistic factors, immobile heavy ions charge state, and densities of the plasma components on the acoustic SWs in such degenerate quantum plasmas. For our purpose, we have derived the K-dV and. K  dV solitary waves with V p   0 .08 ,   0 .6,  ci  0 .5,   20 0 and U o  0.01. Zh= 2.. Fig. 1. The graph shows the variation of the positive potential. and.  for  = 1.3..

(11) 70054. International Journal of Current Research, Vol. 10, Issue, 06, pp.70044-70058, June, 2018. Fig. 2: Showing the variation in the width.  e 0 = 1.33,  i0 = 1.29, . 1 of the solitary waves with  and H for U0 = 0.01. = 40,. Zh  1  , = 1.1. and . = 0.1..  on the K-dV solitary profiles for U0 = 0.01.  e 0 = 1.33,  i0 = 1.29,  = 0.4,  = 40,  c = 0.4, Re  2.5 and Ri  2.2 . The lower red dotted line for  = 1.1. the lower blue solid line for  = 1.13, the upper   = 1.19. green dotdashed line for = 1.16, and the upper black solid line for. Ftg. 3. Showing the effects of. Zh  1,. H = 0.3.. Fig. 4. Showing the effects of Ri on the (K-dV) SWs profiles for U0 = 0.01.. Zh  1. , H = 0.3, Re = 2.5, and.  e 0 = 1.33,  i0 = 1.29, . = 0.4,. . = 40,. c  0.4.  = 1.2. The dotted line for Ri= 1, the solid line for Ri = 2, and the dotdashed line for Ri = 3.. ,.

(12) 70055. Promi Halder et al. Properties of electron-ion acoustic solitary waves in a four component degenerate quantum plasma. Fig. 5. Showing the effects of Re = 2.5, Ri = 2.2, and.  c on the mK-dV SWs profiles for U0 = 0.01.  e 0 = 1.33,  i 0 = 1.29, .  = 1.6. The dotted line for c  0.3 , the solid line for  c. Fig. 6. Showing the variation of the positive potential (mK-dV) solitary waves with 0.4,. . = 60,. Zh  1,. H = 0.3.. Zh  1. . = 60,. Zh  1, H = 0.3,. = 0.4, and the dotdashed line for. c  0.5. ..  and  for U0 = 0.01.  e 0 = 1.33,  i 0 = 1.29,  c =. Re  2.5 and Ri  2.2 and = 1.8.. Fig. 7(a). Showing the variation of the positive potential (mK-dV) solitary waves with 60,. = 0.2,. , H = 0.3. and.  = 1.16..  and. . for U0 = 0.01.. c. = 0.3,. . = 0.1,. . =.

(13) 70056. International Journal of Current Research, Vol. 10, Issue, 06, pp.70044-70058, June, 2018. Fig. 7(b). Showing the variation of the negative potential (mK-dV) solitary waves with.  and.  for. 0.  1  0.05, 2  0.5,  ci  0 .5,   10 ,. Fig. 8. Showing the Variation of. 1 K  dV  with . the non-relativistric case. Fig. 9. Showing the variation of. .  5 / 3. line for. i. = 0.3,. Zh  1  ,. = 60, The blue curve represents. and the red curve represents the ultra-relativistic case. 2 mK dV with  Z h  2,. for U0 = 0.01, p = 0.4,. and U0 = 0.01.. for U0 = 0.01,. the solid line for. Zh  3,.   4 / 3 ..   4 / 3 ,   5 / 3,   0 .2, i and the dotdashed line for. Z h  1.. = 0.3, p = 0.4. The dotted.

(14) 70057. Promi Halder et al. Properties of electron-ion acoustic solitary waves in a four component degenerate quantum plasma. Fig. 10. The graph showing the variation of mK –dVsoliton width.  2  with ci for different values of  2 .. mK-dV equations, and analyzed their stationary SWs solutions based on some typical plasma parameters relevant to different astrophysical and laboratory plasma situations existed in some published works. We consider some typical plasma species density which is consistent with the relativistic degenerate astrophysical plasmas, e.g., nio = 1.1 × 1029 cm-3, ne0= 9.1 × 1029cm-3, nho = 0.4 × 1030 cm-3, (11, 36, 37, 48) and the other quantum parameters e.g.,  = 0.1 to 0.9, H = 0.2 to 0.9 (18, 22, 49), and ambient magnetic fields ~ 1013G (15, 50). The values of these parameters may change depending on different plasma situations. The results, we have found from this investigation can be summarized as follows: We compare the effects of the non-relativestic. (  5 / 3) and ultra relativistic (  4 / 3) limits of the degenerate pressure originating fro the plasma species on V p and.  . amplitude, m of the K-dV SWs in Figs. 9 and 10, respectively. Since the number density of plasma species in non-relativistic case is always less than the ultra relativistic case, plasma particles can move freely in the non-relativistic plasma more than the ultra relativistic plasma. For this reason. Vp. as well as. m. of the EIA SWs are noticeably higher for   5 / 3 than for.  on Vpof the El acoustic waves for different values of  are displayed in Fig. 1. The phase speed of the acoustic waves decreases with the increase of  .This happens due to the increase of the inertia of the acoustic waves with the  increase of the value of  . It is also observed that with the increase in , Vpof the EI acoustic waves increases. The variation of   4 / 3 . The effect. width,. 1 of the K-dV SWs with H is depicted in Fig. 2 for different values of  c . It is found that as the values of both H and.  c increase, 1 of the acoustic SWs decreases significantly. It is observed that for the positive (negative) potential of the solitary waves structures, the magnitude of the amplitude decreases (increases) with the increase in Vpand H shown in Fig. 1 (Fig. 2). It is found that for the positive (negative) potential of the solitary profiles the magnitude of the amplitude increases (decreases) with the increase (decrease) in.  and . shown in Fig.6 (Fig.7). The variation of the K-dV SWs profiles with the.  (Ri) is shown in Fig. 3 (Fig. 4). It is observed that. Our plasma system supports both positive (com-pressive) and negative (rarefactive) SWs structure. In case of and width of the SWs increases with the increase of.  both amplitude.  . Increase of  means the increase of Fermi energy of the electrons and. R. R. ions. But in case of i both amplitude and width of the SWs decreases with the increase of i (increase of relativistic effects of ions). Plasma particles can move freely in the weakly relativistic plasma more than the strong relativistic plasma. 4. The influences of the.  c (  ) on the mK-dV SWs profiles is shown in Fig. 5 (Fig. 6). It seems that there is no effect of  c on the amplitude of. the mK-dV SWs but width decreases with the increase of the.  c . It is also found that both width and amplitude of the mK-dV. SWs decrease with the increase of  . This occurs due to the increase of the inertia of the plasma particles. 5. Fig. 7 compares the the amplitude and width of the quantum El SWs profiles for the relativistic case and the non-relativistic case. Plasma particles can move freely in the non-relativistic plasma more than the ultra-relativistic plasma due to the less number density of plasma species in non-relativistic case than the ultra relativistic case. Therefore, the amplitude as well as width of the SWs are noticeably higher for non-relativistic case than for ultra-relativistic case. The magnitude of the external magnetic field B0has no any effect on the amplitude of the SWs but it does have a direct effect on the width of SWs. It is important to note here that the plasma frequency.   is greater than the ion-neutral collision frequency  i  and dust-neutral collision frequency  d  , i.e.,  p. i. , d   p ,. follows that the collisions do not seriously interfere with plasma oscillations in such a plasma system under consideration.. which.

(15) 70058. International Journal of Current Research, Vol. 10, Issue, 06, pp.70044-70058, June, 2018. . p We found that as the magnitude of bo(i.e., ) increases, the width of SWs decreases, i.e., the magnetic field makes the solitary structures more spiky. We hope that our lower order as well as higher order nonlinear analysis will be helpful for understanding the localized electrostatic disturbances not only in the different astrophysical degenerate compact objects (Shapiro, 1983; GarciaBerro, 2010; Mamun, 2010)), but also in different dense laboratory plasma nonlinear experiments (Berezhiani, 1992; Murkhmd, 2006).. Acknowledgments P. Halder is profoundly grateful to the Ministry of Science and technology (Bangladesh) for awarding the National Science and Technology (NST) fellowship. REFERENCES Abdikian, A. and Mahmood, S. 2016. Phys. Plasmas 23. 122303. " Ali, S., Moslem, W. M., Shukla, P. K. and Schlickeiser, R. 2007. Phys. Plasmas 14, 082307. Berezhiani, V. I., Tskhakaya, D. D. and Shukla, P. K. 1992. Phys. Rev. A 46, 6608. Bhowmik, C., Misra, A. P. and Shukla, P. K. 2007. Phys. Plasmas 14, 122107 Chabrier, G., Saumon, D. and Potekhin, A. Y. 2006. J. Phys. A 39, 4411. Garcia-Berro, E., Torres, S., Althaus, L. G., Renedo, I., Lorn-Aguihar, P., Crsico, A. H., Rohrmann, R. D., Salaris, M. and J. Tsern, Nature 465, 194 (2010). Gardner, C. 1994. SIAM J. Appl. Math.54, 409. Gervino, G., Lavagno, A. and Pigato, D. 2012. Cent. Eur. J, Phys. 10. 594. Haas, F., Garcia, L. G., Goedert, J. and Manfredi.G. 2003. Phys. Plasmas 10, 3858. Harding A. K. and Lai, D. 2006. Rep. Prog. Phys. 69. 2631. Hossain, M. M., Mamun, A. A. and Aahrafi, K. S. 2011. Phys. Plasmas 18, 103704. Hossen M. A. and Mamun. A. A. 2015. Phys. Plasmas 22, 073505 Lai, D.2001. Rev. Mod. Phys. 73, 629. Mamun A.A. and Shukla, P. K. 2010. Phys. Lett. A 374, 4238. Manfredi, G. 2005. Fields Inst. Commun. 46, 263. Manfredi, G. and Haas, F. 2001. Phys. Rev. B 64, 075316 . Markowich, A., Ringhofer, C. and Sehmeiser, C. 1990. Semi-conductor Equations Springer, Vienna, Chaps. 1, 2. Massey, H. 1976. Negative. Ions (Cambridge Universfly Press, Cambridge. Miller, H. R. and Witta, P. J. 1987. Active Galactic Nuclei (Springer-Verlag, Berlin. Murkhmd, M. and Shukla, P. K. 2006. Rev. Mod. Phys. 78, 591. " Pines, D. 1961. J. Nucl. Energy Part C 2, 5. Plastino, A. R. and Plastino, A. 1993. Phys. Lett. A 174, 384. Rouhani, M. R., Mohammadi, Z. and Akbarian, A. 2014. Astrophys. Space Sci. 349, 265. Saeed-ur-Rehman, 2010.Phys. Plasmas 17, 062303. Shapiro S. L.and S. A- Teukolsky, Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects (John Wiley & Sons, New York, 1983).(3) E. Garcia-Berro, S. Torres, L. G. Althaus, I. Renedo, P. Shiikla, P. K., Mamun, A. A. and Mendis, D. A. 2011. Phys. Rev. E 84, 026405. Woolsey, N. C., Courtois, C. and Bendy, R.O. 2004. Plasma Phys. Control. Fusion 46. B397.. *******.

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